## Nontrivial solutions for some fourth order semilinear elliptic problems.(English)Zbl 0929.35053

From the introduction: Let us consider the following problem: $\begin{cases} \Delta^2u+ a^2\Delta u= b[(u+ 1)^+- 1]\quad & \text{in }\Omega,\\ \Delta u= 0,\quad u=0\quad & \text{on }\partial\Omega,\end{cases}$ where $$\Delta^2$$ is the biharmonic operator, $$u^+= \max\{u,0\}$$, $$\Omega\subset \mathbb{R}^N$$ is a smooth open bounded set and $$a$$, $$b$$ are constants.
We study the problem, when the nonlinearity $$(u+1)^+-1$$ is replaced by a more general function $$g$$, by using a variational approach. If $$\lambda_1< a^2$$ and $$b< \lambda_1$$ $$(\lambda_1- a^2)$$ the existence of two solutions is proved by the classical mountain pass theorem. In two different cases we get the existence of two solutions using a “variation of linking” theorem, by studying the geometry of the functional. The first case is when $$\lambda_{j+1}\leq a^2$$ and $$b$$ is close to $$\lambda_{j+1}$$ $$(\lambda_{j+1}- a^2)$$ with $$b<\lambda_{j+1}$$ $$(\lambda_{j+1}- a^2)$$ for some $$j\geq 1$$. The second case is when $$\lambda_1\leq a^2\leq \lambda_i$$ and $$b$$ is close to $$\lambda_i$$ $$(\lambda_i- a^2)$$ with $$b> \lambda_i$$ $$(\lambda_i- a^2)$$ for some $$i\geq 2$$. Finally, we give some uniqueness results.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35A15 Variational methods applied to PDEs 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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