Brezis–Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem

Abstract

We study Brezis-Nirenberg type theorems for the equationwhere Ω is a bounded domain in R N , g(x, ·) is increasing and f is a dissipative nonlinearity. We apply such theorems for studying existence and multiplicity of positive solutions for the equationwhere q > 0, p > 1 and λ > 0.

Semilinear problems for the fractional laplacian with a singular nonlinearity

“…Such elliptic problems with a singular nonlinearity have a large history in the local case, that is, with a principal part of Laplacian type. The seminal paper by Crandall, Rabinowitz and Tartar [13] is the starting point of a large literature, see for instance [3,4,7,8,12,14,19,21,23,25,26,34,35]. The philosophy to deal with this framework is similar to the one used for problems with a concave or a concave-convex nonlinearity.…”

“…where , p and are positive numbers (see, among other papers, [3,4,8,12,14,21,35] for an extensive analysis of this kind of problems). The multiplicity behavior in this case is essentially the same as in concave-convex type 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

“…Recently, Shi and Yao [25], Sun, Wu and Long [26], Yang [27] and Hirano et al [28] established the existence, uniqueness and multiplicity of solutions to problem (1.4). In particular, when γ < 1, p < 1, Shi and Yao [25] proved that problem (1.4) has a unique solution u in…”