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

We study the Schr\"odinger equation   $\left(\mathrm{Q}_{\varepsilon}\right)$: $- \varepsilon^{2(p-1)} \Delta_p v + V(x)\, |v|^{p-2} v - |v|^{q-1}v = 0$, $x \in \mathbb{R}^N$, with $v(x) \rightarrow 0$ as $|x| \rightarrow+\infty$, for the infinite case, as given by Byeon and Wang for a situation of critical frequency,  $\displaystyle \{x\in \mathbb{R}^N \, / \: V(x) = \inf V = 0\} \neq \emptyset$. In the semiclassical limit, $\varepsilon \rightarrow 0$, the corresponding limit problem is $\left(\mathrm{P}\right)$: $\Delta_p w+|w|^{q-1} w=0$, $x \in \Omega$, with $w(x)=0, x \in \partial \Omega$, 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 $\mathrm{L}^{q+1}$-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 $\mathrm{W}^{1,p}(\mathbb{R}^N)$ 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.