Multiple solutions for a p-Laplacian elliptic problem
Boundary Value Problems volume 2014, Article number: 124 (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 .
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:
where (), is bounded from below by a positive constant . The growth condition of will be given in the following.
The corresponding functional is
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 shall use the norm
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 of a topological group G on a normed space Z is a continuous map , such that for all , ,
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 ()
Let Z be a Banach space, and . The functional I satisfies the condition if any sequence such that
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 .
Then I possesses an unbounded sequence of critical values . can be characterized as
where , .
In fact, for each , if , then there exists a critical value .
3 Proof of theorems
Definition 3.1 ([, Definition A.3])
On the space , we define the norm
On the space , we define the norm
Lemma 3.2 ([, Theorem A.4])
Assume , and , then for every , , the operator
Lemma 3.3 Let be the mapping given by
then is bounded and continuous.
Proof By the definition of ,
Therefore is bounded. If in , by (3.1), then . Hence is continuous. □
Lemma 3.4 Suppose the nonlinearity f satisfies (f1)-(f3), then the functional , and
where , here is compact. In addition, each critical point of J is a weak solution of problem (1.1).
Proof By Lemma 3.3, we only need to prove is continuous. By Hölder inequality
where , . If in , then in . It follows from (f3) and Lemma 3.2 that
Assume in . Since is compact, then in . By Lemma 3.2 and (3.2), is compact. □
Lemma 3.5 ([, Lemma 2.1])
There exist constants and , such that for all , , we have
Lemma 3.6 Let be defined in Lemma 3.3. If in and , then in .
Proof If in , then is bounded in .
If , by Lemma 3.5,
so in .
If , by Lemma 3.5 and the Hölder inequality,
Lemma 3.7 Assume that f satisfies (f1)-(f3). Let be a sequence such that
then has a subsequence which converges to a critical point of the functional J.
Proof First we show that each sequence satisfying , , as , is bounded. By (f2) and (f3),
where in the assumption (f2), so is bounded in .
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
By Lemma 3.6, .
Next we want to show that is a critical point of J, i.e. . By Lemma 3.3, ,
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 .
Lemma 3.8 If , then
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.
By the assumptions (f1) and (f3), for ,
where is the lower bound of . Choose , by Lemma 3.8, for , , , and as ,
This proves (A2).
Now we want to show that the condition (A1) is satisfied. By integrating, we obtain from (f2) and (f3) that, there exist two constants , such that for any , . Hence,
Since is finite dimensional, all norms are equivalent on . Therefore, and imply that
So there exists such that (A1) is satisfied.
condition is proved above. By Theorem 2.8, we find, for , that
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).
Let be the involution defined on by
The action of on is defined by
It is clear that 0 is the only radial function on the set
Moreover, the embedding is compact. As in the proof of Theorem 2.1, we obtain a sequence of non-radial solutions of (1.1). □
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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.
The authors declare that they have no competing interests.
Both authors contributed equally. Both authors read and approved the final manuscript.
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Zeng, J., Cai, S. Multiple solutions for a p-Laplacian elliptic problem. Bound Value Probl 2014, 124 (2014). https://doi.org/10.1186/1687-2770-2014-124