Electron. J. Diff. Equ., Vol. 2014 (2014), No. 181, pp. 1-12.

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

Daisuke Naimen

We consider the nonlinear Neumann boundary-value problem
 - \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.

Submitted May 30, 2014. Published August 27, 2014.
Math Subject Classifications: 35J20, 35J60, 35J65.
Key Words: Nonlinear Neumann; elliptic; variational method; critical point.

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Daisuke Naimen
Faculty of Science, Graduate School of Science
Osaka City University
3-3-138 Sugimoto Sumiyoshi-ku, Osaka-shi
Osaka 558-8585, Japan
email: d12sax0J51@ex.media.osaka-cu.ac.jp

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