Existence of Three Solutions for a Doubly Eigenvalue Fourth-order Boundary Value Problem

Abstract

In this paper we consider the existence of at least three solutions for the Dirichlet problem $$\left\{\begin{array}{ll} u^{i\upsilon} + \alpha u'' + \beta u = \lambda f(x,u) + \mu g(x,u), \hspace{1cm} x \in (0,1) \\ u(0) = u(1) = 0, \\ u''(0) = u''(1) = 0 \end{array}\right.$$ where $\alpha,\ \beta$ are real constants, $f,g: [0,1] \times R \to R$ are $L^{2}$-Carathéodory functions and $\lambda,\mu \gt 0$. The approach is based on variational methods and critical points.