On multi-bump solutions of nonlinear Schrödinger equation with electromagnetic fields and critical nonlinearity in $$\mathbb {R}^N$$ R N

Abstract

In this paper, we study a class of nonlinear Schrodinger equations with electromagnetic fields and critical nonlinearity in \(\mathbb {R}^N: -\Delta _Au + (\lambda V(x) + Z(x))u = \beta f(|u|^2)u + |u|^{2^*-2}u\), where f is a continuous function, \(V, Z: \mathbb {R}^N \rightarrow \mathbb {R}\) are continuous functions verifying suitable hypotheses. We show that if the zero set of V(x) has several isolated connected components \(\Omega _1,\ldots ,\Omega _k\) such that the interior of \(\Omega _i\) is not empty and \(\partial \Omega _i\) is smooth, then for \(\lambda > 0\) large there exists, for any non-empty subset \(\Gamma \subset \{1,\ldots ,k\}\), a bump solution trapped in a neighborhood of \(\cup _{j\in \Gamma }\Omega _j\).