Normalized solutions for Schrödinger equations with potential and general nonlinearities involving critical case on large convex domains

Abstract

In this paper, we study the following Schrödinger equations with potentials and general nonlinearities \begin{equation*} \begin{cases} -\Delta u+V(x)u+\lambda u=|u|^{q-2}u+\beta f(u), \\ \int |u|^2dx=\Theta, \end{cases} \end{equation*} both on $\mathbb{R}^N$ as well as on domains $\Omega_r$ where $\Omega_r \subset \mathbb{R}^N$ is an open bounded convex domain and $r>0$ is large. The exponent satisfies $2+\frac{4}{N}\leq q\leq2^*=\frac{2 N}{N-2}$ and $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies $L^2$-subcritical or $L^2$-critical growth. This paper generalizes the conclusion of Bartsch et al. in [4]. Moreover, we consider the Sobolev critical case and $L^2$-critical case of the above problem.

Keywords

• schrödinger equations
• normalized solutions
• variational methods
• mixed nonlinearity