Solutions for a class of singular quasilinear equations involving critical growth in R2$\mathbb {R}^2$

Manassés X. de Souza, Uberlandio B. Severo and Gilberto F. Vieira

Mathematische Nachrichten, 2022, vol. 295, issue 1, 103-123

Abstract: Using a variational approach, we study the existence of solutions for the following class of quasilinear Schrödinger equations: −Δu+V(x)u−Δ(|u|2β)|u|2β−2u=g(u)|x|ainR2,\begin{equation*} \hspace*{6.5pc}-\Delta u+V(x)u-\Delta \big (|u|^{2\beta }\big )|u|^{2\beta -2}u=\frac{g(u)}{|x|^a}\quad \mbox{in}\quad \mathbb {R}^2,\hspace*{-6.5pc} \end{equation*}where β>1/2$\beta >1/2$, a∈[0,2)$a\in [0,2)$, V(x)$V(x)$ is a positive potential bounded away from zero and can be “large” at infinity, the nonlinearity g(s)$g(s)$ is allowed to satisfy the exponential critical growth with respect to the Trudinger–Moser inequality. Precisely, g(s)$g(s)$ behaves like exp(α0|s|4β)$\exp \big (\alpha _0 |s|^{4 \beta }\big )$ as |s|→∞$|s| \rightarrow \infty$ for some α0>0$\alpha _0 >0$. This model of equation has been proposed in the theory of superfluid films in plasma physics. As for as we know, this the first work involving this class of operators and singular nonlinearities with exponential critical growth. Moreover, we are able to deal with exponents β>1/2$\beta > 1/2$.

Date: 2022
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