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

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]).

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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