Soliton solutions for quasilinear Schrödinger equation with critical exponential growth in ℝ N

Caisheng Chen; Hongxue Song

Applications of Mathematics (2016)

  • Volume: 61, Issue: 3, page 317-337
  • ISSN: 0862-7940

Abstract

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In this work, we study the existence of nonnegative and nontrivial solutions for the quasilinear Schrödinger equation - Δ N u + b | u | N - 2 u - Δ N ( u 2 ) u = h ( u ) , x ∈ ℝ N , where Δ N is the N -Laplacian operator, h ( u ) is continuous and behaves as exp ( α | u | N / ( N - 1 ) ) when | u | → ∞ . Using the Nehari manifold method and the Schwarz symmetrization with some special techniques, the existence of a nonnegative and nontrivial solution u ( x ) ∈ W 1 , N ( ℝ N ) with u ( x ) → 0 as | x | → ∞ is established.

How to cite

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Chen, Caisheng, and Song, Hongxue. "Soliton solutions for quasilinear Schrödinger equation with critical exponential growth in $\mathbb {R}^N$." Applications of Mathematics 61.3 (2016): 317-337. <http://eudml.org/doc/276992>.

@article{Chen2016,
abstract = {In this work, we study the existence of nonnegative and nontrivial solutions for the quasilinear Schrödinger equation \[ -\Delta \_Nu+b|u|^\{N-2\}u-\Delta \_N(u^2)u=h(u), \quad x\in \mathbb \{R\}^N, \] where $\Delta _N$ is the $N$-Laplacian operator, $h(u)$ is continuous and behaves as $\exp (\alpha |u|^\{\{N\}/\{(N-1)\}\})$ when $|u|\rightarrow \infty $. Using the Nehari manifold method and the Schwarz symmetrization with some special techniques, the existence of a nonnegative and nontrivial solution $u(x)\in W^\{1,N\}(\mathbb \{R\}^N)$ with $u(x)\rightarrow 0$ as $|x|\rightarrow \infty $ is established.},
author = {Chen, Caisheng, Song, Hongxue},
journal = {Applications of Mathematics},
keywords = {$N$-Laplacian equation; critical exponential growth; Schwarz symmetrization; Nehari manifold; -Laplacian equation; critical exponential growth; Schwarz symmetrization; Nehari manifold},
language = {eng},
number = {3},
pages = {317-337},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Soliton solutions for quasilinear Schrödinger equation with critical exponential growth in $\mathbb \{R\}^N$},
url = {http://eudml.org/doc/276992},
volume = {61},
year = {2016},
}

TY - JOUR
AU - Chen, Caisheng
AU - Song, Hongxue
TI - Soliton solutions for quasilinear Schrödinger equation with critical exponential growth in $\mathbb {R}^N$
JO - Applications of Mathematics
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 3
SP - 317
EP - 337
AB - In this work, we study the existence of nonnegative and nontrivial solutions for the quasilinear Schrödinger equation \[ -\Delta _Nu+b|u|^{N-2}u-\Delta _N(u^2)u=h(u), \quad x\in \mathbb {R}^N, \] where $\Delta _N$ is the $N$-Laplacian operator, $h(u)$ is continuous and behaves as $\exp (\alpha |u|^{{N}/{(N-1)}})$ when $|u|\rightarrow \infty $. Using the Nehari manifold method and the Schwarz symmetrization with some special techniques, the existence of a nonnegative and nontrivial solution $u(x)\in W^{1,N}(\mathbb {R}^N)$ with $u(x)\rightarrow 0$ as $|x|\rightarrow \infty $ is established.
LA - eng
KW - $N$-Laplacian equation; critical exponential growth; Schwarz symmetrization; Nehari manifold; -Laplacian equation; critical exponential growth; Schwarz symmetrization; Nehari manifold
UR - http://eudml.org/doc/276992
ER -

References

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