The radial symmetry of positive solutions for semilinear problems involving weighted fractional Laplacians

This paper deals with the radial symmetry of positive solutions to the nonlocal problem \(( - \Delta )_\gamma ^su = b(x)f(u)\,\,\,\,\,{\rm{in}}\,\,\,{B_1}\backslash \{ 0\} ,\,\,\,\,\,\,u = h\,\,\,\,{\rm{in}}\,\,{\mathbb{R}^N}\backslash {B_1},\) $$(-\mathrm{\Delta }{)}_{\gamma }^{s}u=b(x)f(u)\mathrm{i}\mathrm{n}{B}_1\mathrm{\setminus }\{0\},u=h\mathrm{i}\mathrm{n}{\mathbb{R}}^N\mathrm{\setminus }{B}_1,$$ where b: B₁ → ℝ is locally Holder continuous, radially symmetric and decreasing in the ∣x∣ direction, f: ℝ → ℝ is a Lipschitz function, h: B₁ → ℝ is radially symmetric, decreasing with respect to ∣x∣ in ℝ^NB₁, B₁ is the unit ball centered at the origin, and \(( - \Delta )_\gamma ^s\) $$(-\mathrm{\Delta }{)}_{\gamma }^s$$ is the weighted fractional Laplacian with s ∈ (0, 1), γ ∈ [0, 2s) defined by \(( - \Delta )_\gamma ^su(x) = {c_{N,s}}\mathop {\lim }\limits_{\delta \to {0^ + }} \int_{{\mathbb{R}^N}\backslash {B_\delta }(x)} {{{u(x) - u(y)} \over {|x - y{|^{N + 2s}}}}|y{|^\gamma }{\rm{d}}y.} \) $$(-\mathrm{\Delta }{)}_{\gamma }^{s}u(x)=c_{N,s}\underset{\delta \to 0^{+}}{lim}{\int }_{{\mathbb{R}}^N\mathrm{\setminus }{B}_{\delta }(x)}\frac{u(x)-u(y)}{|x-y{|}^{N+2s}}|y{|}^{\gamma }\mathrm{d}y.$$

We consider the radial symmetry of isolated singular positive solutions to the nonlocal problem in whole space \(( - \Delta )_\gamma ^su(x) = b(x)f(u)\,\,\,\,\,{\rm{in}}\,\,{\mathbb{R}^N}\backslash \{ 0\} ,\) $$(-\mathrm{\Delta }{)}_{\gamma }^{s}u(x)=b(x)f(u)\mathrm{i}\mathrm{n}{\mathbb{R}}^N\mathrm{\setminus }\{0\},$$ under suitable additional assumptions on b and f. Our symmetry results are derived by the method of moving planes, where the main difficulty comes from the weighted fractional Laplacian. Our results could be applied to get a sharp asymptotic for semilinear problems with the fractional Hardy operators \({( - \Delta )^s}u + {\mu \over {|x{|^{2s}}}}u = b(x)f(u)\,\,\,\,{\rm{in}}\,\,\,{B_1}\backslash \{ 0\} ,\,\,\,\,\,\,\,\,u = h\,\,\,\,\,\,{\rm{in}}\,\,\,{\mathbb{R}^N}\backslash {B_1},\) $$(-\mathrm{\Delta }{)}^{s}u+\frac{\mu }{|x{|}^{2s}}u=b(x)f(u)\mathrm{i}\mathrm{n}{B}_1\mathrm{\setminus }\{0\},u=h\mathrm{i}\mathrm{n}{\mathbb{R}}^N\mathrm{\setminus }{B}_1,$$ under suitable additional assumptions on b, f and h.