Existence of positive solutions for an equation involving supercritical exponent in \(\mathbb{R}^N\) (Q1582446)

The existence of classical positive solutions for the problem (P) \(-\varepsilon^2 \Delta u + V(z) u = u^p\) in \({\mathbb{R}}^N\) (\(N\geq 3\) and \(p>1\)) is considered, where \(V\colon {\mathbb{R}}^N \to {\mathbb{R}}\) is a nonnegative radial function of class \({\mathcal C}^1\) of the following form: There exist \(0<R_1 <r_1<r_2 <R_2\) and \(\alpha >0\) such that \(V(z) =0\) in \(\{z\in {\mathbb{R}}^N;\;r_1<z<r_2 \}\) and \(V(z) \geq \alpha \) outside of \(\{z\in {\mathbb{R}}^N;\;R_1<z<R_2 \}\). The difficulty is, that no condition is imposed on \(p\) (with respect to the critical Sobolev exponent). This causes problems with the Palais-Smale condition. The author overrides this inconvenience by seeking solutions of a perturbed functional on the ``good'' space of radial functions \(H^{1,2}_{\text{rad}} ({\mathbb{R}}^N)\) restauring this way the lost compactness. The tool used is the mountain pass theorem of \textit{A. Ambrosetti} and \textit{P. H. Rabinowitz} [J. Funct. Anal. 14, 349-381 (1973; Zbl 0273.49063)].