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Infinitely many solutions for p-harmonic equation with singular term
Journal of Inequalities and Applications volume 2013, Article number: 9 (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.
The main purpose of this paper is to show the existence of infinitely many solutions for the following p-harmonic equation:
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:
(f1) is continuous on and limits subcritical growing at infinity; that is,
where is the critical exponent of Sobolev’s embedding .
(f2) For any , satisfies
(f3) Denote . For all , there exists a constant such that
(f4) is odd with respect to u.
Obviously, for any ,
In , for, we define
this norm is equivalent to .
A weak solution of the problem (1.1) is a critical point of the energy functional
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]).
Li and Squassina in  considered the superlinear p-harmonic equation with Navier boundary conditions
where Ω is an open bounded domain in with a smooth boundary ∂ Ω. , and is a Carathéodory function such that for some positive constant C,
By means of the Morse theory, they proved the existence of two nontrivial solutions to (1.7). After that, Li and Tang  considered a more general problem than (1.7). They got three solutions by the three critical points theorem which was obtained in . If , Candito and Bisci  established a well-determined interval of values of the parameter λ for which the problem (1.7) admits at least two distinct weak solutions. The authors in  considered the p-harmonic equation with Dirichlet conditions and obtained the existence of a nontrivial solution. Under the condition of (1.8),  proved the existence and multiplicity of weak solutions for the nonuniformly nonlinear problem
where with , . and are odd with respect to the second variable. Furthermore, is subcritical and satisfies the Ambrosetti-Rabinowitz condition, that is, there exists a constant such that
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.
This paper proceeds as follows. In the next section, we prove the energy functional satisfies the Palais-Smale condition. In Section 3, by using the symmetric mountain pass theorem (see ), we get the main result of this paper. Throughout the paper, denote
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 .
Lemma 2.1 The condition (f1) implies that for any , there exists a positive constant such that
(f2) implies that as . Furthermore, for any , there exists a small positive constant θ
Lemma 2.2 Under the conditions (f1), (f2) and (f3), if the sequence satisfies
then the sequence is bounded in .
Proof We prove this lemma by contradiction. Without loss of generality, we assume that
Set , then for any . As , there exists such that
If , we denote , set is nonempty. (2.3) implies that
As , . So, for . It follows from (f2) that
Since , then
Using (2.1) and (2.2), we derive that
From (2.6) and (2.7), we get
This contradicts (2.5).
If , then strongly in . Since , we know that
In (2.9), choose , then
From (2.4), it follows that as . Denote
where M is any fixed positive constant. It is clear that
Therefore, for n large enough, .
For every , we define a sequence of as follows:
We claim that
In fact, by (2.1)-(2.10) and the definition of , we have
If as , then (2.11) follows from the above estimate. If is bounded for any n, that is, there exists a constant such that , then
where we use the fact that as . The positive constant ε is arbitrary, thus
(2.11) follows from the above inequality.
From (2.3), we know that
By the definition of , we obtain and , then . The condition (f3) and (2.11) derive that
Which is a contradiction to (2.12). Therefore, the sequence is bounded in . □
Under the conditions of Lemma 2.2, we know . Therefore, there exists a subsequence, still denoted by , and some such that
Obviously, u is a weak solution of (1.1). Now we prove that . According to the concentration compactness principle (see [17, 18]), there exists a subsequence, still denoted by , at most countable set J, a set of different points and two positive number sequences , such that
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 .
Let be small enough such that and for . is a cutting-off function in . for , for , and . Since
we deduce that
By Hölder’s inequality,
Using (f1), (2.14) and (2.15), we have
By the above estimates, (2.13) becomes
Therefore, for any , , which implies that J is finite.
If the concentration is at the origin, let be a cutting-off function in . for , for , and . Choose small enough such that for all . Then
Similarly, we can get
From equalities (2.17)-(2.20), we get
On the other hand, , thus . The proof is complete. □
Lemma 2.4 Assume satisfies (f1), the sequence satisfying (2.3) is bounded in . Then there exist some and a subsequence, still denoted by , such that
where , .
Proof Lemma 2.3 implies that J is finite. Choose with for any . Then
Since a.e. in Ω, we get that
By using the Lebesgue decomposition theorem, we have
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 .
Proof We first prove that
Claim for any closed set , , . In fact, denote . Then, by using a finite covering theorem, there exist finite open balls , with , , such that . Let be a smooth cutting-off function. for any . for , and for . Denote , then
The sequence is still bounded in . Define
then and . Furthermore,
where denotes the support of . As , (2.3) implies
where we have used (2.23). Lemma 2.2 implies that . Hölder’s inequality, (2.25)-(2.28) and Lemma 2.4 deduce that
By (2.1), for any , we have
Therefore, as ,
Since the set F is arbitrary, (2.24) is obtained.
Lemma 2.3 means that for any and , so , which implies that as ,
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)
Assume functional I satisfies the following conditions:
is even and satisfies the Palais-Smale condition.
There exists a finite dimensional subspace such that .
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.
Proof of Theorem 1.1 We know the energy functional is continuous and even. Lemma 2.5 implies that satisfies the PS condition in . In any finite dimensional subspace , all norms are equivalent. For any , since as , there exists a constant such that
Thus, there exists such that . On the other hand, by using (2.2), we have
There exists a sequence of the finite dimensional subspace , and such that
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. □
Remark 3.1 Under the same conditions of Theorem 1.1, we can also prove that the following p-harmonic type equation with Navier boundary conditions:
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 .
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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).
The authors declare that they have no competing interests.
HX carried out the existence of infinitely many solutions for the p-harmonic equation studies and drafted the manuscript. JW participated in its coordination. All authors read and approved the final manuscript.
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Xie, H., Wang, J. Infinitely many solutions for p-harmonic equation with singular term. J Inequal Appl 2013, 9 (2013). https://doi.org/10.1186/1029-242X-2013-9
- infinitely many solutions
- Hardy term