Combined effects of singular and critical nonlinearities in elliptic problems

Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity

“…Many authors have extensively considered problem (1.4) (see [2,[4][5][6][7][8][9]12,14,13,21,26,22,25,[27][28][29]). When λ ≡ 0, the existence of solutions for problem (1.4) has been studied (see [5,8,9,14,13,21]).…”

“…When 1 < p < 5, problem (1.4) has at least two solutions for all μ > 0 and λ > 0 small enough, such as [4,7,26,22,29]. In particular, when λ = 1, p = 5, problem (1.4) has at least two solutions for μ > 0 small enough, such as [2,6,12,25,[27][28][29]. However, the singular Kirchhoff type problems have few been considered, except for [15] and [17].…”

The second order expansion of solutions to a singular Dirichlet boundary value problem

“…Moreover, they also analyzed the asymptotic behavior of solutions andλ as p → 1. Later, Wang et al [29], Wang et al [30] further studied the existence and multiplicity of solutions for a more general model to problem…”

Existence and uniqueness of weak positive solution for essential singular elliptic problem involving the square root of the Laplacian

“…Therefore, it cannot considered by using critical points theory directly. Inspired by [16,18], we carry out a direct analysis in an H 1 0,L -neighborhood and get a unique minimizer which is a weak positive solution of problem (1.2). Moreover, since I(|v|) = I(v), we may assume that v 0 in C.…”

“…In pioneering papers of Crandall et al [5] and Stuart [13] proved that problem (1.4) has a unique classical positive solution u ∈ C 2 (Ω) ∩ C 0 (Ω), using the method of sub-and super-solutions when λ = 0, p ∈ C 1 (Ω). By the Ekeland's variational principle [12] and careful estimates, Sun [16] and my previous work [18] showed that the problem (1.4) possesses at least two weak solutions, provided 1 < β N+2 N−2 , 0 < γ < 1 and λ > 0 small enough.…”