Weighted Sobolev spaces and ground state solutions for quasilinear elliptic problems with unbounded and decaying potentials
© Zhang; licensee Springer 2013
Received: 13 May 2013
Accepted: 8 August 2013
Published: 23 August 2013
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.
Keywordsweighted Sobolev spaces unbounded and decaying potentials quasilinear elliptic problems
where , , , and are nonnegative measurable functions, and may be unbounded, decaying and vanishing.
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.
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.
We write and is the corresponding subspace of a radial function for .
Now, we state some Hardy inequalities.
Lemma 2.1 
Lemma 2.2 
Lemma 2.3 
where is the ball in centered at 0 with radius R, 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 .
- (a)If there exists such that
- (b)If there exist and such that
- (b)For any fixed , let be the ball in Ω with . Since there exist and such that
Hence, we obtain that is compact. □
Now, we state our main theorem in this section.
Consider a finite partition of and is unbounded.
- (a)there exist such that
- (b)there exist and such that
then the embedding is compact.
- (b), such that
Hence, we have is compact. □
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 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 .
where and .
- (a) (1)for , then there exists such that(4.3)
- (2)for , then there exists such that(4.4)
- (b) (1)
for and , , , then (4.3) holds.
for and , , , then (4.4) holds.
where is the volume of the unit sphere in .
and this yields (4.4).
(b)(1) If for .
If for and , we can argue as in the above proof. □
Let , we consider the Sobolev space .
- (a)for and , then there exists such that
- (b)for and , then there exists such that
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 ,
Our results extend Lemma 4 and Lemma 5 in , 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.
If and , then the embedding is continuous.
and , or
If , , and , then the embedding is continuous.
- (2)If , by (4.11) and conditions (V) and (K), we obtain(4.14)
From Lemma 6 in , we have the following.
which yields a contradiction. □
If , . or then the embedding is compact.
as and as ,
as and as ,
If , , as and as , then the embedding is compact.
then we have (strongly). Hence the embedding is compact. □
5 Ground and bound state solutions
Theorem 5.1 Under the assumptions of Theorem 3.2, i.e., is compact embedded into , then problem (1.1) has a ground state solution.
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.
where . Similarly to Theorem 5.1, we obtain the following theorem. □
Theorem 5.2 Under the assumptions of Theorem 4.6, i.e., is a compact embedding into , then problem (1.5) has a ground state solution.
then we have the following theorem.
Theorem 5.3 Under the above conditions and assumptions of Theorem 4.6, problem (5.1) has a positive solution. If, in addition, f is odd in u, then problem (5.1) has infinitely many solutions in .
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. 1992, 43: 270-291. 10.1007/BF00946631MathSciNetView ArticleMATHGoogle Scholar
- del Pino M, Felmer P: Local Mountain Passes for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 1998, 4: 121-137.MathSciNetView ArticleMATHGoogle Scholar
- del Pino M, Felmer P: Semi-classical states of nonlinear Schrödinger equations: a variational reduction method. Math. Ann. 2002, 324: 1-32. 10.1007/s002080200327MathSciNetView ArticleMATHGoogle Scholar
- Ambrosetti A, Felli V, Malchiodi A: Ground states of nonlinear Schrödinger equation with potentials vanishing at infinity. J. Eur. Math. Soc. 2005, 7: 117-144.MathSciNetView ArticleMATHGoogle Scholar
- Ambrosetti A, Malchiodi A, Ruiz D: Bound states of nonlinear Schrödinger equation with potentials vanishing at infinity. J. Anal. Math. 2006, 98: 317-348. 10.1007/BF02790279MathSciNetView ArticleMATHGoogle Scholar
- Bonheure D, Schaftingen JV: Ground states for the nonlinear Schrödinger equation with potentials vanishing at infinity. Ann. Mat. Pura Appl. 2010, 189: 273-301. 10.1007/s10231-009-0109-6MathSciNetView ArticleMATHGoogle Scholar
- Bonheure D, Mercuri C: Embedding theorems and existence results for nonlinear Schrodinger-Poisson systems with unbounded and vanishing potentials. J. Differ. Equ. 2011, 251: 1056-1085. 10.1016/j.jde.2011.04.010MathSciNetView ArticleMATHGoogle Scholar
- Strauss WA: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 1977, 55: 149-162. 10.1007/BF01626517View ArticleMathSciNetMATHGoogle Scholar
- Berestycki H, Lions P-L: Nonlinear scalar field equation, I. Existence of a ground state. Arch. Ration. Mech. Anal. 1993, 82: 313-345.MathSciNetMATHGoogle Scholar
- Su J, Wang Z, Willem M: Nonlinear Schrödinger equations with unbounded and decaying radial potentials. Commun. Contemp. Math. 2007, 9: 571-583. 10.1142/S021919970700254XMathSciNetView ArticleMATHGoogle Scholar
- do Ó JM, Medeiros ES:Remarks on least energy solutions for quasilinear elliptic problems in . Electron. J. Differ. Equ. 2003, 83: 1-14.MathSciNetMATHGoogle Scholar
- Zhang G:Ground state solution for quasilinear elliptic equation with critical growth in . Nonlinear Anal. TMA 2012, 75: 3178-3187. 10.1016/j.na.2011.12.015View ArticleMathSciNetMATHGoogle Scholar
- Su J, Wang Z, Willem M: Weighted Sobolev embedding with unbounded and decaying radial potentials. J. Differ. Equ. 2007, 238: 201-219. 10.1016/j.jde.2007.03.018MathSciNetView ArticleMATHGoogle Scholar
- Garcia Azorero JP, Alonso IP: Hardy inequalities and some critical elliptic and parabolic problems. J. Differ. Equ. 1998, 144: 441-476. 10.1006/jdeq.1997.3375View ArticleMathSciNetMATHGoogle Scholar
- Adimurthi , Chaudhuri N, Ramaswamy M: Improved Hardy-Sobolev inequality and its applications. Proc. Am. Math. Soc. 2002, 130: 489-505. 10.1090/S0002-9939-01-06132-9MathSciNetView ArticleMATHGoogle Scholar
- Dieudonné J: Calcul Infinitesimal. Herman, Paris; 1968.MATHGoogle Scholar
- Lieb EH, Loss M Grad. Stud. Math. 14. In Analysis. 2nd edition. Am. Math. Soc., Providence; 2001.Google Scholar