On biharmonic equations with $p$-Laplacian and indefinite potentials or critical nonlinearity

Abstract

In this paper we consider nonlinear biharmonic equations with $p$-Laplacian ($p\ge 2$) of the form $$\{\begin{array}{l}{\Delta }^{2}u-{\Delta }_{p}u+V(x)u=f(x,u)\text{,}u\in H^2({\mathbb{R}}^N)\text{,}\end{array}$$ where the potential $V(x)$ may be indefinite. Using local linking and Morse theory, nontrivial solutions are obtained. In case the nonlinearity $f(x,\cdot )$ is odd, we obtain a sequence of large energy solutions. In the second part of the paper, for bounded positive potential, we get multiple solutions for the case that $$f(x,u)=\lambda g(x)|u{|}^{q-2}u+|u{|}^{m-2}u$$ with exponent $m$ critical or subcritical.