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

We study the Schr\"odinger equation   \left(\mathrm{Q}_{\varepsilon}\right): - \varepsilon^{۲(p-۱)} \Delta_p v + V(x)\, |v|^{p-۲} v - |v|^{q-۱}v = ۰, x \in \mathbb{R}^N, with v(x) \rightarrow ۰ as |x| \rightarrow+\infty, for the infinite case, as given by Byeon and Wang for a situation of critical frequency,  \displaystyle \{x\in \mathbb{R}^N \, / \: V(x) = \inf V = ۰\} \neq \emptyset. In the semiclassical limit, \varepsilon \rightarrow ۰, the corresponding limit problem is \left(\mathrm{P}\right): \Delta_p w+|w|^{q-۱} w=۰, x \in \Omega, with w(x)=۰, x \in \partial \Omega, where \Omega \subseteq \mathbb{R}^N is a smooth bounded strictly star-shaped region related to the potential V. We prove  that for \left(\mathrm{Q}_{\varepsilon}\right) there exists a non-trivial solution with any prescribed \mathrm{L}^{q+۱}-mass.Applying a Ljusternik-Schnirelman scheme, shows  that  \left(\mathrm{Q}_{\varepsilon}\right) and \left(\mathrm{P}\right) have infinitely many pairs of solutions. Fixed a topological level k \in \mathbb{N}, we show that a solution of \left(\mathrm{Q}_{\varepsilon}\right), v_{k, \varepsilon}, sub converges, in \mathrm{W}^{۱,p}(\mathbb{R}^N) and up to scaling, to a corresponding solution of \left(\mathrm{P}\right). We also prove that the energy of each solution, v_{k,\eps} converges to the corresponding energy of the limit problem  \left(\mathrm{P}\right) so that the critical values of the functionals associated, respectively, to  \left(\mathrm{Q}_{\varepsilon}\right) and \left(\mathrm{P}\right) are topologically equivalent.