Positive solutions of 1-D half Laplacian equation with singular and exponential nonlinearity

Abstract

In this article, we study the existence, multiplicity, regularity and asymptotic behavior of the positive solutions to the problem of half-Laplacian with singular and exponential growth nonlinearity in one dimension (see below $\mathrm{(}{\mathit{P}}_{\mathit{\lambda }}\mathrm{)}$ ). We prove two results regarding the existence and multiplicity of solutions to the problem $\mathrm{(}{\mathit{P}}_{\mathit{\lambda }}\mathrm{)}$ . In the first result, existence and multiplicity have been proved for classical solutions via bifurcation theory while in the latter result multiplicity has been proved for critical exponential nonlinearity by variational methods. An independent question of symmetry and monotonicity properties of classical solution has been answered in the paper. To characterize the behavior of large solutions, we further study isolated singularities for the singular semi linear elliptic equation in $\mathrm{\Omega }\subset {\mathbb{R}}^{\mathit{n}}$ , $\mathit{n}⩾2\mathit{s}$ involving exponential growth nonlinearities in the more general framework of ${\mathrm{(}-\mathrm{\Delta }\mathrm{)}}^{\mathit{s}}$ operator and for all $0\mathrm{<}\mathit{s}\mathrm{<}1$ (see below $\mathrm{(}{\mathit{P}}_{\mathit{s}}\mathrm{)}$ ).