Concentration phenomena for nonlinear magnetic Schrödinger equations with critical growth
The paper is concerned with nonlinear magnetic Schrödinger equations involving critical growth of the form \[ \begin{cases} \left(\frac{\varepsilon}{i}\nabla - A(x) \right)^2u+V(x)u=f\left(|u|^2\right)u+|u|^{2^*-2}u \quad\text{in }\mathbb{R}^N,\\ u\in H^1\left(\mathbb{R}^N; \mathbb{C}\right), \end{cases} \] where \(\varepsilon>0\) is a parameter, \(N \geq 3\), \(2^*=\frac{2N}{N-2}\) is the critical Sobolev exponent, \(V: \mathbb{R}^N\to\mathbb{R}\) and \(A: \mathbb{R}^N\to\mathbb{R}^N\) are continuous potentials and \(f: \mathbb{R}\to\mathbb{R}\) is a subcritical nonlinear term. Under a local assumption on the potential \(V\) along with variational methods, the penalization technique and the Ljusternik-Schnirelmann theory, the authors prove the multiplicity and concentration of nontrivial solutions provided \(\varepsilon>0\) is sufficiently small.
multiplicity and concentration of solutions
scientific article; zbMATH DE number 7341458
scientific article