Biharmonic equations with asymptotically linear nonlinearities. (English) Zbl 1150.35037

Summary: This article considers the equation \(\Delta^2u=f (x, u)\) with boundary conditions either \(u|_{\partial\Omega}=\frac{\partial u}{\partial n} |_{\partial\Omega}=0\) or \(u|_{\partial\Omega}=\Delta u|_{\partial\Omega}=0\), where \(f (x, t)\) is asymptotically linear with respect to \(t\) at infinity, and \(\Omega\) is a smooth bounded domain in \(\mathbb R^N\), \(N>4\). By a variant version of Mountain Pass Theorem, it is proved that the above problems have a nontrivial solution under suitable assumptions on \(f (x, t)\).


35J60 Nonlinear elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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