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

Abstract

In this paper we consider nonlinear biharmonic equations with $p$-Laplacian ($p\ge2$) of the form $$ \left\{ \begin{array}{l} \Delta^2 u - \Delta_p u + V (x) u = f (x, u) \text{,} u \in H^2 (\mathbb{R}^N) \text{,} \end{array} \right. $$ 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.