SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Weighted Sobolev spaces and ground state solutions for quasilinear elliptic problems with unbounded and decaying potentials

Guoqing Zhang

Author Affiliations

College of Sciences, University of Shanghai for Science and Technology, Shanghai, 200093, P.R. China

Boundary Value Problems 2013, 2013:189  doi:10.1186/1687-2770-2013-189


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/189


Received:13 May 2013
Accepted:8 August 2013
Published:23 August 2013

© 2013 Zhang; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we prove some continuous and compact embedding theorems for weighted Sobolev spaces, and consider both a general framework and spaces of radially symmetric functions. In particular, we obtain some a priori Strauss-type decay estimates. Based on these embedding results, we prove the existence of ground state solutions for a class of quasilinear elliptic problems with potentials unbounded, decaying and vanishing.

MSC: 35J20, 35J60, 35Q55.

Keywords:
weighted Sobolev spaces; unbounded and decaying potentials; quasilinear elliptic problems

1 Introduction

In this paper, we consider the following quasilinear elliptic problems:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6">View MathML</a> are nonnegative measurable functions, and may be unbounded, decaying and vanishing.

Recently, these type elliptic equations have been widely studied. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M7">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6">View MathML</a> satisfied

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M10">View MathML</a>

(1.2)

Rabinowitz [1] proved the existence of a ground state solution for problem (1.1). Further, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5">View MathML</a> has a positive lower bound and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6">View MathML</a> is bounded, using critical point theory, del Pino and Felmer [2,3] obtained that problem (1.1) might also not have a ground state solution. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6">View MathML</a> satisfied

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M15">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M17">View MathML</a>, Ambrosetti, Felli and Malchiodi [4], Ambrosetti, Malchiodi and Ruiz [5] obtained the ground and bound state solutions for problem (1.1). In fact, condition (1.3) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5">View MathML</a> tends to zero at infinity. In particular, when the potentials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6">View MathML</a> are neither bound away from zero nor bounded from above, Bonheure and Mercuri [7] proved the existence of the ground state solution for problem (1.1) and obtained the decay estimates by using the Moser iteration scheme. For the radially symmetric space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M21">View MathML</a>, Strauss [8] obtained the famous Strauss inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M22">View MathML</a>

(1.4)

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M24">View MathML</a>. Berestycki and Lions [9] proved the existence of a ground state solution for some scalar equation. In 2007, as the potentials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6">View MathML</a> are radially symmetric, Su, Wang and Willem [10] obtained the existence of a ground state solution for problem (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6">View MathML</a> unbounded and decaying.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M29">View MathML</a>, to the best of our knowledge, it seems to be little work done. do Ó and Medeiros [11] obtained the existence of a ground state solution for some p-Laplacian elliptic problems in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M30">View MathML</a>. Zhang [12] considered a mountain pass characterization of the ground state solution for p-Laplacian elliptic problems with critical growth. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6">View MathML</a> are radially symmetric, Su, Wang and Willem [13] considered the following quasilinear elliptic problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M33">View MathML</a>

(1.5)

and proved some embedding results of a weighted Sobolev space for a radially symmetric function, and obtained the existence of ground and bound state solutions for problem (1.5).

In this paper, for the general potentials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6">View MathML</a> allowing to be unbounded or vanish at infinity, we obtain some necessary and sufficient conditions about some continuous and compact embeddings for the weighted Sobolev space. Based on variational methods and some compact embedding results, we obtain the existence of ground and bounded state solutions for problem (1.1). On the other hand, for the radial potentials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M37">View MathML</a>, in [13] various conditions have been considered for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M38">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M39">View MathML</a>. Our first purpose is to consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M37">View MathML</a> whose behavior can be described by a more general class of functions. Furthermore, we obtain some a priori Strauss-type decay estimates and some continuous and compact embedding results for the radial symmetric weighted Sobolev space. The results then are used to obtain ground and bound state solutions for problem (1.5).

It is worth pointing out that we provide here a unified approach what conditions the potentials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6">View MathML</a> should satisfy so that problem (1.1) and problem (1.5) have ground and bound state solutions, respectively. We extend the results in [13] to a large class of weighted Sobolev embeddings and obtain some new embedding theorems for the general potentials and radially symmetric potentials.

The paper is organized as follows. In Section 2, we collect some results. In Section 3, we obtain some embedding results for the general potentials. In Section 4, we focus on radially symmetric potentials and prove the continuous and compact embeddings. Section 5 is devoted to the existence of ground and bound state solutions for problem (1.1) and problem (1.5), respectively.

2 Preliminaries

In this section, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M44">View MathML</a> denote the collection of smooth functions with compact support. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M45">View MathML</a> be the completion of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M46">View MathML</a> under the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M47">View MathML</a>

(2.1)

We write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M48">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M49">View MathML</a> is the corresponding subspace of a radial function for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M45">View MathML</a>.

Define, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M51">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M52">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M53">View MathML</a>

(2.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M54">View MathML</a>

(2.3)

Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M55">View MathML</a>, which is a Banach space under the uniformly convex norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M56">View MathML</a>

(2.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M57">View MathML</a>.

Now, we state some Hardy inequalities.

Lemma 2.1[14]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M59">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M60">View MathML</a>

Lemma 2.2[13]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M63">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M64">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M65">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M66">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M67">View MathML</a>

Lemma 2.3[15]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M69">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M70">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M71">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M72">View MathML</a>is the ball in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M73">View MathML</a>centered at 0 with radiusR, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M74">View MathML</a>denotes the complement of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M72">View MathML</a>.

3 Embedding results for general potentials

In this section, we derive a tool giving the embedding results on a piece of the partition. We consider the possible relation between the behavior of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6">View MathML</a>.

Lemma 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M78">View MathML</a>be smooth possibly unbounded and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M79">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M81">View MathML</a>be measure nonnegative functions. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M82">View MathML</a>a.e. in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M73">View MathML</a>.

(a) If there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M84">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M85">View MathML</a>

then the embedding

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M86">View MathML</a>

is continuous;

(b) If there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M87">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M88">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M89">View MathML</a>

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M90">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M91">View MathML</a>

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M93">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M94">View MathML</a>

then the embedding

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M95">View MathML</a>

is compact.

Proof (a) Since there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M84">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M97">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M98">View MathML</a>

By Hölder’s inequality and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M99">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M100">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M101">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M102">View MathML</a> is the critical Sobolev exponent, by the Sobolev embedding theorem, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M103">View MathML</a>

Hence, we obtain that the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M104">View MathML</a> is continuous.

(b) For any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M105">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M106">View MathML</a> be the ball in Ω with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M107">View MathML</a>. Since there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M108">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M88">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M110">View MathML</a>

arguing as in the proof of (3.1), by the compact embedding of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M111">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M112">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M113">View MathML</a>

Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M114">View MathML</a>

(3.2)

On the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M115">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M116">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M117">View MathML</a>

Assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M118">View MathML</a> (weakly) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M119">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M120">View MathML</a>

(3.3)

Combining (3.2) and (3.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M121">View MathML</a>

Hence, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M104">View MathML</a> is compact. □

Now, we state our main theorem in this section.

Consider a finite partition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M123">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M73">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M125">View MathML</a> is unbounded.

Theorem 3.2If condition (H) is satisfied for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M126">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M125">View MathML</a>, assume that

(a) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M128">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M129">View MathML</a>

then the embedding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M130">View MathML</a>is continuous;

(b) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M131">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M132">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M133">View MathML</a>

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M134">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M135">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M136">View MathML</a>

then the embedding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M130">View MathML</a>is compact.

Proof (a) Arguing as in the proof of (a) in Lemma 3.1, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M138">View MathML</a>

Hence, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M139">View MathML</a> is continuous.

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M141">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M142">View MathML</a>

then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M143">View MathML</a>

(3.4)

Arguing as in the proof of (b) in Lemma 3.1, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M144">View MathML</a> (weakly) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M145">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M146">View MathML</a>

(3.5)

By (3.5) and the local compactness in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M147">View MathML</a>, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M148">View MathML</a>

Hence, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M130">View MathML</a> is compact. □

Remark 3.3 (1) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M150">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M151">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M152">View MathML</a>, we obtain the standard local Sobolev embedding.

(2) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M7">View MathML</a>, we obtain that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M154">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M155">View MathML</a>, the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M156">View MathML</a> is compact. This has already been obtained in [6].

4 Embedding theorem for a radially symmetric function space

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M37">View MathML</a> are radial weights. In [13], Su, Wang and Willem considered for potentials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M159">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M160">View MathML</a> and obtained some embedding theorems. In this section, we extend some results in [13] to a more general class of functions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M161">View MathML</a>, 0+. In particular, we also obtain some embedding theorems for the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M162">View MathML</a>. Theorem 4.5 and Theorem 4.6 are new embedding results.

Following [16,17], we shall refer to this class as the Hardy-Dieudonne comparison class. Define

Then we take the set of all the finite products

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M164">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M165">View MathML</a> is not closed with respect to the operation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M166">View MathML</a>, we consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M167">View MathML</a>

Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M168">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M169">View MathML</a>. The process can be iterated, we have the following.

Definition 4.1 The set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M170">View MathML</a> is called Hardy-Dieudonne class of functions at +∞. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M171">View MathML</a> is called Hardy-Dieudonne class of functions at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M172">View MathML</a>.

Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M37">View MathML</a> be continuous nonnegative functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M175">View MathML</a>, and

(V) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M176">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M177">View MathML</a>;

(K) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M178">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M179">View MathML</a>,

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M180">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M181">View MathML</a>.

By conditions (V) and (K), we obtain that there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M185">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M186">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M187">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M188">View MathML</a>

(4.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M189">View MathML</a>

(4.2)

Now, we define the following two radially symmetric Sobolev spaces:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M190">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M191">View MathML</a> under the uniformly convex norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M192">View MathML</a>

Lemma 4.2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M193">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5">View MathML</a>satisfies condition (V), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M59">View MathML</a>. If

(a)

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M196">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M197">View MathML</a>, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M66">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M199">View MathML</a>

(4.3)

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M200">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M201">View MathML</a>, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M66">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M203">View MathML</a>

(4.4)

(b)

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M204">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M197">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M208">View MathML</a>, then (4.3) holds.

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M204">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M201">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M213">View MathML</a>, then (4.4) holds.

Proof (a)(1) By density, it is enough to prove it for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M214">View MathML</a> with support in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M215">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M216">View MathML</a>

By Hölder’s inequality and (4.1), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M66">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M218">View MathML</a>

(4.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M219">View MathML</a> is the volume of the unit sphere in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M73">View MathML</a>.

On the other hand, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M221">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M197">View MathML</a>. By a simple computation, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M223">View MathML</a>

(4.6)

Combining (4.5) and (4.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M224">View MathML</a>

(2) By density, it is enough to prove it for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M214">View MathML</a> with support in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M226">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M227">View MathML</a>

By a similar computation as for (4.5) and(4.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M228">View MathML</a>

and this yields (4.4).

(b)(1) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M196">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M197">View MathML</a>.

By Hölder’s inequality and Lemma 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M231">View MathML</a>

(4.7)

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M232">View MathML</a>

(4.8)

Combining (4.5), (4.7) and (4.8), we obtain that (4.3) holds.

(2) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M233">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M201">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M235">View MathML</a>, we can argue as in the above proof. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M236">View MathML</a>, we consider the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M237">View MathML</a>.

Lemma 4.3Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M238">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5">View MathML</a>satisfies condition (V). If

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M240">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M197">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M242">View MathML</a>, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M66">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M244">View MathML</a>

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M245">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M201">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M247">View MathML</a>, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M66">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M249">View MathML</a>

Proof (a) Arguing as in the proof of (2) of (a) in Lemma 4.2, by Hölder’s inequality and Lemma 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M250">View MathML</a>

(4.9)

On the other hand, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M251">View MathML</a>

(4.10)

Combining (4.9) and (4.10), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M252">View MathML</a>

(b) Similarly, we can argue as in the proof of (a) in Lemma 4.3. □

Remark 4.4

(1) The previous estimates should be compared with Lemma 1 in [13],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M253">View MathML</a>

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M254">View MathML</a>;

(2) Our results extend Lemma 4 and Lemma 5 in [13], and we obtain the general Strauss-type decay estimates;

(3) Under the conditions of Lemma 4.2 and Lemma 4.3, we obtain that there exist two comparison functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M255">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M256">View MathML</a>

(4.11)

Now, we state our main embedding theorems in this section.

Theorem 4.5If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6">View MathML</a>satisfy (V) and (K), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M259">View MathML</a>is relatively compact, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M260">View MathML</a>.

(a) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M59">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M63">View MathML</a>, then the embedding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M267">View MathML</a>is continuous.

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M263">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M264">View MathML</a>, or

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M265">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M266">View MathML</a>,

(b) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M269">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M270">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M271">View MathML</a>, then the embedding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M272">View MathML</a>is continuous.

Proof (a)(1) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M273">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M274">View MathML</a>

By Lemma 2.2, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M275">View MathML</a>

Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M276">View MathML</a>

(4.12)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M264">View MathML</a>, arguing as previously, similarly we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M278">View MathML</a>

(4.13)

(2) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M279">View MathML</a>, by (4.11) and conditions (V) and (K), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M280">View MathML</a>

(4.14)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M266">View MathML</a>, we obtain similarly

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M282">View MathML</a>

(4.15)

(b) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M284">View MathML</a>. By Hölder’s inequality and Lemma 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M285">View MathML</a>

(4.16)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M286">View MathML</a>, we obtain similarly

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M287">View MathML</a>

(4.17)

From Lemma 6 in [13], we have the following.

Under the conditions of Theorem 4.5, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M288">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M289">View MathML</a>, the embedding

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M290">View MathML</a>

(4.18)

Now, we prove that the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M291">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M292">View MathML</a> is continuous. It suffices to show

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M293">View MathML</a>

Assume to the contrary that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M294">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M295">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M296">View MathML</a>

But from (4.12) and (4.13), or (4.14) and (4.15), or (4.16) and (4.17), and (4.18), let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M183">View MathML</a> be large enough, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M298">View MathML</a>

which yields a contradiction. □

Theorem 4.6If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M5">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M6">View MathML</a>are nonnegative measurable functions satisfying (V) and (K). <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M301">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M302">View MathML</a>is relatively compact.

(a) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M63">View MathML</a>. orthen the embedding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M313">View MathML</a>is compact.

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M305">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M306">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M305">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M308">View MathML</a>,

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M309">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M306">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M311">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M308">View MathML</a>,

(b) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M269">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M316">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M317">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M318">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M308">View MathML</a>, then the embedding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M320">View MathML</a>is compact.

Proof (a) Arguing as in the proof of (a) and (b) of Theorem 4.5, we obtain that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M105">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M322">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M323">View MathML</a>.

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M118">View MathML</a> (weakly) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M325">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M292">View MathML</a>), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M327">View MathML</a>

then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M328">View MathML</a> (strongly). Hence the embedding is compact. □

5 Ground and bound state solutions

Now, consider problem (1.1) with general potentials

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M329">View MathML</a>

Theorem 5.1Under the assumptions of Theorem 3.2, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M145">View MathML</a>is compact embedded into<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M331">View MathML</a>, then problem (1.1) has a ground state solution.

Proof Now, we define the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M332">View MathML</a> on the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M145">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M334">View MathML</a>

It is obvious that the critical point of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M332">View MathML</a> is exactly the weak solution of problem (1.1). The existence of a ground state solution follows from the compact embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M336">View MathML</a> immediately.

Further, consider problem (1.5) with radially symmetric potentials

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M337">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M292">View MathML</a>. Similarly to Theorem 5.1, we obtain the following theorem. □

Theorem 5.2Under the assumptions of Theorem 4.6, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M339">View MathML</a>is a compact embedding into<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M340">View MathML</a>, then problem (1.5) has a ground state solution.

For a more general equation than (1.5),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M341">View MathML</a>

(5.1)

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M342">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M343">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M344">View MathML</a>and there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M345">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M346">View MathML</a>

then we have the following theorem.

Theorem 5.3Under the above conditions and assumptions of Theorem 4.6, problem (5.1) has a positive solution. If, in addition, fis odd inu, then problem (5.1) has infinitely many solutions in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M339">View MathML</a>.

Competing interests

The author declares that he has no competing interests.

Authors’ contributions

The author read and approved the final manuscript.

Acknowledgements

The author gives his sincere thanks to the referees for their valuable suggestions. This paper was supported by Shanghai Natural Science Foundation Project (No. 11ZR1424500) and Shanghai Leading Academic Discipline Project (No. XTKX2012).

References

  1. Rabinowitz, PH: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys.. 43, 270–291 (1992). Publisher Full Text OpenURL

  2. del Pino, M, Felmer, P: Local Mountain Passes for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ.. 4, 121–137 (1998)

  3. del Pino, M, Felmer, P: Semi-classical states of nonlinear Schrödinger equations: a variational reduction method. Math. Ann.. 324, 1–32 (2002). Publisher Full Text OpenURL

  4. Ambrosetti, A, Felli, V, Malchiodi, A: Ground states of nonlinear Schrödinger equation with potentials vanishing at infinity. J. Eur. Math. Soc.. 7, 117–144 (2005)

  5. Ambrosetti, A, Malchiodi, A, Ruiz, D: Bound states of nonlinear Schrödinger equation with potentials vanishing at infinity. J. Anal. Math.. 98, 317–348 (2006). Publisher Full Text OpenURL

  6. Bonheure, D, Schaftingen, JV: Ground states for the nonlinear Schrödinger equation with potentials vanishing at infinity. Ann. Mat. Pura Appl.. 189, 273–301 (2010). Publisher Full Text OpenURL

  7. Bonheure, D, Mercuri, C: Embedding theorems and existence results for nonlinear Schrodinger-Poisson systems with unbounded and vanishing potentials. J. Differ. Equ.. 251, 1056–1085 (2011). Publisher Full Text OpenURL

  8. Strauss, WA: Existence of solitary waves in higher dimensions. Commun. Math. Phys.. 55, 149–162 (1977). Publisher Full Text OpenURL

  9. Berestycki, H, Lions, P-L: Nonlinear scalar field equation, I. Existence of a ground state. Arch. Ration. Mech. Anal.. 82, 313–345 (1993)

  10. Su, J, Wang, Z, Willem, M: Nonlinear Schrödinger equations with unbounded and decaying radial potentials. Commun. Contemp. Math.. 9, 571–583 (2007). Publisher Full Text OpenURL

  11. do Ó, JM, Medeiros, ES: Remarks on least energy solutions for quasilinear elliptic problems in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M73">View MathML</a>. Electron. J. Differ. Equ.. 83, 1–14 (2003)

  12. Zhang, G: Ground state solution for quasilinear elliptic equation with critical growth in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/189/mathml/M73">View MathML</a>. Nonlinear Anal. TMA. 75, 3178–3187 (2012). Publisher Full Text OpenURL

  13. Su, J, Wang, Z, Willem, M: Weighted Sobolev embedding with unbounded and decaying radial potentials. J. Differ. Equ.. 238, 201–219 (2007). PubMed Abstract | Publisher Full Text OpenURL

  14. Garcia Azorero, JP, Alonso, IP: Hardy inequalities and some critical elliptic and parabolic problems. J. Differ. Equ.. 144, 441–476 (1998). Publisher Full Text OpenURL

  15. Adimurthi, Chaudhuri, N, Ramaswamy, M: Improved Hardy-Sobolev inequality and its applications. Proc. Am. Math. Soc.. 130, 489–505 (2002). Publisher Full Text OpenURL

  16. Dieudonné, J: Calcul Infinitesimal, Herman, Paris (1968)

  17. Lieb, EH, Loss, M: Analysis, Am. Math. Soc., Providence (2001)