Soliton solutions for a class of critical Schrödinger equations with Stein–Weiss convolution parts in $\mathbb {R}^2$

Abstract

We consider the following class of quasilinear Schrödinger equations introduced in plasma physics and nonlinear optics with Stein–Weiss convolution parts

$$\begin{aligned} -\Delta u+V(x) u+\frac{\kappa }{2} u\Delta (u^2)=\frac{1}{|x|^\beta }\Bigg (\int _{\mathbb {R}^2}\frac{H(u)}{|x-y|^\mu |y|^\beta }dy\Bigg ) h(u),~x\in \mathbb {R}^2, \end{aligned}$$

where $\kappa \in \mathbb {R}\backslash \{0\}$ is a parameter, $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$ and H is the primitive of h that fulfills the critical exponential growth in the Trudinger–Moser sense. For $\kappa <0$: (i) via using a change of variable argument and the mountain-pass theorem, we investigate the existence of ground state solutions only assuming that $V\in C^0(\mathbb {R}^2,\mathbb {R}^+)$ and $\inf _{x \in \mathbb {R}^2}V(x)>0$, which complements and generalizes the problems proposed in our recent work in Alves and Shen (J Differ Equ 344:352–404, 2023); (ii) by developing a new type of Trudinger–Moser inequality, we establish a Pohoz̆aev type ground solution by the constraint minimization approach when $V\equiv 1$. Moreover, if $\kappa >0$ is small, combining the mountain-pass theorem and Nash–Moser iteration procedure, we obtain the existence of nontrivial solutions, where the asymptotical behavior is also considered when $\kappa \rightarrow 0^+$. It seems that the results presented above are even new for the case $\kappa =0$.