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 Strausstype 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 problems1 Introduction
In this paper, we consider the following quasilinear elliptic problems:
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
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
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
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 pLaplacian elliptic problems in . Zhang [12] considered a mountain pass characterization of the ground state solution for pLaplacian elliptic problems with critical growth. When and are radially symmetric, Su, Wang and Willem [13] considered the following quasilinear elliptic problem:
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 Strausstype 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
We write and is the corresponding subspace of a radial function for .
and
Then we have , which is a Banach space under the uniformly convex norm
Now, we state some Hardy inequalities.
Lemma 2.1[14]
Lemma 2.2[13]
If, , andfor some, there existssuch that
Lemma 2.3[15]
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.
then the embedding
is continuous;
(b) If there existandsuch that
then the embedding
is compact.
Proof (a) Since there exists such that , we have
By Hölder’s inequality and , we obtain
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
On the domain , since , we have
Assume (weakly) in , then we have
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
then the embeddingis continuous;
and
Proof (a) Arguing as in the proof of (a) in Lemma 3.1, we obtain
Hence, we obtain that is continuous.
then we have
Arguing as in the proof of (b) in Lemma 3.1, when (weakly) in , we have
By (3.5) and the local compactness in , we obtain that
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 HardyDieudonne 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 HardyDieudonne class of functions at +∞. is called HardyDieudonne class of functions at .
Now, let , be continuous nonnegative functions in , and
By conditions (V) and (K), we obtain that there exist positive constants , , , , , such that
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
(2) for, then there existssuch that
(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
where is the volume of the unit sphere in .
On the other hand, for . By a simple computation, we obtain
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).
By Hölder’s inequality and Lemma 2.1, we have
Similarly, we have
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
On the other hand, we have
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],
(2) Our results extend Lemma 4 and Lemma 5 in [13], and we obtain the general Strausstype decay estimates;
(3) Under the conditions of Lemma 4.2 and Lemma 4.3, we obtain that there exist two comparison functions such that
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.
(b) If, , and, then the embeddingis continuous.
By Lemma 2.2, we obtain
Hence, we have
If , arguing as previously, similarly we obtain
(2) If , by (4.11) and conditions (V) and (K), we obtain
(b) If , . By Hölder’s inequality and Lemma 2.3, we have
From Lemma 6 in [13], we have the following.
Under the conditions of Theorem 4.5, for and , the embedding
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
which yields a contradiction. □
Theorem 4.6Ifandare nonnegative measurable functions satisfying (V) and (K). andis relatively compact.
(a) If, . orthen the embeddingis compact.
(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 ,
Assume that (weakly) in (), we obtain
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),
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).
References

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

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

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

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)

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

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

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

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

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

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

do Ó, JM, Medeiros, ES: Remarks on least energy solutions for quasilinear elliptic problems in . Electron. J. Differ. Equ.. 83, 1–14 (2003)

Zhang, G: Ground state solution for quasilinear elliptic equation with critical growth in . Nonlinear Anal. TMA. 75, 3178–3187 (2012). Publisher Full Text

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

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

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

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