Concentration phenomena for nonlinear magnetic Schrödinger equations with critical growth

Abstract

In this paper, we are concerned with the following nonlinear magnetic Schrödinger equation with critical growth: $\left\{ {\matrix{ {{{\left( {{\varepsilon \over i}\nabla - A\left( x \right)} \right)}^2}u + V\left( x \right)u = f\left( {{{\left| u \right|}^2}} \right)u + {{\left| u \right|}^{{2^*} - 2}}u\,\,\,\,\,\,{\rm{in}}\,{\mathbb{R}^N},} \hfill \cr {u \in {H^1}\left( {{\mathbb{R}^N},\mathbb{C}} \right),} \hfill \cr } } \right.$ where ∊ > 0 is a parameter, N ≥ 3 and ${2^*} = {{2N} \over {N - 2}}$ is the Sobolev critical exponent, V: ℝ^N → ℝ and A: ℝ^N → ℝ^N are continuous potentials, f: ℝ → ℝ is a subcritical nonlinear term. Under a local assumption on the potential V, by the variational methods, the penalization technique and the Ljusternic—Schnirelmann theory, we prove the multiplicity and concentration of nontrivial solutions of the above problem for ε small. For the problem, the function f is only continuous, which allows to consider larger classes of nonlinearities in the reaction.