Multiple Sign-Changing Solutions to Non-local Supercritical Exponent Problem in Symmetric Domains

In this article we establish the existence of a positive solution and multiple sign-changing solutions to the following supercritical exponent problem involving fractional Laplace operator \(\begin{aligned} (P_{b,\Omega }^{s})\left\{ \begin{array}{ll} (-\Delta )^s u = b(x)|u|^{q-2}\, u\, & \text { in }\Omega , \\ u =0 \, & \text { on }\partial \Omega , \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill (P_{b,\mathrm{\Omega }}^s)\left\{\begin{array}{cc}{(-\mathrm{\Delta })}^{s}u=b(x){|u|}^{q-2}u\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=0\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a bounded smooth domain with non-trivial topology in \({\mathbb {R}}^{N}\) ${\mathbb{R}}^N$ which is invariant under a group G of orthogonal transformations of \({\mathbb {R}}^N\) ${\mathbb{R}}^N$ , \(N\ge 2s\) $N\ge 2s$ , \(s\in (0,1)\) $s\in (0,1)$ , \(b\in C^{0,\alpha }({\overline{\Omega }})\) $b\in C^{0,\alpha }(\overline{\mathrm{\Omega }})$ and positive, \(q> 2^{*}_{s}:=\frac{2N}{N-2\,s}\) $q>2_s^{\ast }:=\frac{2N}{N-2s}$ .