Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions (Q1969980)

The existence of positive solutions of the problem \[ -h^2\Delta u+V(x)u=K(x)u^{p-1}+Q(x) u^{q-1}, \quad x\in \mathbb{R}^n \tag \(P_h\) \] satisfying the condition \(\underset{|x|\to\infty}\lim u(x)=0\) is studied. Under the conditions \(2<q<p<\frac{2N}{N-2}\), \(\inf \{V(x):x\in \mathbb{R}^n\}>0,\) \(K(x)>0\) for \(x\in \mathbb{R}^n\) and some other conditions it is proved the existence of \(\text{cat}_{M_\delta}(M)\) solutions of the problem \((P_h)\) for enough small \(h.\) Here cat is the Ljusternik-Schnirelman category, \(M\) is the set of global minimum points of a suitable ground energy function, \(M_\delta\) is a \(\delta\)-neighbourhood of \(M\).