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WEAK POTENTIAL CONDITIONS FOR SCHRÖDINGER EQUATIONS WITH CRITICAL NONLINEARITIES
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 100 / Issue 2 / April 2016
- Published online by Cambridge University Press:
- 28 October 2015, pp. 272-288
- Print publication:
- April 2016
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- Article
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In this paper, we prove the existence of nontrivial solutions to the following Schrödinger equation with critical Sobolev exponent:
under assumptions that (i)
$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}-{\rm\Delta}u+V(x)u=K(x)|u|^{2^{\ast }-2}u+f(x,u),\quad x\in \mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N})\end{array}\right.\end{eqnarray}$$
$V(x_{0})<0$ for some 
$x_{0}\in \mathbb{R}^{N}$ and (ii) there exists 
$b>0$ such that the set 
${\mathcal{V}}_{b}:=\{x\in \mathbb{R}^{N}:V(x)<b\}$ has finite measure, in addition to some common assumptions on 
$K$ and 
$f$, where 
$N\geq 3$, 
$2^{\ast }=2N/(N-2)$.
