Multiple Solutions for Generalized Biharmonic Equations with Two Singular Terms

Abstract

In this article, we investigate more general nonlinear biharmonic equation

$$\Delta ^2u+V_\lambda (x)u=\mu f(x)u^{-\gamma }+g(x)u^{p-1}~~~\textrm{in}~\mathbb {R}^N,$$

where $\Delta ^2:=\Delta (\Delta )$ is the biharmonic operator, $N\ge 1$, $\lambda >0$ is a parameter, $0<\gamma <1$. Different from previous works on biharmonic problems, we suppose that $V(x)=\lambda a(x)-b(x)$ with $\lambda >0$ and b(x) could be singular at the origin. Under suitable conditions on $V_\lambda (x)$, f(x) and g(x), the multiplicity of solutions is obtained for $\lambda >0$ sufficiently large and some new estimates will be established. Our analysis is based on the Nehari manifold as well as the fibering map.