Critical growth fractional systems with exponential nonlinearity

TL;DR

This paper investigates the existence of multiple positive solutions for a fractional elliptic system with exponential nonlinearities, employing fibering maps and Nehari manifold analysis, and extends results to superlinear systems with critical exponential growth.

Contribution

It introduces new methods for establishing multiple solutions in fractional systems with exponential nonlinearities, including analysis of fibering maps and Nehari manifolds.

Findings

Multiple positive solutions exist for certain parameter ranges.

Existence results extend to superlinear systems with critical exponential growth.

The methods can handle fractional Laplacian operators with exponential nonlinearities.

Abstract

We study the existence of positive solutions for the system of fractional elliptic equations of the type, \begin{equation*} \begin{array}{rl} (-\Delta)^{\frac{1}{2}} u &=\frac{p}{p+q}\lambda f(x)|u|^{p-2}u|v|^q + h_1(u,v) e^{u^2+v^2},\;\textrm{in}\; (-1, 1),\\ (-\Delta)^{\frac{1}{2}} v &=\frac{q}{p+q}\lambda f(x)|u|^p|v|^{q-2}v + h_2(u,v) e^{u^2+v^2},\;\textrm{in}\; (-1, 1), u,v&>0 \;\textrm{in } \; (-1,1), u&=v=0 \; \text{in} \; \mathbb R\setminus (-1,1). \end{array} \end{equation*} where {$1<p+q<2$}, $h_1(u,v)=(\alpha{+}2u^2)|u|^{\alpha-2}u|v|^\beta, h_2(u,v)=(\beta{+}2v^2) |u|^\alpha |v|^{\beta-2}v$ and ${\alpha+\beta>2}$. Here $(-\Delta)^{\frac{1}{2}}$ is the fractional Laplacian operator. We show the existence of multiple solutions for suitable range of $\lambda$ by analyzing the fibering maps and the corresponding Nehari manifold. We also study the existence of positive solutions for a superlinear system with critical growth exponential nonlinearity.