Quasilinear Schrödinger equations with unbounded or decaying potentials in dimension 2

Gilson M. de Carvalho, Rodrigo G. Clemente and José Carlos de Albuquerque

Mathematische Nachrichten, 2023, vol. 296, issue 9, 4357-4373

Abstract: We establish the existence of nontrivial solutions for the following class of quasilinear Schrödinger equations: −Δu+V(|x|)u+κ2[Δ(u2)]u=Q(|x|)h(u),x∈R2,u(x)→0,as|x|→∞,$$\begin{equation*} {\left\lbrace \begin{aligned} &-\Delta u+V(|x|)u+ \frac{\kappa }{2}[\Delta (u^{2})]u=Q(|x|)h(u), \quad x \in \mathbb {R}^2,\\ &u(x)\rightarrow 0, \quad \textrm {as}\quad |x|\rightarrow \infty , \end{aligned} \right.} \end{equation*}$$where κ is a positive parameter, V(|x|)$V(|x|)$ and Q(|x|)$Q(|x|)$ are continuous functions that can be singular at the origin, unbounded or vanishing at infinity, and the nonlinearity h(s)$h(s)$ has critical exponential growth motivated by the Trudinger–Moser inequality. To prove our main result, we apply variational methods together with careful L∞$L^{\infty }$‐estimates.

Date: 2023
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