N -Laplacian Coupled Systems With Stein–Weiss Type Convolutions Parts: Existence of Solutions,Asymptotic Behavior and Regularity

Abstract

In this article, we study the following class of Stein–Weiss coupled system $$\begin{array}{r}\{\begin{array}{ll}-{\mathrm{\Delta }}_{N}u+|u{|}^{N-2}u=\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\displaystyle \frac{F(u(y))}{|y{|}^{\beta }|x-y{|}^{\mu }}}\mathrm{d}y}\right){\displaystyle \frac{f(u(x))}{|x{|}^{\beta }}}+\lambda p|u{|}^{p-2}u|v{|}^q,& \text{in }{\mathbb{R}}^N,\\ -{\mathrm{\Delta }}_{N}v+|v{|}^{N-2}v=\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\displaystyle \frac{G(v(y))}{|y{|}^{\beta }|x-y{|}^{\mu }}}\mathrm{d}y}\right){\displaystyle \frac{g(v(x))}{|x{|}^{\beta }}}+\lambda q|u{|}^p|v{|}^{q-2}v,& \text{in }{\mathbb{R}}^N,\end{array}\end{array}$$ where $N\ge 2$ , $0<\mu <N$ , ${\mathrm{\Delta }}_{N}u=\text{div}(|\mathrm{\nabla }u{|}^{N-2}\mathrm{\nabla }u)$ is the $N$ -Laplacian operator, $\beta \ge 0$ , $0<2\beta +\mu <N$ and $F(s),G(s)$ are the primitives of $f(s)$ and $g(s)$ , respectively. We deal with two classes of coupled systems separately: The nonlinear coupled case when $N>2$ , $p,q>N/2$ , and the linearly coupled case when $N=2$ and $p=q=N/2$ . Assuming that the nonlinearities $f(s)$ and $g(s)$ have critical exponential growth in the sense of Trudinger–Moser inequality, we study the existence of positive solutions for the above system. Moreover, asymptotic behavior and regularity of the solutions enrich the study of the system. In our approach we introduce an alternative to the standard arguments based on Lions’ vanishing–nonvanishing and shifted sequences argument (not applicable if $\beta \ne 0$ ) by utilizing a variant of Palais Principle of symmetric criticality.