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The eigenvalue problems of biharmonic equations. (Chinese. English summary) Zbl 1199.35113

Summary: In this paper, we consider the existence of the solution for the biharmonic eigenvalue problem under Navier boundary condition \[ \begin{cases} \Delta^2u=\lambda a(x)u+f(x, u),&x\in \Omega,\\u=\Delta u=0,&x\in \partial \Omega , \end{cases} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N(N\geq 5),\Delta^2\) is the biharmonic operator, and the weight function \(a(x)\in L^r(\Omega)\) \((r\geq \frac N4)\) with \(a(x)>0\) a.e. in \(\Omega\). By variational method, we obtain the second eigenvalue of this problem when \(f(x, u)=0\) and study the structure of it, and discuss the existence of the nonzero solutions under resonance and nonresonance conditions.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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