Normalized solutions for the nonlinear Schrödinger equation with potential and Sobolev critical nonlinearity

Abstract

This paper investigates the existence of normalized solutions to the following Schrödinger equation $$ \begin{cases} -\Delta u + [V(x) + \lambda] u = \mu |u|^{q-2}u + |u|^{2^*-2}u, & x\in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}u^2\mathrm{d}x = c, \end{cases} $$ where $N\ge 3,$ $2 < q < 2^*=\frac{2N}{N-2}$, $c > 0$ is a prescribed mass, and $\lambda\in\mathbb{R}$ is a Lagrange multiplier dependent on the solution $u$. Existing studies have confirmed the existence of local minimizers for the equation when $2 < q < 2+\frac{4}{N}$, but the existence of a second Mountain-Pass type solution remains unresolved. By improving variational methods and introducing a refined constraint set $A_{s_0}$, we prove the existence of a positive local minimizer $\tilde{u}_c$ for $c\in(0,c_0)$ under a mild potential assumption, with $\Phi(\tilde{u}_c) < 0$. Further, under additional assumptions on the radiality and decay of $V(x)$, leveraging the compactness of the radial Sobolev space, we construct a non-standard Mountain-Pass geometry and accurately estimate the minimax energy level, establishing the existence of a second radial Mountain-Pass type solution $u_c$ with $0 < \Phi(u_c) < m(c)+\frac{1}{N}\mathcal{S}^{\frac{N}{2}}$. Our results fill the gap in the multi-solution research of the equation for $2 < q < 2+\frac{4}{N}$, and the proposed variational framework provides a new approach for the study of constrained elliptic equations.