Existence and multiplicity of solutions for a supercritical elliptic problem in unbounded cylinders
- Ronaldo B Assunção^{1},
- Olimpio H Miyagaki^{2} and
- Bruno M Rodrigues^{3}Email author
Received: 17 November 2016
Accepted: 31 March 2017
Published: 13 April 2017
Abstract
Keywords
MSC
1 Introduction
Equations of this type arise in existence problems of stationary anisotropic solutions for the Schrödinger equation [2], in theory of non-Newtonian fluids [3], in problems of flow through porous media [4], in study of pseudoplastic fluids [5], in dynamic models for galaxies with cylindrical symmetry [6], and several other models. Variants of problem (2) in the radial setting were initially treated by Clément, de Figueiredo and Mitidieri [7] who proved, for example, the Brézis and Nirenberg [8] result for this radial operator. In recent years, several researchers have studied variants of problem (2) in the radial setting; see references [9–12].
In this present work, as the domain is unbounded, the lack of compactness of the Sobolev embedding \(W_{0}^{1,p}(\Omega)\hookrightarrow L^{q}(\Omega)\) (\(p\leq q < p^{*}:=\frac{pN}{N-p} \)) makes standard variational techniques more delicate.
Generally speaking, some geometrical and topological properties of the domain can help us to show existence results for elliptic problems; for example, the symmetry of the domain can be used to improve the Sobolev embedding. However, since we consider unbounded domains, the lack of compactness of the Sobolev embedding does not follow immediately from the standard variational techniques. This is one of the main difficulties we have to deal with in this work.
First we consider problem (6) in the case where \(f\equiv0\), and we get the following existence result. Note that in its statement, \(p_{N,m}^{*}\) is the critical Sobolev exponent in dimension \(N-m\), which is greater than the usual critical Sobolev exponent \(p^{*}=\frac{pN}{N-p}\).
Theorem 1
If \(1\leq m < N-p\), \(f\equiv0\) and \(p< q< p^{*}_{N,m}\), then problem (6) has at least one invariant solution.
A natural question is to check what happens to the previous problem under the presence of certain perturbations. For this purpose, we shall consider the perturbed problem by a function f belonging to the dual space of \(W_{0}^{1,p}(\Omega)\), denoted by \(W_{0}^{-1,p}(\Omega)\), and we get the following existence and multiplicity result.
Theorem 2
If \(1\leq m < N-p\) and \(p< q< p^{*}_{N,m}\), then there is a constant \(\overline{\varepsilon} >0\) such that for any \(f\in W_{0}^{-1,p}(\Omega)\) with \(0< \Vert f \Vert _{-1} < \overline{\varepsilon}\), problem (6) has at least two invariant solutions.
To prove these results, we study an auxiliary problem and show that its solutions are axially symmetric and belong to the space \(W_{0}^{1,p}(S)\subset W_{0}^{1,p}(\Omega)\), where \(S:=(A,B)\times \mathbb {R}^{N-m-1}\). As usual, this is done by defining an energy functional \(I \colon W_{0}^{1,p}(S) \to \mathbb {R}\) and by showing the existence of critical points for I in the space \(W_{0}^{1,p}(S)\). These critical points are the weak solutions of the auxiliary problem and, by our setting, they also solve problem (6).
Since S is an unbounded domain, the difficulty to prove Theorems 1 and 2 lies in the fact that \(W_{0}^{1,p}(S)\) cannot be compactly embedded into \(L^{q}(S)\) for any \(q\in(p, p_{N,m}^{*})\). In order to solve the lack of compactness, we construct a subspace of invariant functions \(W_{0,G}^{1,p}(S)\subset W_{0}^{1,p}(S)\) with compact embedding \(W_{0,G}^{1,p}(S)\hookrightarrow L^{q}(S)\) for \(q\in (p,p_{N,m}^{*})\) (see [19, 20]).
Using the principle of symmetric criticality [21], we can look for critical points of I restricted on \(W_{0,G}^{1,p}(S)\). In this way we obtain a weak solution in \(W_{0,G}^{1,p}(S)\) for our problem using the mountain pass theorem of Ambrosetti and Rabinowitz [22]. Finally, to show the existence of a second solution, we use Ekeland’s variational principle [23].
Since \(q\in(p,p_{N,m}^{*})\) and \(p_{N,m}^{*}>p^{*}\), in problem (6) we consider not only the subcritical and critical cases but also the supercritical one.
Note that the p-Laplacian operator \(\Delta_{p} u= \operatorname{div} ( \vert \nabla u \vert ^{p-2} \nabla u )\) is a special case of the operator \(\operatorname{div} (\frac{ \vert \nabla u \vert ^{p-2} \nabla u}{ \vert y \vert ^{ap}} )\); therefore, Theorems 1 and 2 improve the results of Hashimoto, Ishiwata and Ôtani [15].
This work is organized as follows. In Section 2 we introduce some notation and state some well-known results, such as the principle of symmetric criticality and the mountain pass theorem. In Section 3 we introduce the auxiliary problem, whose solutions are also solutions to problem (6). To ensure the existence of solutions to the auxiliary problem, we use the results of the previous section as well as Ekeland’s variational principle.
2 Preliminaries
In this section, we give some results which are used in the proofs of our main theorems. First, we denote by \(O(N)\) the group of linear isometries of \(\mathbb {R}^{N}\). Recall that if G is a closed subgroup of \(O(N)\), then an open subset Ω of \(\mathbb {R}^{N}\) is G-invariant if \(g\Omega=\Omega\) for every \(g\in G\). Furthermore, a function \(u: \Omega\rightarrow \mathbb {R}\) is called G-invariant if \(u(gx)=u(x)\) for all \(g\in G\), \(x \in\Omega\).
Definition 1
Now we can state a result by Palais [21].
Lemma 1
Principle of symmetric criticality
Assume that the action of the topological group G on the Banach space X is isometric. If \(\varphi\in C^{1}(X,\mathbb {R})\) is invariant and if u is a critical point of φ on \(\operatorname{Fix}(G)\), then u is a critical point of φ.
A frequently used compactness criterion is the Palais-Smale condition (PS condition, in short).
Definition 2
Palais-Smale condition
We finish this section with the statement of the well-known result by Ambrosetti and Rabinowitz [22].
Lemma 2
Ambrosetti-Rabinowitz mountain pass theorem
- (i)
there are constants \(\rho,\alpha>0\) such that \(\Phi \vert_{\partial B_{\rho}}>\alpha\),
- (ii)
there is \(e\in X\setminus\overline{B_{\rho}}\) such that \(\Phi(e)<0\),
3 Proof of the main results
In our arguments, the proof of Theorem 2 contains the existence result which is stated as in Theorem 1. Therefore, for the sake of brevity, we will deal only with problem (6) in the case where f is not necessarily identical to zero.
We denote by \(W_{0}^{1,p}(S)\) the subspace of axially symmetric functions of \(W_{0}^{1,p}(\Omega)\) with the norm defined by \(\Vert v \Vert = ( \int_{S} \vert \nabla v \vert ^{p} \,dx )^{\frac{1}{p}}\). This norm \(\Vert \cdot \Vert \) is equivalent to the standard norm on \(W_{0}^{1,p}(S)\) (see [24], pp.158-159).
In [19, 20], we have an important result of compactness which ensures that the embedding \(W_{0,G}^{1,p}(S)\hookrightarrow L^{q}(S)\) is compact for \(1\leq m < N-p\) and \(q\in(p,p_{N,m}^{*})\), where \(p_{N,m}^{*}:=\frac{p(N-m)}{N-m-p}\). So, \(W_{0,G}^{1,p}(S)\) can be compactly embedded into \(L^{q}(S)^{G}\) for the norms defined in (8).
Note that when \(b=a+1\) and \(b= a- \frac{m}{N-m}\), we have \(q=p\) and \(q=p_{N,m}^{*}\), respectively. Hence, we will consider \(a- \frac{m}{N-m} < b < a+1\), so that the compactness result and Maz’ja’s inequality are both satisfied.
The following lemma shows that the functional I verifies the geometry conditions of the mountain pass theorem.
Lemma 3
- (i)
there are \(\overline{\varepsilon},\rho, \alpha>0\) such that \(I \vert_{\partial B_{\rho}}\geq\alpha\), since \(0< \Vert f \Vert _{E^{-1}}<\overline{\varepsilon}\);
- (ii)
there is \(e\in E\setminus\overline{B_{\rho}}\) such that \(I(e)<0\).
Proof
Lemma 4
The functional I satisfies the Palais-Smale condition in E.
Proof
- 1.
\(\vert I(v_{n}) \vert \leq M\) for some \(M>0\) and
- 2.
\(I'(v_{n})\to0\) in \(E^{-1}\), where \(E^{-1}\) is the dual space of E.
Lemma 5
The functional I is weakly lower semicontinuous in E, i.e., if \(\{v_{n} \}\) converges weakly to v in E, then \(I(v)\leq\liminf I(v_{n})\).
Proof
Proof of Theorem 2
By Lemmas 3 and 4, all the assumptions of the mountain pass theorem in [22] are satisfied. Hence, we deduce the existence of \(v_{1}^{*} \in W_{0,G}^{1,p}(S)\) which is a weak solution to problem (7) and \(I(v_{1}^{*})=\overline{c}>0\).
By Lemma 5, the functional I is weakly lower semicontinuous and bounded from below by relation (9).
Declarations
Acknowledgements
OHM was supported by INCTmat/MCT/Brazil, CNPq/Brazil Proc. 304015/2014-8 and Fapemig/Brazil CEX APQ-00063/15. BMR was supported by Capes/DS.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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