An exact estimate result for a class of singular equations with critical exponents

Abstract

We consider the singular boundary value problem, γ ∈ (0, 1). It is well known that there exists λ * > 0 such that the problem has a solution for all λ ∈ (0, λ * ) and no solution for λ > λ * . We obtain an exact result for λ * (Ω, p, γ, h).

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].…”

On quasi-linear equation problems involving critical and singular nonlinearities

“…They constructed lower-upper solutions to show the problem (1.3) has one weak solution in W 1,p 0 (Ω). We note that when p = 2, the multiplicity of positive solutions for problem (1.1) has been considered by Sun and Wu [22], Sun and Li [21], and Chen and Chen [6,7].…”

Kirchhoff type equations with strong singularities

“…(see, e.g. [1][2][3][4][5][6][7][8][9][10][11][12][14][15][16][17][18][19][20][21][22][23][24][25][26][28][29][30][31][32][33][34][36][37][38][39][40][41][42] and references therein). For the singular boundary theory, we refer the reader to the books by Agarwal and O'Regan [1], and Hernández and Mancebo [25] for an excellent introduction to the subject.…”