Normalized solutions for Schrödinger equations with potential and general nonlinearities involving critical case on large convex domains - Jun Wang; Zhaoyang Yin (Electronic Journal of Qualitative Theory of Differential Equations)
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.