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Existence of solutions for a class of biharmonic equations with the Navier boundary value condition
Boundary Value Problems volume 2012, Article number: 154 (2012)
Abstract
In this paper, the existence of at least one nontrivial solution for a class of fourthorder elliptic equations with the Navier boundary value conditions is established by using the linking methods.
1 Introduction
Consider the following Navier boundary value problem:
where ${\mathrm{\u25b3}}^{2}$ is the biharmonic operator, $l\in R$ and Ω is a bounded smooth domain in ($N>4$).
The conditions imposed on $f(x,t)$ are as follows:
(H_{1}) , and there are constants ${C}_{1},{C}_{2}\ge 0$ such that
where ${p}^{\ast}=\frac{2N}{N4}$;
(H_{2}) $f(x,t)=\circ (t)$, $t\to 0$, uniformly on Ω;
(H_{3}) ${lim}_{t\to \mathrm{\infty}}\frac{f(x,t)}{t}=+\mathrm{\infty}$ uniformly on Ω;
(H_{4}) There is a constant $\theta \ge 1$ such that for all $(x,t)\in \mathrm{\Omega}\times R$ and $s\in [0,1]$,
where $F(x,t)={\int}_{0}^{t}f(x,s)\phantom{\rule{0.2em}{0ex}}ds$;
(H_{5}) For some $\delta >0$, either
or
This fourthorder semilinear elliptic problem has been studied by many authors. In [1], there was a survey of results obtained in this direction. In [2], Micheletti and Pistoia showed that (1.1) admits at least two solutions by a variation of linking if $f(x,u)$ is sublinear. And in [3], the authors proved that the problem (1.1) has at least three solutions by a variational reduction method and a degree argument. In [4], Zhang and Li showed that (1.1) admits at least two nontrivial solutions by the Morse theory and local linking if $f(x,u)$ is superlinear and subcritical on u. In [5], Zhang and Wei obtained the existence of infinitely many solutions for the problem (1.1) where the nonlinearity involves a combination of superlinear and asymptotically linear terms. As far as the problem (1.1) is concerned, existence results of signchanging solutions were also obtained. See, e.g., [6, 7] and the references therein.
We will use linking methods to give the existence of at least one nontrivial solution for (1.1).
Let X be a Banach space with a direct sum decomposition
The function $I\in {C}^{1}(X,R)$ has a local linking at 0, with respect to $({X}^{1},{X}^{2})$ if for some $r>0$,
It is clear that 0 is a critical point of I.
The notion of local linking generalizes the notions of local minimum and local maximum. When 0 is a nondegenerate critical point of a functional of class ${C}^{2}$ defined on a Hilbert space and $I(0)=0$, I has local linking at 0.
The condition of local linking was introduced in [8] under stronger assumptions
Under those assumptions, the existence of nontrivial critical points was proved for functionals which are
The cases (a), (b) and (c) were considered in [10] with only conditions (1.2) and (1.3), and three theorems about critical points were proved. Using these theorems, the author in [10] proved the existence of at least one nontrivial solution for the secondorder elliptic boundary value problem with the Dirichlet boundary value condition. In the present paper, we also use the three theorems in [10] and the linking technique to give the existence of at least one nontrivial solution for the biharmonic problem (1.1) under a weaker condition. The main results are as follows.
Theorem 1.1 Assume the conditions (H_{1})(H_{4}) hold. If l is an eigenvalue of −△ (with the Dirichlet boundary condition), assume also (H_{5}). Then the problem (1.1) has at least one nontrivial solution.
We also consider asymptotically quadratic functions.
Let $0<{\lambda}_{1}<{\lambda}_{2}<\cdots <{\lambda}_{k}<\cdots $ be the eigenvalues of $(\mathrm{\u25b3},{H}_{0}^{1}(\mathrm{\Omega}))$. Then ${\mu}_{j}$ ($j\in {N}_{+}$) is the eigenvalue of $({\mathrm{\u25b3}}^{2}+l\mathrm{\u25b3},{H}^{2}(\mathrm{\Omega})\cap {H}_{0}^{1}(\mathrm{\Omega}))$, where ${\mu}_{j}={\lambda}_{j}({\lambda}_{j}l)$. We assume that
(H_{6}) $f(x,u)={f}_{\mathrm{\infty}}u+\circ (u)$, $u\to \mathrm{\infty}$, uniformly in Ω, and ${\mu}_{k}<{f}_{\mathrm{\infty}}<{\mu}_{k+1}$.
Theorem 1.2 Assume the conditions (H_{1}), (H_{6}) and one of the following conditions:
(A_{1}) ${\lambda}_{j}<l<{\lambda}_{j+1}$, $j\ne k$;
(A_{2}) ${\lambda}_{j}=l<{\lambda}_{j+1}$, $j\ne k$, for some $\delta >0$,
(A_{3}) ${\lambda}_{j}<l={\lambda}_{j+1}$, $j\ne k$, for some $\delta >0$,
Then the problem (1.1) has at least one nontrivial solution.
2 Preliminaries
Let X be a Banach space with a direct sum decomposition
Consider two sequences of a subspace:
such that
For every multiindex $\alpha =({\alpha}_{1},{\alpha}_{2})\in {N}^{2}$, let ${X}_{\alpha}={X}_{{\alpha}_{1}}\oplus {X}_{{\alpha}_{2}}$. We know that
A sequence $({\alpha}_{n})\subset {N}^{2}$ is admissible if for every $\alpha \in {N}^{2}$, there is $m\in N$ such that $n\ge m\Rightarrow {\alpha}_{n}\ge \alpha $. For every $I:X\to R$, we denote by ${I}_{\alpha}$ the function I restricted ${X}_{\alpha}$.
Definition 2.1 Let I be locally Lipschitz on X and $c\in R$. The functional I satisfies the ${(C)}_{c}^{\ast}$ condition if every sequence $({u}_{{\alpha}_{n}})$ such that $({\alpha}_{n})$ is admissible and
contains a subsequence which converges to a critical point of I.
Definition 2.2 Let I be locally Lipschitz on X and $c\in R$. The functional I satisfies the ${(C)}^{\ast}$ condition if every sequence $({u}_{{\alpha}_{n}})$ such that $({\alpha}_{n})$ is admissible and
contains a subsequence which converges to a critical point of I.
Remark 2.1

1.
The ${(C)}^{\ast}$ condition implies the ${(C)}_{c}^{\ast}$ condition for every $c\in R$.

2.
When the ${(C)}_{c}^{\ast}$ sequence is bounded, then the sequence is a ${(PS)}_{c}^{\ast}$ sequence (see [11]).

3.
Without loss of generality, we assume that the norm in X satisfies
$${\parallel {u}_{1}+{u}_{2}\parallel}^{2}={\parallel {u}_{1}\parallel}^{2}+{\parallel {u}_{2}\parallel}^{2},\phantom{\rule{1em}{0ex}}{u}_{j}\in {X}_{j},j=1,2.$$
Definition 2.3 Let X be a Banach space with a direct sum decomposition
The function $I\in {C}^{1}(X,R)$ has a local linking at 0, with respect to $({X}^{1},{X}^{2})$, if for some $r>0$,
Lemma 2.1 (see [10])
Suppose that $I\in {C}^{1}(X,R)$ satisfies the following assumptions:
(B_{1}) I has a local linking at 0 and ${X}^{1}\ne \{0\}$;
(B_{2}) I satisfies ${(PS)}^{\ast}$;
(B_{3}) I maps bounded sets into bounded sets;
(B_{4}) for every $m\in N$, $I(u)\to \mathrm{\infty}$, $\parallel u\parallel \to \mathrm{\infty}$, $u\in X={X}_{m}^{1}\oplus {X}^{2}$. Then I has at least two critical points.
Remark 2.2 Assume I satisfies the ${(C)}_{c}^{\ast}$ condition. Then this theorem still holds.
Let X be a real Hilbert space and let $I\in {C}^{1}(X,R)$. The gradient of I has the form
where A is a bounded selfadjoint operator, 0 is not the essential spectrum of A, and B is a nonlinear compact mapping.
We assume that there exist an orthogonal decomposition,
and two sequences of finitedimensional subspaces,
such that
For every multiindex $\alpha =({\alpha}_{1},{\alpha}_{2})\in {N}^{2}$, we denote by ${X}_{\alpha}$ the space
by ${p}_{\alpha}:X\to {X}_{\alpha}$ the orthogonal projector onto ${X}_{\alpha}$, and by ${M}^{}(L)$ the Morse index of a selfadjoint operator L.
Lemma 2.2 (see [10])
I satisfies the following assumptions:

(i)
I has a local linking at 0 with respect to $({X}^{1},{X}^{2})$;

(ii)
there exists a compact selfadjoint operator ${B}_{\mathrm{\infty}}$ such that
$$B(u)={B}_{\mathrm{\infty}}(u)+\circ (\parallel u\parallel ),\phantom{\rule{1em}{0ex}}\parallel u\parallel \to \mathrm{\infty};$$ 
(iii)
$A+{B}_{\mathrm{\infty}}$ is invertible;

(iv)
for infinitely many multipleindices $\alpha :=(n,n)$,
$${M}^{}((A+{P}_{\alpha}{B}_{\mathrm{\infty}}){}_{{X}_{\alpha}})\ne dim{X}_{n}^{2}.$$
Then I has at least two critical points.
3 The proof of main results
Proof of Theorem 1.1 (1) We shall apply Lemma 2.1 to the functional
defined on $X={H}_{0}^{1}(\mathrm{\Omega})\cap {H}^{2}(\mathrm{\Omega})$. We consider only the case $l={\lambda}_{j}$, and
Then other case is similar and simple.
Let ${X}^{2}$ be the finite dimensional space spanned by the eigenfunctions corresponding to negative eigenvalues of ${\mathrm{\u25b3}}^{2}+l\mathrm{\u25b3}$ and let ${X}^{1}$ be its orthogonal complement in X. Choose a Hilbertian basis ${e}_{n}$ ($n\ge 0$) for X and define
By the condition (H_{1}) and Sobolev inequalities, it is easy to see that the functional I belongs to ${C}^{1}(X,R)$ and maps bounded sets to bounded sets.
(2) We claim that I has a local linking at 0 with respect to $({X}^{1},{X}^{2})$. Decompose ${X}^{1}$ into $V+W$ when $V=ker({\mathrm{\u25b3}}^{2}+l\mathrm{\u25b3})$, $W={({X}^{2}+V)}^{\mathrm{\perp}}$. Also, set $u=v+w$, $u\in {X}^{1}$, $v\in V$, $w\in W$. By the equivalence of norm in the finitedimensional space, there exists $C>0$ such that
It follows from (H_{1}) and (H_{2}) that for any $\u03f5>0$, there exists ${C}_{\u03f5}$ such that
Hence, we obtain
where $m>0$, ${c}^{\ast}$ is a constant and hence, for $r>0$ small enough,
Let $u=v+w\in {X}^{1}$ be such that ${\parallel u\parallel}_{X}\le {r}_{1}=\frac{\delta}{2C}$ and let
From (3.2), we have
for all $\parallel u\parallel \le {r}_{1}$ and $x\in \mathrm{\Omega}$. On the one hand, one has $u(x)\le v(x)+w(x)\le {\parallel v\parallel}_{\mathrm{\infty}}+\frac{\delta}{2}\le \delta $ for all $x\in {\mathrm{\Omega}}_{1}$. Hence, from (H_{5}), we obtain
On the other hand, we have
for all $x\in {\mathrm{\Omega}}_{2}$. It follows from (3.3) that
for all $x\in {\mathrm{\Omega}}_{2}$ and all $u\in {X}_{1}$ with $\parallel u\parallel \le {r}_{1}$, which implies that
where ${C}_{3}$ is a constant. Hence, there exist positive constants ${C}^{\ast \ast}$, ${C}_{4}$ and ${C}_{5}$ such that
for all $u\in {X}^{1}$ with $\parallel u\parallel \le {r}_{1}$, which implies that
for $0<r$ small enough.
(3) We claim that I satisfies ${(C)}_{c}^{\ast}$. Consider a sequence $({u}_{{\alpha}_{n}})$ such that $({u}_{{\alpha}_{n}})$ is admissible and
and
Let ${w}_{{\alpha}_{n}}={\parallel {u}_{{\alpha}_{n}}\parallel}^{1}{u}_{{\alpha}_{n}}$. Up to a subsequence, we have
If $w=0$, we choose a sequence $\{{t}_{n}\}\subset [0,1]$ such that
For any $m>0$, let ${v}_{{\alpha}_{n}}=2\sqrt{m}{w}_{{\alpha}_{n}}$. By the Sobolev imbedded theory, we have
So, for n large enough, $2\sqrt{m}{\parallel {u}_{{\alpha}_{n}}\parallel}^{1}\in (0,1)$, and combining EhrlingNirenbergGagliardo inequality, we have
where ϵ is a small enough constant.
That is, $I({t}_{n}{u}_{{\alpha}_{n}})\to \mathrm{\infty}$. Now, $I(0)=0$, $I({u}_{{\alpha}_{n}})\to c$, we know that ${t}_{n}\in [0,1]$ and
Therefore, using (H_{3}), we have
This contradicts (3.5).
If $w\ne 0$, then the set $\u229d=\{x\in \mathrm{\Omega}:w(x)\ne 0\}$ has a positive Lebesgue measure. For $x\in \u229d$, we have ${u}_{{\alpha}_{n}}(x)\to \mathrm{\infty}$. Hence, by (H_{3}), we have
From (3.4), we obtain
By (3.8), the righthand side of (3.9) $\to +\mathrm{\infty}$. This is a contradiction.
In any case, we obtain a contradiction. Therefore, $\{{u}_{{\alpha}_{n}}\}$ is bounded.
Finally, we claim that for every $m\in N$,
By (H_{2}) and (H_{3}), there exists large enough M such that
So, for any $u\in {X}_{m}^{1}\oplus {X}^{2}$, we have
Hence, our claim holds. □
Proof of Theorem 1.2 We omit the proof which depends on Lemma 2.2 and is similar to the preceding one. □
Author’s contributions
The author read and approved the final manuscript.
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Acknowledgements
The author would like to thank the referees for valuable comments and suggestions in improving this paper. This work was supported by the National NSF (Grant No. 10671156) of China.
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Pei, R. Existence of solutions for a class of biharmonic equations with the Navier boundary value condition. Bound Value Probl 2012, 154 (2012) doi:10.1186/168727702012154
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Keywords
 biharmonic
 Navier boundary value problems
 local linking