Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians - Mao Anmin; Yan Changchang (Advances in Nonlinear Analysis, 2025)
We consider the following elliptic problem with mixed local and nonlocal operators: (−Δ)s1u+(−Δ)s2u+λu=f(u)inRN,∫RN∣u(x)∣2dx=c,u∈Hs2(RN),\left\{\begin{array}{l}{\left(-\Delta )}^{{s}_{1}}u+{\left(-\Delta )}^{{s}_{2}}u+\lambda u=f\left(u)\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u\left(x)| }^{2}{\rm{d}}x=c,\hspace{1.0em}\\ u\in {H}^{{s}_{2}}\left({{\mathbb{R}}}^{N}),\hspace{1.0em}\end{array}\right. where 0<s1<s2<10\lt {s}_{1}\lt {s}_{2}\lt 1, (−Δ)si{\left(-\Delta )}^{{s}_{i}} is fractional Laplacian. We get a sharp description of the existence and nonexistence of the global minimizer on the mass constraint, which is called energy ground state. By using of concentration-compactness principle, we show that there exists a constant c0{c}_{0} such that there exists an energy ground state if c>c0c\gt {c}_{0} and there exists no energy ground state if 0<c<c00\lt c\lt {c}_{0}. Some almost critical conditions which determine c0=0{c}_{0}=0 or c0>0{c}_{0}\gt 0 are established. In the case c=c0c={c}_{0}, under certain conditions, we establish the existence result of energy ground state. Finally, by using the method of mountain pass type characterization, we show that any energy ground state is action ground state of the corresponding action functional.