Normalized solutions of NLS equations with mixed nonlocal nonlinearities

Abstract

Abstract We study the existence and nonexistence of normalized solutions for the nonlinear Schrödinger equation with mixed nonlocal nonlinearities: − Δ u = λ u + μ ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u + ( I α ∗ ∣ u ∣ q ) ∣ u ∣ q − 2 u in R N , ∫ R N ∣ u ∣ 2 d x = c , \left\{\begin{array}{ll}-\Delta u=\lambda u+\mu \left({I}_{\alpha }\ast {| u| }^{p}){| u| }^{p-2}u+\left({I}_{\alpha }\ast {| u| }^{q}){| u| }^{q-2}u& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x=c,& \end{array}\right. where N ≥ 1 , N + α N ≤ p < q ≤ N + α + 2 N N\ge 1,\frac{N+\alpha }{N}\le p\lt q\le \frac{N+\alpha +2}{N} , the parameter μ ∈ R \mu \in {\mathbb{R}} and λ \lambda is a Lagrange multiplier. Furthermore, we prove the relationship between minimizers and ground state solutions under the Nehari manifold, which seems to be the first result in the nonlocal context.