Some multiplicity results for a superlinear elliptic problem in $\mathbb R^N$
Keywords
Nonlinear Schrödinger equation, weighted Sobolev spaces, critical point theory, fibering methodAbstract
In this paper we shall study the semilinear elliptic problem $$ \cases -\Delta u +\sigma(x)u= |u|^{p-2}u + f(x) & \text{in }\mathbb R^N,\\ u\rightarrow 0\quad\text{as } |x| \rightarrow\infty, \endcases $$ where $\sigma(x) \rightarrow\infty$ as $| x| \rightarrow\infty$, $p> 2$ and $f\in L^{2}(\mathbb R^{N})$. Thanks to a compact embedding of a suitable weigthed Sobolev space in $L^{2}(\mathbb R^{N})$, a direct use of the Symmetric Mountain Pass Theorem (if $f=0$) and of the fibering method (if $f\neq0$) allows to extend some multiplicity results, already known in the case of bounded domains.Downloads
Published
2003-03-01
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SALVATORE, Addolorata. Some multiplicity results for a superlinear elliptic problem in $\mathbb R^N$. Topological Methods in Nonlinear Analysis. Online. 1 March 2003. Vol. 21, no. 1, pp. 29 - 39. [Accessed 29 March 2024].
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