Non-radial normalized solutions for a nonlinear Schrodinger equation

This article concerns the existence of multiple non-radial positive solutions of the L2-constrained problem $$\displaylines{-\Delta{u}-Q(\varepsilon x)|u|^{p-2}u=\lambda{u},\quad \text{in }\mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=1,}$$ where \(Q(x)\) is a radially symmetric function, ε>0 is a small parameter, \(N\geq 2\), and \(p \in (2, 2+4/N)\) is assumed to be mass sub-critical. We are interested in the symmetry breaking of the normalized solutions and we prove the existence of multiple non-radial positive solutions as local minimizers of the energy functional.