Multi-bump solutions for singularly perturbed Schrödinger equations in $\mathbb{R}^2$ with general nonlinearities

Abstract

We are concerned with the following equation: $$ -\varepsilon^2\Delta u+V(x)u=f(u),\quad u(x)> 0\quad \mbox{in } \mathbb{R}^2. $$ By a variational approach, we construct a solution $u_\varepsilon$ which concentrates, as $\varepsilon \to 0$, around arbitrarily given isolated local minima of the confining potential $V$: here the nonlinearity $f$ has a quite general Moser's critical growth, as in particular we do not require the {\it monotonicity} of $f(s)/s$ nor the {\it Ambrosetti-Rabinowitz} condition.