Connected components of positive solutions of biharmonic equations with the clamped plate conditions in two dimensions

Abstract

This article concerns the clamped plate equation $$\displaylines{ \Delta^2 u=\lambda a(x)f(u), \quad \text{in } \Omega,\cr u=\frac {\partial u}{\partial \nu}= 0 \quad \text{on } \partial \Omega, }$$ where \(\Omega\) is a bounded domain in \(\mathbb{R}^2\) of class \(C^{4, \alpha}\), \(a\in C(\bar \Omega, (0, \infty))\), \(f: [0, \infty)\to [0,\infty)\) is a locally H\"older continuous function with exponent \(\alpha\), and \(\lambda\) is a positive parameter. We show the existence of S-shaped connected component of positive solutions under suitable conditions on the nonlinearity. Our approach is based on bifurcation techniques.
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