Consider the following Navier boundary value problem:

$\{\begin{array}{cc}{\mathrm{\u25b3}}^{2}u(x)+l\mathrm{\u25b3}u=f(x,u),\hfill & \text{in}\mathrm{\Omega};\hfill \\ u=\mathrm{\u25b3}u=0\hfill & \text{on}\partial \mathrm{\Omega},\hfill \end{array}$

(1.1)

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

$|f(x,t)|\le {C}_{1}+{C}_{2}{|t|}^{s-1},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in \mathrm{\Omega},\mathrm{\forall}t\in R,s\in (2,{p}^{\ast})(N>4),$

where ${p}^{\ast}=\frac{2N}{N-4}$;

(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

$s\in [0,1]$,

$\theta (f(x,t)t-2F(x,t))\ge (sf(x,st)t-2F(x,st)),$

where $F(x,t)={\int}_{0}^{t}f(x,s)\phantom{\rule{0.2em}{0ex}}ds$;

(H

_{5}) For some

$\delta >0$, either

$F(x,t)\ge 0,\phantom{\rule{1em}{0ex}}\text{for}|t|\le \delta ,x\in \mathrm{\Omega},$

or

$F(x,t)\le 0,\phantom{\rule{1em}{0ex}}\text{for}|t|\le \delta ,x\in \mathrm{\Omega}.$

This fourth-order 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 sign-changing 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

$X={X}^{1}\oplus {X}^{2}.$

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 non-degenerate 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

$I(u)\ge c>0,\phantom{\rule{1em}{0ex}}u\in {X}^{1},\parallel u\parallel =r,dim{X}^{2}<\mathrm{\infty}.$

Under those assumptions, the existence of nontrivial critical points was proved for functionals which are

- (a)

- (b)

- (c)
asymptotically quadratic [9].

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 second-order 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* $\delta >0$,

$F(x,u)\ge 0,\phantom{\rule{1em}{0ex}}\mathit{\text{for}}|u|>\delta ,x\in \mathrm{\Omega};$

(A

_{3})

${\lambda}_{j}<l={\lambda}_{j+1}$,

$j\ne k$,

*for some* $\delta >0$,

$F(x,u)\ge 0,\phantom{\rule{1em}{0ex}}\mathit{\text{for}}|u|\le \delta ,x\in \mathrm{\Omega}.$

*Then the problem* (1.1) *has at least one nontrivial solution*.