Multiple solutions for a semilinear elliptic equation

Abstract:

Let $\Omega$ be a bounded, smooth domain in ${\mathbb {R}^N}$, $N \geqslant 1$. We consider the problem of finding nontrivial solutions to the elliptic boundary value problem \[ \begin {array}{*{20}{c}} {\Delta u + \lambda u = h(x)|u{|^{p - 1}}u\quad {\text {in }}\Omega } \\ {u = 0\quad {\text {on}}\partial \Omega } \\ \end {array} \] where $h \geqslant 0$, $h\not \equiv 0$ is Hölder continuous on $\overline \Omega$ and $p > 1$, $\lambda$ are constants. Let ${\Omega _0}$ denote the interior of the set where $h$ vanishes in $\Omega$. We assume $h > 0$ a.e. on $\Omega \backslash {\Omega _0}$ and consider the eigenvalues ${\lambda _i}(\Omega )$ and ${\lambda _i}({\Omega _0})$ of the Dirichlet problem in $\Omega$ and ${\Omega _0}$ respectively. We prove that no nontrivial solution of the equation exists if $\lambda$ satisfies, for some $k \geqslant 1$, \[ {\lambda _k}({\Omega _0}) \leqslant \lambda \leqslant {\lambda _{k + 1}}(\Omega )\] On the other hand, if, for some nonnegative integers $s$, $k$ with $s \geqslant k + 1$, $\lambda$ satisfies \[ {\lambda _s}(\Omega ) < \lambda < {\lambda _{k + 1}}({\Omega _0})\] then the equation above possesses at least $s - k$ pairs of nontrivial solutions. For the proof of these results we use a variational approach. In particular, the existence result takes advantage of the even character of the associated functional.