Soliton solutions for quasilinear Schrödinger equation with critical exponential growth in \(\mathbb{R}^N\). (English) Zbl 1413.35124

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.


35D30 Weak solutions to PDEs
35J20 Variational methods for second-order elliptic equations
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
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