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

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

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:

(1.1)

where , , , and are nonnegative measurable functions, and may be unbounded, decaying and vanishing.

Recently, these type elliptic equations have been widely studied. As , if and satisfied

(1.2)

Rabinowitz [1] proved the existence of a ground state solution for problem (1.1). Further, when has a positive lower bound and 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 and satisfied

(1.3)

where , , 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 tends to zero at infinity. In particular, when the potentials and 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 , Strauss [8] obtained the famous Strauss inequality

(1.4)

for a.e. and . Berestycki and Lions [9] proved the existence of a ground state solution for some scalar equation. In 2007, as the potentials and are radially symmetric, Su, Wang and Willem [10] obtained the existence of a ground state solution for problem (1.1) with and unbounded and decaying.

For , 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 . Zhang [12] considered a mountain pass characterization of the ground state solution for p-Laplacian elliptic problems with critical growth. When and are radially symmetric, Su, Wang and Willem [13] considered the following quasilinear elliptic problem:

(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 and 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 and , in [13] various conditions have been considered for with . Our first purpose is to consider and 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 and 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 denote the collection of smooth functions with compact support. Let be the completion of under the norm

(2.1)

We write and is the corresponding subspace of a radial function for .

Define, for and ,

(2.2)

and

(2.3)

Then we have , which is a Banach space under the uniformly convex norm

(2.4)

where .

Now, we state some Hardy inequalities.

Lemma 2.1[14]

If, , we have

Lemma 2.2[13]

If, , andfor some, there existssuch that

Lemma 2.3[15]

If, or, then

whereis the ball incentered at 0 with radiusR, denotes the complement of.

### 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 and .

Lemma 3.1Letbe smooth possibly unbounded and

andbe measure nonnegative functions. a.e. in.

(a) If there existssuch that

then the embedding

is continuous;

(b) If there existandsuch that

andsuch that

and, such that

then the embedding

is compact.

Proof (a) Since there exists such that , we have

By Hölder’s inequality and , we obtain

(3.1)

where . Since is the critical Sobolev exponent, by the Sobolev embedding theorem, we have

Hence, we obtain that the embedding is continuous.

(b) For any fixed , let be the ball in Ω with . Since there exist and such that

arguing as in the proof of (3.1), by the compact embedding of into , we have

Hence, we have

(3.2)

On the domain , since , we have

Assume (weakly) in , then we have

(3.3)

Combining (3.2) and (3.3), we have

Hence, we obtain that is compact. □

Now, we state our main theorem in this section.

Consider a finite partition of and is unbounded.

Theorem 3.2If condition (H) is satisfied for anyand, assume that

(a) there existsuch that

then the embeddingis continuous;

(b) there existandsuch that

andsuch that

and

then the embeddingis compact.

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

Hence, we obtain that is continuous.

(b) , such that

then we have

(3.4)

Arguing as in the proof of (b) in Lemma 3.1, when (weakly) in , we have

(3.5)

By (3.5) and the local compactness in , we obtain that

Hence, we have is compact. □

Remark 3.3 (1) Let and , and , we obtain the standard local Sobolev embedding.

(2) Let , we obtain that if with , the embedding is compact. This has already been obtained in [6].

### 4 Embedding theorem for a radially symmetric function space

Assume that and are radial weights. In [13], Su, Wang and Willem considered for potentials with and obtained some embedding theorems. In this section, we extend some results in [13] to a more general class of functions for , 0+. In particular, we also obtain some embedding theorems for the Sobolev space . 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

Since is not closed with respect to the operation , we consider

Then we have and . The process can be iterated, we have the following.

Definition 4.1 The set is called Hardy-Dieudonne class of functions at +∞. is called Hardy-Dieudonne class of functions at .

Now, let , be continuous nonnegative functions in , and

(V) and ;

(K) and ,

where and .

By conditions (V) and (K), we obtain that there exist positive constants , , , , , such that

(4.1)

(4.2)

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

and under the uniformly convex norm

Lemma 4.2Assume thatandsatisfies condition (V), . If

(a)

(1) for, then there existssuch that

(4.3)

(2) for, then there existssuch that

(4.4)

(b)

(1) forand, , , then (4.3) holds.

(2) forand, , , then (4.4) holds.

Proof (a)(1) By density, it is enough to prove it for with support in . We have

By Hölder’s inequality and (4.1), there exists such that

(4.5)

where is the volume of the unit sphere in .

On the other hand, for . By a simple computation, we obtain

(4.6)

Combining (4.5) and (4.6), we have

(2) By density, it is enough to prove it for with support in , we have

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

and this yields (4.4).

(b)(1) If for .

By Hölder’s inequality and Lemma 2.1, we have

(4.7)

Similarly, we have

(4.8)

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

(2) If for and , we can argue as in the above proof. □

Let , we consider the Sobolev space .

Lemma 4.3Assume thatandsatisfies condition (V). If

(a) forand, then there existssuch that

(b) forand, then there existssuch that

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

(4.9)

On the other hand, we have

(4.10)

Combining (4.9) and (4.10), we obtain

(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],

for every ;

(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 such that

(4.11)

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

Theorem 4.5Ifandsatisfy (V) and (K), is relatively compact, .

(a) Ifand, then the embeddingis continuous.

(1) and, or

(2) and,

(b) If, , and, then the embeddingis continuous.

Proof (a)(1) If , we obtain

By Lemma 2.2, we obtain

Hence, we have

(4.12)

If , arguing as previously, similarly we obtain

(4.13)

(2) If , by (4.11) and conditions (V) and (K), we obtain

(4.14)

If , we obtain similarly

(4.15)

(b) If , . By Hölder’s inequality and Lemma 2.3, we have

(4.16)

If , we obtain similarly

(4.17)

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

Under the conditions of Theorem 4.5, for and , the embedding

(4.18)

Now, we prove that the embedding for is continuous. It suffices to show

Assume to the contrary that , then there exists such that

But from (4.12) and (4.13), or (4.14) and (4.15), or (4.16) and (4.17), and (4.18), let be large enough, we obtain

Theorem 4.6Ifandare nonnegative measurable functions satisfying (V) and (K). andis relatively compact.

(a) If, . orthen the embeddingis compact.

(1) asandas,

(2) asandas,

(b) If, , asandas, then the embeddingis compact.

Proof (a) Arguing as in the proof of (a) and (b) of Theorem 4.5, we obtain that there exists ,

and .

Assume that (weakly) in (), we obtain

then we have (strongly). Hence the embedding is compact. □

### 5 Ground and bound state solutions

Now, consider problem (1.1) with general potentials

Theorem 5.1Under the assumptions of Theorem 3.2, i.e., is compact embedded into, then problem (1.1) has a ground state solution.

Proof Now, we define the functional on the Sobolev space ,

It is obvious that the critical point of the functional is exactly the weak solution of problem (1.1). The existence of a ground state solution follows from the compact embedding immediately.

Further, consider problem (1.5) with radially symmetric potentials

where . Similarly to Theorem 5.1, we obtain the following theorem. □

Theorem 5.2Under the assumptions of Theorem 4.6, i.e., is a compact embedding into, then problem (1.5) has a ground state solution.

For a more general equation than (1.5),

(5.1)

Ifand, and there existssuch that

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.

### 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).

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