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
In this paper, we consider the following quasilinear elliptic problems:
Rabinowitz  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 , Ambrosetti, Malchiodi and Ruiz  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  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  obtained the famous Strauss inequality
for a.e. and . Berestycki and Lions  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  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  obtained the existence of a ground state solution for some p-Laplacian elliptic problems in . Zhang  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  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  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  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.
Now, we state some Hardy inequalities.
3 Embedding results for general potentials
then the embedding
then the embedding
Hence, we have
Combining (3.2) and (3.3), we have
Now, we state our main theorem in this section.
Proof (a) Arguing as in the proof of (a) in Lemma 3.1, we obtain
then we have
(2) Let , we obtain that if with , the embedding is compact. This has already been obtained in .
4 Embedding theorem for a radially symmetric function space
Assume that and are radial weights. In , Su, Wang and Willem considered for potentials with and obtained some embedding theorems. In this section, we extend some results in  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.
Then we take the set of all the finite products
Now, we define the following two radially symmetric Sobolev spaces:
Combining (4.5) and (4.6), 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.
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. □
(1) The previous estimates should be compared with Lemma 1 in ,
(2) Our results extend Lemma 4 and Lemma 5 in , and we obtain the general Strauss-type decay estimates;
Now, we state our main embedding theorems in this section.
By Lemma 2.2, we obtain
Hence, we have
From Lemma 6 in , we have the following.
which yields a contradiction. □
5 Ground and bound state solutions
Now, consider problem (1.1) with general potentials
Further, consider problem (1.5) with radially symmetric potentials
For a more general equation than (1.5),
then we have the following theorem.
The author declares that he has no competing interests.
The author read and approved the final manuscript.
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).
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: Semi-classical states of nonlinear Schrödinger equations: a variational reduction method. Math. Ann.. 324, 1–32 (2002). Publisher Full Text
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 Schrodinger-Poisson 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
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
Zhang, G: Ground state solution for quasilinear elliptic equation with critical growth in . Nonlinear Anal. TMA. 75, 3178–3187 (2012). 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 Hardy-Sobolev inequality and its applications. Proc. Am. Math. Soc.. 130, 489–505 (2002). Publisher Full Text