Multiple complex-valued solutions for nonlinear magnetic Schrodinger equations

Abstract

We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations $$ L^{\hbar}_{A,V} u = f(|u|^2)u \quad \mbox{in } R^N $$ where $N\ge 3$, $L_{A,V}^{\hslash }$ is the Schrödinger operator with a magnetic field having source in a $C^1$ vector potential $A$ and a scalar continuous (electric) potential $V$ defined by \begin{equation} L^{\hbar}_{A,V}= -\hbar^2 Δ-\frac{2\hbar}{i} A \cdot \nabla + |A|^2- \frac{\hbar}{i}\operatorname{div}A + V(x). \end{equation} Here $f$ is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain $\Omega \subset R^N$ such that \[ m_0 \equiv \inf_{x \in Ω} V(x) < \inf_{x \in \partial Ω} V(x) \] and we set $K=\{x\in \Omega |V(x)=m_0\}$. For $\hslash >0$ small we prove the existence of at least $cuplenght(K)+1$ geometrically distinct, complex-valued solutions whose modula concentrate around $K$ as $\hslash \to 0$.

arXiv admin note: text overlap with arXiv:1305.3685 Manuscript to appear in Journal of fixed point theory and applications