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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Multiple solutions for a semilinear elliptic equation

Authors: Manuel A. del Pino and Patricio L. Felmer
Journal: Trans. Amer. Math. Soc. 347 (1995), 4839-4853
MSC: Primary 35J65; Secondary 58E05
MathSciNet review: 1303117
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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{displaymath}\begin{array}{*{20}{c}} {\Delta u + \lambda u = h(x)\vert u{\... ...a } \\ {u = 0\quad {\text{on}}\partial \Omega } \\ \end{array} \end{displaymath}

where $ h \geqslant 0$, $ h\not\equiv0$ 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$,

$\displaystyle {\lambda _k}({\Omega _0}) \leqslant \lambda \leqslant {\lambda _{k + 1}}(\Omega )$

On the other hand, if, for some nonnegative integers $ s$, $ k$ with $ s \geqslant k + 1$, $ \lambda $ satisfies

$\displaystyle {\lambda _s}(\Omega ) < \lambda < {\lambda _{k + 1}}({\Omega _0})$

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.

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Keywords: Variational method, multiplicity, uniqueness, blow up of solutions
Article copyright: © Copyright 1995 American Mathematical Society