Normalized solutions for the nonlinear Schrodinger equation with potential and combined nonlinearities

Abstract

In present paper, we study the following nonlinear Schr\"{o}dinger equation with combined power nonlinearities \begin{align*} - \Delta u+V(x)u+\lambda u=|u|^{2^*-2}u+\mu |u|^{q-2}u \quad \quad \text{in} \ \mathbb{ R}^N, \ N\geq 3 \end{align*} having prescribed mass \begin{align*} \int_{ \mathbb{ R}^N}u^2dx=a^2, \end{align*} where $\mu, a>0$, $q\in(2, 2^*)$, $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent, $V$ is an external potential vanishing at infinity, and the parameter $\lambda\in \mathbb{R}$ appears as a Lagrange multiplier. Under some mild assumptions on $V$, for the $L^2$-subcritical perturbation $q\in(2, 2+\frac{4}{N})$, we prove that there exists $a_0>0$ such that the normalized solution with negative energy to the above problem with $\mu>0$ can be obtained for $a\in (0, a_0)$; for the $L^2$-critical perturbation $q=2+\frac{4}{N}$, by limiting the range of $\mu$, the positive ground state normalized solution to the above problem for any $a>0$ is also found with the aid of the Poho\v{z}aev constraint; moreover, for the $L^2$-supercritical perturbation $q\in( 2+\frac{4}{N}, 2^*)$, we get a positive ground state normalized solution for the above problem with $a>0$ and $\mu>0$ by using the Poho\v{z}aev constraint. At the same time, the exponential decay property of the positive normalized solution is established, which is important for the instability analysis of the standing waves. Furthermore, we give a description of the ground state set and obtain the strong instability of the standing waves for $q\in[2+\frac{4}{N}, 2^*)$. This paper can be regarded as a generalization of Soave [J. Funct. Anal. (2020)] in a sense.