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

Abstract

In present paper, we study the following nonlinear Schrödinger equation with combined power nonlinearities $-\Delta u+V\left(x\right)u+\lambda u={\left|u\right|}^{2^{\ast }-2}u+\mu {\left|u\right|}^{q-2}u\text{in}{R}^N,N\ge 3$having prescribed mass ${\int }_{R^N}{u}^{2}dx=a^2,$where $\mu ,a>0$, $q\in (2,2^{\ast })$, $2^{\ast }=\frac{2N}{N-2}$ is the critical Sobolev exponent, $V$ is an external potential vanishing at infinity, and the parameter $\lambda \in R$ appears as a Lagrange multiplier. Under some mild assumptions on $V$, combining the Pohožaev manifold, constrained minimization arguments and some analytical skills, we get the existence of normalized solutions for the problem with $q\in (2,2^{\ast })$. At the same time, the exponential decay property of the solutions 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 $q\in [2+\frac{4}{N},2^{\ast })$.