- Open Access
Multiple solutions for a p-Laplacian elliptic problem
© Zeng and Cai; licensee Springer. 2014
- Received: 17 April 2014
- Accepted: 6 May 2014
- Published: 20 May 2014
We consider the following p-Laplacian elliptic equation on : . For certain , we are interested in the functional on a group invariant subspace, and we obtain the existence of infinitely many radial solutions and non-radial solutions of the equation, which extends the result of (Bartsch and Willem in J. Funct. Anal. 117:447-460, 1993) to the space .
- infinitely many radial solutions and non-radial solutions
- group invariant
originates from different problems in physics and mathematical physics. For , problem (1.1) is interpreted as a stationary state of the reaction-diffusion Klein-Gordon equation in chemical dynamics, and Schrödinger equations in finding certain solitary waves.
In the 1980s, people searched for the spherically symmetric solutions of the autonomous equation (where is continuous and odd in u). Berestycki-Lions [1, 2] advocated it for the first time; they obtained the existence of infinitely many radial solutions of the autonomous equation. Then Struwe  got similar results. Gidas et al.  further demonstrated that any positive solution of the equation with some properties must be radial.
Then Bartsch-Willem  found an unbounded sequence of non-radial solutions of (1.1) in with , under the assumption that b and f satisfy certain growth conditions and f is odd in u.
on a bounded domain with smooth boundary ∂ Ω, provided that the nonlinearity f is superlinear and subcritical. They proved (1.2) has a pair of a subsolution and a supersolution. In  they studied problem (1.2) on a bounded domain , with arbitrary, and proved a nodal solution provided that is subcritical and superlinear. Infinitely many nodal solutions are obtained if, in addition, .
on a bounded domain of with Dirichlet boundary condition, where (, are positive constants). They applied the mountain pass theorem to prove the existence of solutions in for the equation in the superlinear and sublinear cases.
For (1.1), Drábek-Pohozaev  proved the existence of multiple positive solutions of quasilinear problems (1.1) of second order by using the fibering method. They considered solutions both in the bounded domain and in the whole space . Moreover, De Nápoli-Mariani  introduced a notion of uniformly convex functional that generalizes the notion of uniformly convex norm. They proved the existence of at least one solution of (1.1), and the existence of infinitely many solutions under further assumptions.
In the present paper, we aim to find the existence of infinitely many radial and non-radial solutions of problem (1.1), and extend the result of  to the space .
A direct extension to the case is faced with serious difficulties. First the energy functional associated to (1.1) is defined on , which is not a Hilbert space for . Another difficulty is the lack of a powerful regularity theory. For the Laplace operator there exists a sequence of Banach spaces with and . But the imbedding (, for ) is not compact.
We study the functional on a group invariant subspace (where is the group of orthogonal linear transformations in ), then we apply the principle of symmetric criticality [, Theorem 5.4] and the fountain theorem to obtain the existence of multiple solutions.
where (), is bounded from below by a positive constant . The growth condition of will be given in the following.
where , .
We require the following assumptions on the nonlinearity f:
(f1) , and for , uniformly on .
(f2) There exists such that for all , , .
(f3) For , and , there exists a constant , such that for any , , .
The main result of this paper is as follows.
Theorem 2.1 If , the assumptions (f1)-(f3) hold, and f is odd in u, then problem (1.1) possesses infinitely many radial solutions.
Theorem 2.2 Suppose or , if the assumptions (f1)-(f3) hold and f is odd in u, then for problem (1.1) there exist infinitely many non-radial solutions.
(f2) means that the nonlinearity f is superlinear, and (f3) means that f is subcritical. These two conditions enable us to use a variational approach for the study of (1.1).
Condition (f2) corresponds to the standard superlinearity condition of Ambrosetti-Rabinowitz in the case . In the case without the assumption (f3), the above theorems may not be true. It can be seen from Pohozaev’s identity for p-Laplacian equations that (1.1) has only a trivial solution .
Remark 2.4 If we fail to define the action of in the proof of Theorem 2.2.
We denote is the completion of with the norm , where . Denote by the usual norm in .
Let be the group of orthogonal linear transformations in , , and .
Throughout this paper, we will use C and to represent various positive constants.
Now, we recall some definitions for the action of a topological group and the fountain theorem.
Definition 2.5 ([, Definition 1.27])
The action is isometric if .
The set of invariant points is defined by . A set is G-invariant if for every . A function is G-invariant if for every .
A map is G-equivariant if , for every .
Definition 2.6 ([, p.99])
Suppose Z is a G-Banach space, that is, there is a G isometric action on Z. Let be the set of G-invariant subsets of Z, and be the class of G-equivariant mappings of Z.
Definition 2.7 ()
has a convergent subsequence.
Theorem 2.8 (Fountain theorem [, Theorem 3.6])
The compact group G acts isometrically on the Banach space , the spaces are invariant and there exists a finite-dimensional space V such that for every , . The action of G on V is admissible.
Let be an G-invariant functional. If for every , there exists such that
(A1) , where .
(A2) , as , where .
(A3) I satisfies the condition, for every .
where , .
In fact, for each , if , then there exists a critical value .
Definition 3.1 ([, Definition A.3])
Lemma 3.2 ([, Theorem A.4])
then is bounded and continuous.
Therefore is bounded. If in , by (3.1), then . Hence is continuous. □
where , here is compact. In addition, each critical point of J is a weak solution of problem (1.1).
Assume in . Since is compact, then in . By Lemma 3.2 and (3.2), is compact. □
Lemma 3.5 ([, Lemma 2.1])
Lemma 3.6 Let be defined in Lemma 3.3. If in and , then in .
Proof If in , then is bounded in .
so in .
then has a subsequence which converges to a critical point of the functional J.
where in the assumption (f2), so is bounded in .
By Lemma 3.6, .
By (3.3), is a critical point of J. □
Now we give the proof of Theorems 2.1 and 2.2 by applying the fountain theorem and the principle of symmetric criticality. First we recall some properties of Banach space.
According to the results in , there exists a Schauder basis for E. Let , then is a Schauder basis for . Since is reflexive, there are , which are characterized by the relations , forming a basis for .
and define a group action of .
Proof It is clear that , so we assume for , , as . For every , there exists such that and . By the definition of , in . Since the imbedding is compact, then in . Thus we get . □
Proof of Theorem 2.1 Note that J is -invariant, by the principle of symmetric criticality [, Theorem 5.4], any critical point of is a solution of problem (1.1). J is invariant with respect to the action .
Now we claim that satisfies the assumptions of the fountain theorem.
This proves (A2).
So there exists such that (A1) is satisfied.
are critical values of the functional J. So we can get an unbounded sequence of solutions of (1.1), and the solutions are radial. □
Proof of Theorem 2.2 In this proof, we will show that it suffices to find the critical points of J restricted to a subspace of invariant functions. The proof is similar to Theorem 1.31 in .
Let be a fixed integer different from . The action of on E is defined by . For is compatible with , the embedding () is compact (or see  for details).
Moreover, the embedding is compact. As in the proof of Theorem 2.1, we obtain a sequence of non-radial solutions of (1.1). □
The first author is supported by the project of ‘Min-Hong Kong cooperation postdoctoral training’ funded by Fujian Provincial Civil Service Bureau and the Nonlinear Analysis Innovation Team (IRTL1206) funded by Fujian Normal University. The second author is sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
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