Multiple solutions for generalized biharmonic equations with singular potential and two parameters

Abstract

We investigate a more general nonlinear biharmonic equation

$${\mathrm{\Delta }}^{2}u-\beta {\mathrm{\Delta }}_{p}u+V_{\lambda }\left(x\right)u=f\left(x,u\right)\text{ in }{ℝ}^N,$$

where ${\mathrm{\Delta }}^2:=\mathrm{\Delta }\left(\mathrm{\Delta }\right)$ is the biharmonic operator, $N\ge 1$, $\lambda >0$ and $\beta \in ℝ$ are parameters, ${\mathrm{\Delta }}_{p}u=div\left(|\nabla u{|}^{p-2}\nabla u\right)$ with $p\ge 2$. Differently from previous works on biharmonic problems, we replace Laplacian with $p$-Laplacian, and suppose that $V\left(x\right)=\lambda a\left(x\right)-b\left(x\right)$ with $\lambda >0$ and $b\left(x\right)$ can be singular at the origin, in particular we allow $\beta$ to be a real number. Under suitable conditions on $V_{\lambda }\left(x\right)$ and $f\left(x,u\right)$, the multiplicity of solutions is obtained for $\lambda >0$ sufficiently large. Our analysis is based on variational methods as well as the Gagliardo–Nirenberg inequality.