Branch continuation inside the essential spectrum for the nonlinear Schrödinger equation

Abstract

We consider the nonlinear stationary Schrödinger equation \begin{equation*} -Δu -λu= Q(x)|u|^{p-2}u, \qquad \text{in }\mathbb{R}^N \end{equation*} in the case where $N\ge 3$, $p$ is a superlinear, subcritical exponent, $Q$ is a bounded, nonnegative and nontrivial weight function with compact support in ${\mathbb{R}}^N$ and $\lambda \in \mathbb{R}$ is a parameter. Under further restrictions either on the exponent $p$ or on the shape of $Q$, we establish the existence of a continuous branch $\mathcal{C}$ of nontrivial solutions to this equation which intersects $\{\lambda \}\times L^s({\mathbb{R}}^N)$ for every $\lambda \in (-\infty ,{\lambda }_Q)$ and $s>\frac{2N}{N-1}$. Here ${\lambda }_Q>0$ is an explicit positive constant which only depends on $N$ and $\text{diam}(\text{supp }Q)$. In particular, the set of values $\lambda$ along the branch enters the essential spectrum of the operator $-\Delta$.

Revised version; to appear in 'Journal of Fixed Point Theory and Applications', Special issue in honour of Paul Rabinowitz