Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential. (English) Zbl 1480.35131
Summary: We are concerned with the following nonlinear fractional Schrödinger equation:
\[ (-\Delta)^s u+V(x)u+\omega u = |u|^{p-2}u\quad \text{in } \mathbb{R}^N, \tag{P} \]
where \( s\in(0,1) \) and \( p\in\left(2+4s/N,2^*_s\right) \), that is, the mass supercritical and Sobolev subcritical. Under certain assumptions on the potential \( V:{\mathbb{R}}^N\rightarrow{\mathbb{R}} \), positive and vanishing at infinity including potentials with singularities (which is important for physical reasons), we prove that there exists at least one \( L^2 \)-normalized solution \( (u,\omega)\in H^s({\mathbb{R}}^N)\times{\mathbb{R}}^+ \) of equation (P). In order to overcome the lack of compactness, the proof is based on a new min-max argument and splitting lemma for nonlocal version.
MSC:
| 35J10 | Schrödinger operator, Schrödinger equation |
| 35R11 | Fractional partial differential equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
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