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

We study the Schrödinger equation (Qε): −ε2(p−1)∆pv+ V (x) |v|p−2v − |v|q−1v = 0, x ∈ RN , with v(x) → 0 as |x| → +∞, for the infinite case, as given by Byeon and Wang for a situation of critical frequency, {x ∈ RN / 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 ⊆ RN is a smooth bounded strictly star-shaped region related to the po-tential V . We prove that for (Qε) there exists a non-trivial solution with any prescribed Lq+1-mass. Applying a Ljusternik-Schnirelman scheme, shows that (Qε) and (P) have infinitely many pairs of so-lutions. Fixed a topological level k ∈ N, we show that a solution of (Qε), vk,ε, sub converges, in W1,p(RN ) and up to scaling, to a corresponding solution of (P). We also prove that the energy of each solution, vk,ε converges to the corresponding energy of the limit problem (P) so that the critical values of the functionals asso-ciated, respectively, to (Qε) and (P) are topologically equivalent.