Existence of infinitely many solutions for nonlinear Neumann problems with indefinite coefficients

Abstract

We consider the nonlinear Neumann boundary-value problem $$\displaylines{ - \Delta u +u =a(x)| u | ^{p-2}u\quad \text{in }\Omega,\cr \frac{\partial u}{\partial \nu}=\lambda b(x)|u|^{q-2}u\quad \text{on } \partial\Omega, }$$ where $N\ge 3$ and $\Omega \subset \mathbb{R}^N$ is a bounded domain with smooth boundary. We suppose a and b are possibly sign-changing functions in $\overline{\Omega}$ and on $\partial \Omega$ respectively. Under some additional assumptions on a and b, we show that there are infinitely many solutions for sufficiently small $\lambda>0$ if $1<q<2<p\le2^*=2N/(N-2)$. When $p=2^*$, we use the concentration compactness argument to ensure the PS condition for the associated functional. We also consider a general problem including the supercritical case and obtain the existence of infinitely many solutions.

Keywords

• Nonlinear Neumann
• elliptic
• variational method
• critical point