Abstract
We consider the following pLaplacian 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 nonradial solutions of the equation, which extends the result of (Bartsch and Willem in J. Funct. Anal. 117:447460, 1993) to the space .
Keywords:
pLaplacian; infinitely many radial solutions and nonradial solutions; group invariant1 Introduction
The interesting equation
originates from different problems in physics and mathematical physics. For , problem (1.1) is interpreted as a stationary state of the reactiondiffusion KleinGordon 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). BerestyckiLions [1,2] advocated it for the first time; they obtained the existence of infinitely many radial solutions of the autonomous equation. Then Struwe [3] got similar results. Gidas et al.[4] further demonstrated that any positive solution of the equation with some properties must be radial.
Then BartschWillem [5] found an unbounded sequence of nonradial 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 pLaplacian equation has found considerable interest, and different approaches have been developed. BartschLiu [6] studied the pLaplacian 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 [7] 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 JiuSu [8] applied Morse theory to study the existence of nontrivial solutions of pLaplacian type Dirichlet boundary value problems. AgarwalPerera [9] obtained two positive solutions of singular discrete pLaplacian problems using variational methods. ChabrowskiFu [10] studied the pLaplacian 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), DrábekPohozaev [11] 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 NápoliMariani [12] 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 nonradial solutions of problem (1.1), and extend the result of [5] 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 [[13], 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 nonradial solutions for a pLaplacian equation:
where (), is bounded from below by a positive constant . The growth condition of will be given in the following.
The corresponding functional is
We require the following assumptions on the nonlinearity f:
(f_{1}) , and for , uniformly on .
(f_{2}) There exists such that for all , , .
(f_{3}) For , and , there exists a constant , such that for any , , .
The main result of this paper is as follows.
Theorem 2.1If, the assumptions (f_{1})(f_{3}) hold, andfis odd inu, then problem (1.1) possesses infinitely many radial solutions.
Theorem 2.2Supposeor, if the assumptions (f_{1})(f_{3}) hold andfis odd inu, then for problem (1.1) there exist infinitely many nonradial solutions.
Remark 2.3 The assumptions (f_{1})(f_{3}) are from [5]. BartschWillem [5] considered the existence of nonradial solutions for the Euclidean scalar field equation ().
(f_{2}) means that the nonlinearity f is superlinear, and (f_{3}) means that f is subcritical. These two conditions enable us to use a variational approach for the study of (1.1).
Condition (f_{2}) corresponds to the standard superlinearity condition of AmbrosettiRabinowitz in the case . In the case without the assumption (f_{3}), the above theorems may not be true. It can be seen from Pohozaev’s identity for pLaplacian 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 ([[14], Definition 1.27])
The action of a topological group G on a normed space Z is a continuous map , such that for all , ,
The set of invariant points is defined by . A set is Ginvariant if for every . A function is Ginvariant if for every .
A map is Gequivariant if , for every .
Definition 2.6 ([[15], p.99])
Suppose Z is a GBanach space, that is, there is a G isometric action on Z. Let be the set of Ginvariant subsets of Z, and be the class of Gequivariant mappings of Z.
Definition 2.7 ([14])
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 [[14], Theorem 3.6])
The compact groupGacts isometrically on the Banach space, the spacesare invariant and there exists a finitedimensional spaceVsuch that for every, . The action ofGonVis admissible.
Letbe anGinvariant functional. If for every, there existssuch that
(A_{3}) Isatisfies thecondition, for every.
ThenIpossesses an unbounded sequence of critical values. can be characterized as
3 Proof of theorems
Definition 3.1 ([[14], Definition A.3])
On the space , we define the norm
On the space , we define the norm
Lemma 3.2 ([[14], Theorem A.4])
Assume, and, then for every, , the operator
is continuous.
Lemma 3.3Letbe the mapping given by
thenis bounded and continuous.
Therefore is bounded. If in , by (3.1), then . Hence is continuous. □
Lemma 3.4Suppose the nonlinearityfsatisfies (f_{1})(f_{3}), then the functional, and
where, hereis compact. In addition, each critical point ofJis 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 (f_{3}) and Lemma 3.2 that
So
Assume in . Since is compact, then in . By Lemma 3.2 and (3.2), is compact. □
Lemma 3.5 ([[16], Lemma 2.1])
There exist constantsand, such that for all, , we have
Lemma 3.6Letbe defined in Lemma 3.3. Ifinand, thenin.
Proof If in , then is bounded in .
If , by Lemma 3.5 and the Hölder inequality,
□
Lemma 3.7Assume thatfsatisfies (f_{1})(f_{3}). Letbe a sequence such that
thenhas a subsequence which converges to a critical point of the functionalJ.
Proof First we show that each sequence satisfying , , as , is bounded. By (f_{2}) and (f_{3}),
where in the assumption (f_{2}), 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
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 [17], 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 .
We denote
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 [[13], 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 (f_{1}) and (f_{3}), for ,
where is the lower bound of . Choose , by Lemma 3.8, for , , , and as ,
This proves (A_{2}).
Now we want to show that the condition (A_{1}) is satisfied. By integrating, we obtain from (f_{2}) and (f_{3}) 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 (A_{1}) 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 [14].
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 [18] 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 nonradial solutions of (1.1). □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally. Both authors read and approved the final manuscript.
Acknowledgements
The first author is supported by the project of ‘MinHong 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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