Spike solutions for nonlinear Schrödinger equations in 2D with vanishing potentials
| Autor: | João Marcos do Ó, Elisandra Gloss, Federica Sani |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Physics
Applied Mathematics 010102 general mathematics Mathematics::Analysis of PDEs 01 natural sciences Exponential function Schrödinger equation Sobolev space Combinatorics Nonlinear system symbols.namesake 0103 physical sciences Mountain pass theorem symbols 010307 mathematical physics 0101 mathematics Exponential growth Schrödinger equations Semi-classical limit Vanishing potentials Weighted Trudinger–Moser inequality |
| Popis: | We consider $$\varepsilon $$-perturbed nonlinear Schrodinger equations of the form $$\begin{aligned} - \varepsilon ^2\Delta u + V(x)u = Q(x)f(u) \quad \text {in } \mathbb {R}^2, \end{aligned}$$where V and Q behave like $$(1+|x|)^{-\alpha }$$ with $$\alpha \in (0,2)$$ and $$(1+|x|)^{-\beta }$$ with $$\beta \in (\alpha , + \infty )$$, respectively. When f has subcritical exponential growth—by means of a weighted Trudinger–Moser-type inequality and the mountain pass theorem in weighted Sobolev spaces—we prove the existence of nontrivial mountain pass solutions, for any $$\varepsilon >0$$, and in the semi-classical limit, these solutions concentrate at a global minimum point of $$\mathcal A=V/Q$$. Our existence result holds also when f has critical growth, for any $$\varepsilon >0$$. |
| Databáze: | OpenAIRE |
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