Existence for Elliptic Equation Involving Decaying Cylindrical Potentials with Subcritical and Critical Exponent

Abstract

We consider the existence of nontrivial solutions to elliptic equations with decaying cylindrical potentials and subcritical exponent. We will obtain a local minimizer by using Ekeland’s variational principle.

1. Introduction

In this paper, we study the existence of nontrivial solutions of the following problem:where , and let and be integers such that and belongs to . is the critical Sobolev exponent, , , is a continuous function on , and and are parameters which we will specify later.

We denote point in by the pair , , and , the closure of with respect to the norms with for .

From the Hardy inequality, it is easy to see that the norm is equivalent to .

We define the weighted Sobolev space with , which is a Banach space with respect to the norm defined by .

My motivation of this study is the fact that such equations arise in the search for solitary waves of nonlinear evolution equations of the Schrödinger or Klein-Gordon type (cf. [1–3]). Roughly speaking, a solitary wave is a nonsingular solution which travels as a localized packet in such a way that the physical quantities corresponding to the invariances of the equation are finite and conserved in time. Accordingly, a solitary wave preserves intrinsic properties of particles such as the energy, the angular momentum, and the charge, whose finiteness is strictly related to the finiteness of the -norm. Owing to their particle-like behavior, solitary waves can be regarded as a model for extended particles and they arise in many problems of mathematical physics, such as classical and quantum field theory, nonlinear optics, fluid mechanics, and plasma physics (see, e.g., [4]).

Several existence and nonexistence results are available in the case , and we quote, for example, [5–7] and the references therein. When , ; problem has been studied in the famous papers by Brézis and Nirenberg [8] and Xuan [9] which consider the existence and nonexistence of nontrivial solutions to quasilinear Brézis-Nirenberg-type problems with singular weights.

Concerning the existence result in the case , we cite [10, 11] and the references therein. As noticed in [10], for and , Badiale and Rolando have considered the problem . They established the existence of nontrivial nonnegative radial solution when and or and ; in addition, if the function is odd, then has infinitely many radial solutions. In [5], Badiale et al. proved the nonexistence of nonzero classical solutions when and the pair belongs to the light gray region. That is, , where

Since our approach is variational, we define the functional on by We say that is a weak solution of the problem if it is a nontrivial nonnegative function and satisfies Throughout this work, we consider the following regions , , such that with .

Concerning the perturbation , we assume In our work, we prove the existence of at least one critical point of by Ekeland’s variational principle in [12].

We will state our main result.

Theorem 1. Assume that , , , and hold.
If , then there exists such that the problem has at least one nontrivial solution for any .

This paper is organized as follows. In Section 2, we give some preliminaries. Section 3 is devoted to the proof of Theorem 1.

2. Preliminaries

We list here a few integrals inequalities. The first inequality that we need is the weighted Hardy inequality [13] The starting point for studying is the Hardy-Sobolev-Maz’ya inequality that is peculiar to the cylindrical case and that was proved by Gazzini and Musina in [14]. It states that there exists positive constant such that for ; equation of is related to a family of inequalities given by Caffarelli et al. [15], for any . The embedding is compact, where and is the weighted space with respect to the norm

Definition 2. Assume , , and . Then, the infimum defined by is achieved on .

Lemma 3. Let be a Palais-Smale sequence ( for short) of such thatfor some . Then, in and .

Proof. From (10), we have where denotes as . Then, and is bounded in . Going if necessary to a subsequence, we can assume that there exists such that Consequently, we get, for all , which means that

3. Existence Result

Firstly, we require the following lemmas.

Lemma 4. Let be a sequence of for some . Then, and either

Proof. We know that is bounded in . Up to a subsequence if necessary, we have that Denote , and then . As in Brézis and Lieb [16], we have From Lebesgue theorem and by using the assumption , we obtain Then, we deduce that From the fact that in , we can assume that Assuming that , we have by definition of and so Then, we get Therefore, if not, we obtain . That is, in .

Lemma 5. Suppose that , , and hold. If , then there exist and and positive constants such that, for all ,(i)there exist such that ,(ii)we have

Proof. (i) Let where is small, and such that . Choosing , then, if large enough, Thus, if , we obtain that .
(ii) By the Holder inequality and the definition of and since , we get for all If , then there exist and small enough such that We also assume that is small enough such that . Thus, we have Using Ekeland’s variational principle, for the complete metric space with respect to the norm of , we can prove that there exists a sequence such that for some with .
Now, we claim that . If not, by Lemma 4, we have which is a contradiction.
Then, we obtain a critical point of for all large enough satisfying

Proof of Theorem 1. From Lemmas 4 and 5, we can deduce that there exists at least a nontrivial solution for our problem with positive energy.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.