Electronic Journal of Differential Equations,
Vol. 2014 (2014), No. 181, pp. 1-12.
Title: Existence of infinitely many solutions for nonlinear Neumann problems
with indefinite coefficients
Author: Daisuke Naimen (Osaka City Univ., Osaka, Japan)
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