Existence of normalized solutions to nonlinear Schrödinger equations with potential on lattice graphs

2025/07/06 by Guan, Weiqi
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.04204

Abstract

We study the existence of ground state normalized solution of the following Schrödinger equation: \begincases -Δu+V(x)u+λu=f(x,u), amp; x∈ℤ^d
\Vert u\Vert₂²=a \endcases where V(x) is trapping potential or well potential, f(x,u) satisfies Berestycki-Lions type condition and other suitable conditions. We show that there always exists a threshold α∈[0,∞) such that there do not exist ground state normalized solutions for a∈ (0,α), and there exists a ground state normalized solution for a∈(α,∞). Furthermore, we prove sufficient conditions for the positivity of α that α=0 if f(x,u) is mass-subcritical near 0, and α>0 if f(x,u) is mass-critical or mass-supercritical near 0.

Citations

• Normalized solutions for nonlinear Schrödinger equations on graphs
• Existence of ground state solutions to some Nonlinear Schrödinger equations on lattice graphs
• The existence of extremal functions for discrete Sobolev inequalities on lattice graphs
• Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case
• Normalized ground states for the NLS equation with combined nonlinearities
• Optimal Hardy inequalities for Schrödinger operators on graphs
• Kazdan-Warner equation on graph

Related

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• The ground state solutions of nonlinear Schrödinger equations with Hardy weights on lattice graphs
• Existence of normalized solutions to nonlinear Schrödinger equations on lattice graphs
• Normalized solutions for nonlinear Schrödinger equations on graphs
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