Geodesic
Soliton solutions for quasilinear Schrödinger equation with critical exponential growth in $\mathbb {R}^N$
Applications of Mathematics, Tome 61 (2016) no. 3, pp. 317-337.

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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.
DOI : 10.1007/s10492-016-0134-x
Classification : 35D30, 35J20, 35J92
Keywords: $N$-Laplacian equation; critical exponential growth; Schwarz symmetrization; Nehari manifold 
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     title = {Soliton solutions for quasilinear {Schr\"odinger} equation with critical exponential growth in $\mathbb {R}^N$},
     journal = {Applications of Mathematics},
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Chen, Caisheng; Song, Hongxue. Soliton solutions for quasilinear Schrödinger equation with critical exponential growth in $\mathbb {R}^N$. Applications of Mathematics, Tome 61 (2016) no. 3, pp. 317-337. doi : 10.1007/s10492-016-0134-x. http://geodesic.mathdoc.fr/articles/10.1007/s10492-016-0134-x/

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