Some multiplicity results for a superlinear elliptic problem in $\mathbb R^N$

Abstract

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.