Existence and Asymptotic Behaviour for the 2D-Generalized Quasilinear Schrödinger Equations Involving Trudinger–Moser Nonlinearity and Potentials

Abstract

In this work, one of the purposes is concerned with the following generalized quasilinear Schrödinger equation:

$$\begin{aligned} -\text {div}(\text {j}^2(u)\nabla u)+\text {j}(u)\text {j}'(u)|\nabla u|^2+V(x)u=g(x,u),\,\, x\in \mathbb {R}^2, \end{aligned}$$

where $\text {j}\in \mathcal {C}^1(\mathbb {R},\mathbb {R}^{+})$, $V: \mathbb {R}^2\rightarrow \mathbb {R}$ and $g: \mathbb {R}^2\times \mathbb {R}\rightarrow \mathbb {R}$ satisfies Trudinger–Moser type growth. By using a change of variable, we obtain the existence of nontrivial solutions via the monotonicity condition instead of Ambrosetti–Rabinowitz condition. In addition, we intend to show the existence of nontrivial solutions for the following:

$$\begin{aligned} -\text {div}(\text {j}^2(u)\nabla u)+\text {j}(u)\text {j}'(u)|\nabla u|^2+\lambda V(x)u=Q(x)g(u),\,\, x\in \mathbb {R}^2, \end{aligned}$$

where $\lambda \ge 1$, $Q,V: \mathbb {R}^2\rightarrow \mathbb {R}$ and $g: \mathbb {R}\rightarrow \mathbb {R}$ satisfies Trudinger–Moser type growth. Furthermore, the asymptotic properties of the solutions are established as $\lambda \rightarrow +\infty$. Our results complement and extend some well-known ones showed by do Ó and Severo (Calc Var Partial Differ Eq 38:275–315, 2010), do Ó et al. (Commun Contemp Math 11:547–583, 2009), do Ó et al. (Nonlinear Anal 67:3357–3372, 2007).