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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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