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 .
Keywords:p-Laplacian; infinitely many radial solutions and non-radial solutions; group invariant
The interesting equation
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.
In recent years, the existence and structure of solutions for the p-Laplacian equation has found considerable interest, and different approaches have been developed. Bartsch-Liu  studied the p-Laplacian problem
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, .
Furthermore Jiu-Su  applied Morse theory to study the existence of nontrivial solutions of p-Laplacian type Dirichlet boundary value problems. Agarwal-Perera  obtained two positive solutions of singular discrete p-Laplacian problems using variational methods. Chabrowski-Fu  studied the p-Laplacian problem:
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.
2 The main results and preliminaries
This paper is devoted to the study of infinitely many radial and non-radial solutions for a p-Laplacian equation:
The corresponding functional is
We require the following assumptions on the nonlinearity f:
The main result of this paper is as follows.
(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 .
We shall use the norm
Now, we recall some definitions for the action of a topological group and the fountain theorem.
Definition 2.5 ([, Definition 1.27])
Definition 2.6 ([, p.99])
Definition 2.7 ()
has a convergent subsequence.
Theorem 2.8 (Fountain theorem [, Theorem 3.6])
3 Proof of theorems
Definition 3.1 ([, Definition A.3])
Lemma 3.2 ([, Theorem A.4])
Lemma 3.5 ([, Lemma 2.1])
Since E is reflexive, is reflexive, then has a weakly convergent subsequence. Going if necessary to a subsequence, let . By Lemma 3.4 , and by the definition of , , then converges. So we assume . Observe that
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 .
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 .
This proves (A2).
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).
It is clear that 0 is the only radial function on the set
The authors declare that they have no competing interests.
Both authors contributed equally. Both authors read and approved the final manuscript.
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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