Existence of many sign-changing nonradial solutions for semilinear elliptic problems on thin annuli

Alfonso Castro, Marcel B. Finan

DOI: http://dx.doi.org/10.12775/TMNA.1999.014

Abstract


We study the existence of many nonradial sign-changing solutions of
a superlinear Dirichlet boundary value problem in an annulus in $\mathbb R^N$.
We use Nehari-type variational method and group invariance
techniques to prove that the critical points of an action functional on
some spaces of invariant functions in $H_{0}^{1,2}(\Omega_{\varepsilon})$,
where $\Omega_{\varepsilon}$ is an annulus in $\mathbb R^N$ of width
$\varepsilon$, are weak solutions (which in our case are also classical
solutions) to our problem. Our result generalizes an earlier result of
Castro et al. (See [A. Castro, J. Cossio and J. M. Neuberger,
< i> A minmax principle, index of the critical point, and existence of
sign-changing solutions to elliptic boundary value problems< /i> ,
Electron. J. Differential Equations < b> 2< /b> (1998),
1–18]).

Keywords


Dirichlet's problem; superlinear; subcritical; sign-changing nonradial solution; group action; symmetric criticality lemma; variational method

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