The effect of concentrating potentials in some singularly perturbed problems

Abstract.

The equation $-\epsilon^2\Delta u+a_{\epsilon}(x)u= f(u)$ with boundary Dirichlet zero data is considered in a bounded domain $\Omega \subset \mathbb{R}^N$ . Under the assumption that $a_{\epsilon}(x) \geq a_{\infty} \gt 0$ concentrates, as $\epsilon\to 0$, round a manifold $\mathcal M\in \Omega$ and that f is a superlinear function, satisfying suitable growth assumptions, the existence of multiple distinct positive solutions is proved.