Existence and Asymptotic of Solutions for a p-Laplace Schrödinger Equation with Critical Frequency

Abstract:

We study the Schrödinger equation (Qε): −ε^2(p−¹⁾∆pv+ V (x) |v|^p−²v − |v|^q−¹v = 0, x ∈ R^N, with v(x) → 0 as |x| → +∞, for the infinite case, as given by Byeon and Wang for a situation of critical frequency, {x ∈ R^N / V (x) = inf V = 0} ̸= ∅. In the semiclassical limit, ε → 0, the corresponding limit problem is (P): ∆pw + |w|^q−¹w = 0, x ∈ Ω, with w(x) = 0, x ∈ ∂Ω, where Ω ⊆ R^N is a smooth bounded strictly star-shaped region related to the potential V . We prove that for (Qε) there exists a non-trivial solution with any prescribed L^q⁺¹-mass. Applying a Ljusternik-Schnirelman scheme, shows that (Qε) and (P) have infinitely many pairs of solutions. Fixed a topological level k ∈ N, we show that a solution of (Qε), v<inf>k,ε</inf>, sub converges, in W^1,p(R^N) and up to scaling, to a corresponding solution of (P). We also prove that the energy of each solution, v<inf>k,ε</inf> converges to the corresponding energy of the limit problem (P) so that the critical values of the functionals associated, respectively, to (Qε) and (P) are topologically equivalent.

Año de publicación:

2023

Keywords:

• Asymptotic properties
• ground state
• Multiplicty
• p-Laplace operator
• Quasilinear Schrödinger equation

Fuente:

scopus