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Article

Chen, Caisheng ; Song, Hongxue
Soliton solutions for quasilinear Schrödinger equation with critical exponential growth in $\mathbb {R}^N$. (English). Applications of Mathematics, vol. 61 (2016), issue 3, pp. 317-337
MSC: 35D30, 35J20, 35J92 | MR 3502114 | Zbl 06587855 | DOI: 10.1007/s10492-016-0134-x
Keywords:
$N$-Laplacian equation; critical exponential growth; Schwarz symmetrization; Nehari manifold
Summary:
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|\to \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 $u(x)\to 0$ as $|x|\to \infty $ is established.
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