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 [Formula: see text]). We prove two results regarding the existence and multiplicity of solutions to the problem [Formula: see text]. 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 [Formula: see text], [Formula: see text] involving exponential growth nonlinearities in the more general framework of [Formula: see text] operator and for all [Formula: see text] (see below [Formula: see text]).

Symmetry of solutions to singular fractional elliptic equations and applications

“…The proof uses mainly Theorem 1 to get local uniform bound in Ω\{0}, the asymptotic behavior of f and Theorem 5 (proved in [2] and extending some results in [5]) to get the behavior of solutions near isolated singularities. In [2], we give further applications.…”

“…Une étape importante dans la preuve du théorème qui constitue par ailleurs un résultat d'intérêt plus large est l'étude des singularités isolés dans l'esprit du résultat bien connu de Brezis-Lions ( [4]) dans le cas local. Dans ce contexte, le Théorème 5 (voir la preuve dans [2]) étend un résultat récent de Chen and Quaas ( [5]) dans le cas de nonlinéarités exponentielles.…”

“…From Theorem 1 we know that the solutions are radial and radially decreasing, from this we only need to study the behavior near an isolated singularity. For that we exploit the following result proved in [2]:…”

Combined effects of singular and exponential nonlinearities in fractional Kirchhoff problems

“…In the critical case for 0 < q < 1, the question of existence and multiplicity of (weak) solutions of nonlocal singular problems has been answered in [14,26,27], whereas the q ≥ 1 case has been dealt in [13]. We also refer to the recent paper [3] on nonlocal singular problems with exponential nonlinearity. The singular Kirchhoff problems involving the Laplace operator has been investigated in [20,21].…”

Quasilinear nonlocal elliptic problems with variable singular exponent

“…In the critical case for 0 < q < 1, the question of existence and multiplicity of weak solutions to nonlocal singular problems has been answered in [19,27,28] whereas q ≥ 1 case has been dealt in [18]. We also refer a recent article [5] related to nonlocal singular problem with exponential nonlinearity. Moreover, we refer readers to [1,2] concerning the existence and multiplicity results for the fractional p-Laplacian problems.…”