We consider ε-perturbed nonlinear Schrödinger equations of the form -ε2Δu+V(x)u=Q(x)f(u)inR2,where V and Q behave like (1 + | x|) -α with α∈ (0 , 2) and (1 + | x|) -β with β∈ (α, + ∞) , 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 ε> 0 , and in the semi-classical limit, these solutions concentrate at a global minimum point of A= V/ Q. Our existence result holds also when f has critical growth, for any ε> 0.

Spike solutions for nonlinear Schrödinger equations in 2D with vanishing potentials / J.M. do O, E. Gloss, F. Sani. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - (2019). [Epub ahead of print] [10.1007/s10231-019-00856-7]

Spike solutions for nonlinear Schrödinger equations in 2D with vanishing potentials

F. Sani
Ultimo
2019

Abstract

We consider ε-perturbed nonlinear Schrödinger equations of the form -ε2Δu+V(x)u=Q(x)f(u)inR2,where V and Q behave like (1 + | x|) -α with α∈ (0 , 2) and (1 + | x|) -β with β∈ (α, + ∞) , 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 ε> 0 , and in the semi-classical limit, these solutions concentrate at a global minimum point of A= V/ Q. Our existence result holds also when f has critical growth, for any ε> 0.
Exponential growth; Schrödinger equations; Semi-classical limit; Vanishing potentials; Weighted Trudinger–Moser inequality
Settore MAT/05 - Analisi Matematica
2019
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/652661
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