Multiplicity and concentration of positive solutions for a class of quasilinear problems through Orlicz-Sobolev space (Q2836107)

The authors consider quasilinear elliptic problems of the form NEWLINE\[NEWLINE \begin{cases} &-\Delta_\Phi u+V(\varepsilon x)\phi(|u|)u = f(u) \text{ in }\mathbb{R}^N\\ &u \in W^{1,\Phi}(\mathbb{R}^N) \end{cases} NEWLINE\]NEWLINE with \(N\geq 2\) and \(\varepsilon>0\) small. Here \(\phi:\mathbb{R}^+\to\mathbb{R}^+\) is a monotone function, \(\Phi(t)=\int^{|t|}_{0}\phi(s) s\, ds\), \(V\in C^0(\mathbb{R}^N)\) and \(f\in C^1(\mathbb{R})\). It is assumed that \(\liminf_{|x|\to\infty} V(x)>V_0:=\mathrm{inf}_{x\in\mathbb{R}^N} V(x)\). The main result states the existence of multiple positive solutions depending on the topology of the set \(V^{-1}(V_0)\), under various conditions on \(\phi\) and \(f\). The equation \(-\Delta u+V(\epsilon x)u=|u|^{p-2}u\) with \(2<p<2^*\), treated in [\textit{S. Cingolani} and \textit{M. Lazzo}, Topol. Methods Nonlinear Anal. 10, No. 1, 1--13 (1997; Zbl 0903.35018)], is a special case as well as equations involving the \(p\)-Laplacian and more general operators, dealt with in [\textit{C. O. Alves} and \textit{G. M. Figueiredo}, Differ. Integral Equ. 19, No. 2, 143--162 (2006; Zbl 1212.35107); Adv. Nonlinear Stud. 11, No. 2, 265--294 (2011; Zbl 1231.35006)].