Normalized solutions of NLS equations with mixed nonlocal nonlinearities

Abstract

We study the existence and nonexistence of normalized solutions for the nonlinear Schrödinger equation with mixed nonlocal nonlinearities: −Δu=λu+μ(Iα∗∣u∣p)∣u∣p−2u+(Iα∗∣u∣q)∣u∣q−2uinRN,∫RN∣u∣2dx=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+αN≤p<q≤N+α+2NN\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.

Keywords

• normalized solutions
• mixed nonlocal nonlinearities
• minimizers
• ground state solutions
• 35j20
• 35j60