Document Type : Research Paper
Authors
1 Department of Mathematics, Yachay Tech University, Hda. San Jos\'e s/n y Proyecto Yachay, Urcuqu\'i 100119, Ecuador.
2 Technische Universitat Wien, Wiedner Hauptstr. 8, 1040 Wien, Austria
3 Yachay Tech University, Hda. San Jose s/n y Proyecto Yachay, Urcuqui 100119, Ecuador.
4 Eötvös University, Pazmany Peter setany 1/C, 1117 Budapest, Hungary.
Abstract
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.
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Main Subjects
Berlin Heidelberg, 2011.
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