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

Abstract

Using a variational approach, we study the existence of solutions for the following class of quasilinear Schrödinger equations: 78.0pt−normalΔu+V(x)u−normalΔtrue(false|ufalse|2βtrue)false|ufalse|2β−2u=gfalse(ufalse)|x|a1emin1emR2,\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∈false[0,2false)$a\in [0,2)$, Vfalse(xfalse)$V(x)$ is a positive potential bounded away from zero and can be “large” at infinity, the nonlinearity gfalse(sfalse)$g(s)$ is allowed to satisfy the exponential critical growth with respect to the Trudinger–Moser inequality. Precisely, gfalse(sfalse)$g(s)$ behaves like exptrue(α0false|sfalse|4βtrue)$\exp \big (\alpha _0 |s|^{4 \beta }\big )$ as false|sfalse|→∞$|s| \rightarrow \infty$ 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$\beta > 1/2$.

Quasilinear Schrödinger equations with Stein-Weiss type convolution and critical exponential nonlinearity in $\mathbb R^N$

“…In the same spirit, in the critical dimension N = 2, the critical nonlinearity is expected to behave like exp(αs 4 ) as s → ∞( see [14,44]). For more development on this topic, we refer to some recent contemporary works [41,42], where the authors studied the equations of type (P * ) for N = 2, without the convolution term R N F (y, u)|x − y| −µ dy and also, considered some stronger assumption than (V 2 ) on the potential function V .…”