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Fourth order elliptic boundary value problem with nonlinear term decaying at the origin
Journal of Inequalities and Applications volume 2013, Article number: 432 (2013)
Abstract
We consider the number of the weak solutions for some fourth order elliptic boundary value problem with bounded nonlinear term decaying at the origin. We get a theorem, which shows the existence of the bounded solution for this problem. We obtain this result by approaching the variational method and using the generalized mountain pass theorem for the fourth order elliptic problem with bounded nonlinear term.
MSC:35J30, 35J40.
1 Introduction
Let Ω be a bounded domain in {R}^{n} with smooth boundary ∂ Ω. Let c\in {R}^{1} and g:\overline{\mathrm{\Omega}}\times R\to R be a {C}^{1} function. In this paper, we consider the number of the weak solutions for the following fourth order elliptic problem with the Dirichlet boundary condition
We assume that g\in {C}^{1}(\overline{\mathrm{\Omega}}\times R,R) satisfies the following:
(g1) g\in {C}^{1}(\overline{\mathrm{\Omega}}\times R,R),
(g2) g(x,0)=0, g(x,\xi )=o(\xi ) uniformly with respect to x\in \overline{\mathrm{\Omega}},
(g3) there exists C>0 such that g(x,\xi )<C\mathrm{\forall}(x,\xi )\in \overline{\mathrm{\Omega}}\times R.
The eigenvalue problem
has infinitely many eigenvalues {\lambda}_{j}, j\ge 1, which is repeated as often as its multiplicity, and the corresponding eigenfunctions {\varphi}_{j}, j\ge 1 suitably normalized with respect to {L}^{2}(\mathrm{\Omega}) inner product. The eigenvalue problem
has also infinitely many eigenvalues {\mathrm{\Lambda}}_{j}={\lambda}_{j}({\lambda}_{j}c), j\ge 1 and corresponding eigenfunctions {\varphi}_{j}, j\ge 1. We note that
Furthermore, we assume that c\in {R}^{1} satisfies {\lambda}_{j}<c<{\lambda}_{j+1}.
Jung and Choi [1] proved that (1.1) has at least one nontrivial solution, when c<{\lambda}_{1} and g satisfies the condition (g1), (g2) and additional conditions
(g3)′ there exists \xi \ge 0 such that p(x,\xi )\le 0\mathrm{\forall}x\in \overline{\mathrm{\Omega}},
(g4)′ there exist a constant r>0 and an element e\in H such that \parallel e\parallel =r, e<\xi and \frac{1}{2}{r}^{2}{\int}_{\mathrm{\Omega}}P(x,e)<0,
by reducing problem (1.1) to the problem with bounded nonlinear term and then applying the maximum principle for the elliptic operator −Δ and \mathrm{\Delta}c two times and the mountain pass theorem in the critical point theory. Jung and Choi [2] showed the existence of at least two solutions, one of which is a bounded solution and a large norm solution of (1.1), when g(u) is polynomial growth or exponential growth nonlinear term. The authors proved these results by the variational method and the mountain pass theorem. For the constant coefficient semilinear case Choi and Jung [3] showed that the problem
has at least two nontrivial solutions, when c<{\lambda}_{1}, {\mathrm{\Lambda}}_{1}<b<{\mathrm{\Lambda}}_{2} and s<0 or when {\lambda}_{1}<c<{\lambda}_{2}, b<{\mathrm{\Lambda}}_{1} and s>0. The authors obtained these results by using the variational reduction method. The authors [4] also proved that when c<{\lambda}_{1}, {\mathrm{\Lambda}}_{1}<b<{\mathrm{\Lambda}}_{2} and s<0, (1.2) has at least three nontrivial solutions by using the degree theory. Tarantello [5] also studied the problem
She showed that if c<{\lambda}_{1} and b\ge {\mathrm{\Lambda}}_{1}, then (1.3) has a negative solution. She obtained this result by the degree theory. Micheletti and Pistoia [6] also proved that if c<{\lambda}_{1} and b\ge {\mathrm{\Lambda}}_{2}, then (1.3) has at least three solutions by the variational linking theorem and LeraySchauder degree theory.
In this paper, we are trying to find weak solutions of (1.1), that is,
where H is introduced in Section 2.
We consider the associated functional of (1.1)
where G(x,s)={\int}_{0}^{s}g(x,\tau )\phantom{\rule{0.2em}{0ex}}d\tau. By (g1), I is well defined.
Our main result is the following.
Theorem 1.1Assume that{\lambda}_{j}<c<{\lambda}_{j+1}, j\ge 1, andgsatisfies the conditions (g1)(g3). Then (1.1) has at least one bounded weak solution.
We prove Theorem 1.1 by approaching the variational method and using the mountain pass theorem for the reduced fourth order elliptic problem with bounded nonlinear term. The outline of the proof of Theorem 1.1 is as follows: In Section 2, we prove that functional I(u)\in {C}^{1} and the functional I satisfies the PalaisSmale condition. In Section 3, we show that the functional I satisfies the generalized mountain pass theorem, and so, prove that I has at least one nontrival critical point, from which we prove Theorem 1.1.
2 Variational approach
Let {L}^{2}(\mathrm{\Omega}) be a square integrable function space defined on Ω. Any element u in {L}^{2}(\mathrm{\Omega}) can be written as
We define a subspace H of {L}^{2}(\mathrm{\Omega}) as follows
Then this is a complete normed space with a norm
Since {\lambda}_{k}\to +\mathrm{\infty} and c is fixed, we have {\mathrm{\Lambda}}_{k}\to \mathrm{\infty} and

(i)
{\mathrm{\Delta}}^{2}u+c\mathrm{\Delta}u\in H implies u\in H,

(ii)
\parallel u\parallel \ge C{\parallel u\parallel}_{{L}^{2}(\mathrm{\Omega})} for some C>0,

(iii)
{\parallel u\parallel}_{{L}^{2}(\mathrm{\Omega})}=0 if and only if \parallel u\parallel =0,
which is proved in [7].
Let
Then H={H}_{}\oplus {H}_{+}, for u\in H, u={u}^{}+{u}^{+}\in {H}_{}\oplus {H}_{+}. Let {P}_{+} be the orthogonal projection from H onto {H}_{+} and {P}_{} be the orthogonal projection from H onto {H}_{}. We can write {P}_{+}u={u}^{+}, {P}_{}u={u}^{}, for u\in H.
By the following Lemma 2.1, the weak solutions of (1.1) coincide with the critical points of the associated functional I(u).
Lemma 2.1Assume that{\lambda}_{j}<c<{\lambda}_{j+1}, j\ge 1, andgsatisfies the conditions (g1)(g3). ThenI(u)is continuous, and Fréchet differentiable inHwith Fréchet derivative
If we set
then{F}^{\prime}(u)is continuous with respect to weak convergence, {F}^{\prime}(u)is compact and
this implies thatI\in {C}^{1}(H,R)andF(u)is weakly continuous.
The proof of Lemma 2.1 has the similar process to that of the proof in Appendix B in [8].
Now, we shall show that I(u) satisfies the PalaisSmale condition.
Lemma 2.2Assume that{\lambda}_{j}<c<{\lambda}_{j+1}, j\ge 1, andgsatisfies the conditions (g1)(g3). Then the functionalIsatisfies the PalaisSmale condition: Any sequence({u}_{m})inH, for whichI({u}_{m})\le Mand{I}^{\prime}({u}_{m})\to 0asm\to \mathrm{\infty}, possesses a convergent subsequence.
Proof Let us choose u\in H. By g\in {C}^{1} and (g1), G(x,u) is bounded. Then we have
Since u is bounded and {\int}_{\mathrm{\Omega}}G(x,u)\phantom{\rule{0.2em}{0ex}}dx is bounded, I(u) is bounded from below. Thus, I satisfies the (\mathit{PS}) condition. □
3 Proof of Theorem 1.1
Now, we recall the generalized mountain pass theorem (cf. Theorem 5.3 in [8]).
Let
Theorem 3.1 (Generalized mountain pass theorem)
LetHbe a real Banach space withH=V\oplus X, whereV\ne \{0\}and is finitedimensional. Suppose thatI\in {C}^{1}(H,R)satisfies(\mathit{PS})condition, and

(i)
there are constants\rho ,\alpha >0and a bounded neighborhood{B}_{\rho}of 0 such thatI{}_{\partial {B}_{\rho}\cap X}\ge \alpha, and

(ii)
there is ane\in \partial {B}_{1}\cap XandR>\rhosuch that ifQ=({\overline{B}}_{R}\cap V)\oplus \{re\mid 0<r<R\}, thenI{}_{\partial Q}\le 0.
ThenIpossesses a critical valueb\ge \alpha. Moreover, bcan be characterized as
where
We shall show that the functionalIsatisfies the generalized mountain pass geometrical assumptions.
Let{H}_{j}=span\{{\varphi}_{1},\dots ,{\varphi}_{j}\}. Then{H}_{j}is a subspace ofHsuch that
Let
Lemma 3.1Assume that{\lambda}_{j}<c<{\lambda}_{j+1}andgsatisfies (g1)(g3). Then

(i)
there are constants\rho >0,\alpha >0and a bounded neighborhood{B}_{\rho}of 0 such thatI{}_{\partial {B}_{\rho}\cap {H}_{j}^{\perp}}\ge \alpha, and

(ii)
there is ane\in \partial {B}_{1}\cap {H}_{j}^{\perp}andR>\rhosuch that ifQ=({\overline{B}}_{R}\cap {H}_{j})\oplus \{re\mid 0<r<R\}, thenI{}_{\partial Q}\le 0, and

(iii)
there exists{u}_{0}\in H\mathrm{\setminus}Qsuch that\parallel {u}_{0}\parallel >RandI({u}_{0})\le 0.
Proof (i) Let u\in {H}_{j}^{\perp}. We note that
Since G(x,u(x)) is bounded, there exists a constant C>0 such that C\le G(x,u(x))\le C. Thus, we have
for C>0. There exist \rho >o and \alpha >o such that if u\in \partial {B}_{\rho}\cap {H}_{j}^{\perp}, then I(u)\ge \alpha.

(ii)
Let us choose an element e\in \partial {B}_{1}\cap {H}_{j}^{\perp}. Let u\in (\overline{{B}_{r}}\cap {H}_{j})\oplus \{re\mid 0<r\}. Then u=v+w, v\in {B}_{r}\cap {H}_{j}, w=re. We note that
\text{if}v\in {B}_{r}\cap {H}_{j},\phantom{\rule{1em}{0ex}}{\int}_{\mathrm{\Omega}}({\mathrm{\Delta}}^{2}v+c\mathrm{\Delta}v)v\phantom{\rule{0.2em}{0ex}}dx\le {\lambda}_{j}({\lambda}_{j}c){\parallel v\parallel}_{{L}^{2}(\mathrm{\Omega})}^{2}0.
Thus, we have
for C>0. Then there exists R>0 such that if u\in Q=({\overline{B}}_{R}\cap {H}_{j})\oplus \{re\mid 0<r<R\}, then I(u){}_{\partial Q}\le 0, from which we can choose an element {u}_{0}\in H\mathrm{\setminus}{B}_{R} such that I({u}_{0})\le 0.

(iii)
If we choose {u}_{0}\in H\mathrm{\setminus}Q, then by (ii), I({u}_{0})\le 0. □
Proof of Theorem 1.1 We will show that I(u) has a nontrivial critical point by the generalized mountain pass theorem. By Lemma 2.1, I(u) is continuous and Fréchet differentiable in H. By Lemma 2.2, the functional I satisfies (\mathit{PS}) condition. We note that I(0)=0. By Lemma 3.1, there are constants \rho >0,\alpha >0 and a bounded neighborhood {B}_{\rho} of 0 such that I{}_{\partial {B}_{\rho}\cap {H}_{j}^{\perp}}\ge \alpha, and there is an e\in \partial {B}_{1}\cap {H}_{j}^{\perp} and R>\rho such that if Q=({\overline{B}}_{R}\cap {H}_{j})\oplus \{re\mid 0<r<R\}. Let us set
By the generalized mountain pass theorem, I possesses a critical value b\ge \alpha. Moreover, b can be characterized as
Thus, we prove that I has at least one nontrivial critical point. We denote by \tilde{u} a critical point of I such that I(\tilde{u})=b. We claim that b is bounded. In fact, by (iii) of Lemma 3.1, we have
and by (g3),
for some constant {C}_{2}>0. Since 0\le t\le 1, b is bounded:
We claim that \tilde{u} is bounded. In fact, by contradiction, {\mathrm{\Delta}}^{2}\tilde{u}+c\mathrm{\Delta}\tilde{u}=g(x,\tilde{u}) and for any K>0, {max}_{\mathrm{\Omega}}\tilde{u}(x)>K imply that
is not bounded, which is absurd to the fact that b=I(\tilde{u}) is bounded. Thus, \tilde{u} is bounded, so (1.1) has at least one bounded weak solution. Thus, we prove Theorem 1.1. □
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Acknowledgements
This work (Tacksun Jung) was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (KRF20100023985).
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Authors’ contributions
TJ carried out the studies for the existence of weak solutions of the fourth order elliptic boundary value problem, participated in the sequence alignment and drafted the manuscript. QC participated in the sequence alignment and drafted the manuscript.
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Jung, T., Choi, QH. Fourth order elliptic boundary value problem with nonlinear term decaying at the origin. J Inequal Appl 2013, 432 (2013). https://doi.org/10.1186/1029242X2013432
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DOI: https://doi.org/10.1186/1029242X2013432
Keywords
 fourth order elliptic boundary value problem
 nonlinear term decaying at the origin
 bounded nonlinear term
 variational method
 generalized mountain pass theorem
 {(\mathit{PS})}_{c} condition