Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities
This paper is concerned with equations of the form: \[ -\varepsilon ^2\Delta u + V(x) u =f(u),\quad u\in H^1(\mathbb{R}^n), \] where \(f\in C^1(\mathbb{R},\mathbb{R}),\) and \(V(x)\) is locally Hölder continuous and bounded below away from \(0.\) Positive solutions concentrating, as \(\varepsilon \to 0,\) in a given set of local minima of \(V(x),\) are studied. The aim of the authors is to extend the result of \textit{M. del Pino} and \textit{P. Felmer} [Calc. Var. Partial Differ. Equ. 4, No. 2, 121--137 (1996; Zbl 0844.35032)] to a wider class of nonlinearities. They show, by means of new techniques developed in their recent works, that the del Pino-Felmer's result is still valid if the following conditions are satisfied: i) There exists \(a \in (0,+\infty]\) such that \(\frac{f(\xi)}{\xi} \to a\) as \(\xi \to \infty,\) ii) There exists \(D\geq 1\) such that \(\widehat F (s)\leq D \widehat F (t)\) for all \(0 \leq s \leq t,\) where \(\widehat F(\xi)= \frac{1}{2} f(\xi) \xi -F(\xi),\) with \(F(\xi)= \int _0^{\xi} f(\tau) \,d\tau.\) Concrete examples of nonlinearities \(f\) which satisfy these conditions are presented.
scientific article; zbMATH DE number 2123514