Infinitely many solutions for p-harmonic equation with singular term
© Xie and Wang; licensee Springer 2013
Received: 19 May 2012
Accepted: 13 November 2012
Published: 6 January 2013
In this paper, we study the following p-harmonic problem involving the Hardy term:
where Ω is an open bounded domain containing the origin in , and . By using the variational method, we prove that the above problem has infinitely many solutions with positive energy levels.
AMS Subject Classification:35J60, 35J65.
where Ω is an open bounded domain containing the origin in , the boundary ∂ Ω is smooth. , . is the outer normal derivative. Nonlinearity satisfies the following conditions:
where is the critical exponent of Sobolev’s embedding .
(f4) is odd with respect to u.
this norm is equivalent to .
where . It is easy to check that is a continuous even functional.
Biharmonic equations can describe the static form change of a beam or the sport of a rigid body. For example, this type of equation furnishes a model for studying traveling wave in suspension bridges (see ). By using variational arguments, many authors investigated nonlinear biharmonic equations under Dirichlet boundary conditions or Navier boundary conditions and got interesting results (see [4–10]).
The aim of this paper is to obtain infinitely many solutions for the problem (1.1) when . The main difficulty lies in the fact that the embeddings and are not compact. Furthermore, the assumption (f1) is not the usual subcritical growth (1.8). The condition (f3) is weaker than the A-R condition (1.10). We use the concentration compactness principle (see [17, 18]) to overcome those difficulties. The following theorem is our main result.
Theorem 1.1 Assume satisfies (f1)-(f4), then the problem (1.1) possesses infinitely many weak solutions and the corresponding critical values are positive.
We use to denote the norm of , C, stand for universal constants. We omit dx and Ω in the integrals if there is no other indication.
2 Palais-Smale condition
To show the (PS) sequence of the variational functional is compact in , we first prove the boundedness of by the analytic argument which has been used in . Then, using the concentration compactness principle, we get the compactness of .
then the sequence is bounded in .
This contradicts (2.5).
Therefore, for n large enough, .
(2.11) follows from the above inequality.
Which is a contradiction to (2.12). Therefore, the sequence is bounded in . □
where is the unit Dirac measure at .
Lemma 2.3 Assume satisfies (f1), the sequence is bounded in and satisfies (2.3). Then the set is finite, .
Proof We first prove that for any .
Therefore, for any , , which implies that J is finite.
On the other hand, , thus . The proof is complete. □
where , .
where . Then .
Lemma 2.5 Assume satisfies (f1)-(f3), the sequence satisfies (2.3). Then there exist and a subsequence, still denoted by , such that converges to u strongly in .
Since the set F is arbitrary, (2.24) is obtained.
Together with a.e. in Ω, we complete the proof. □
3 The proof of the main result
In this section, by using the following symmetric mountain pass theorem (see ), we give the proof of Theorem 1.1.
Lemma 3.1 (Symmetric mountain pass theorem)
is even and satisfies the Palais-Smale condition.
There exists a finite dimensional subspace such that .
- (3)There exists a sequence of the finite dimensional subspace , and such that
Then I has infinitely many different critical points, and the corresponding energy values are positive.
Finally, all the assumptions of Lemma 3.1 are satisfied. Hence, the problem (1.1) possesses infinitely many weak solutions, and the corresponding critical values are positive. □
has infinitely many solutions , where Ω containing the origin is an open bounded domain in , ∂ Ω is smooth. , .
Remark 3.2 All the results obtained above obviously hold if we choose with .
The authors are thankful to the referees for their helpful suggestions and necessary corrections in the completion of this paper. This research is supported by the Education Department of Henan Province (12B110002) and NSF of China (61143002).
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