Singular elliptic problems: Existence, non-existence and boundary behavior

The existence and non-existence of positive solutions to a singular quasilinear elliptic problem in

“…With this super-solution, we may also obtain a solution of the problem. So we can extend the result of [13] to R N .…”

“…We also refer the reader to the work of Goncalves, Santos [13], where they found a super-solution v ∈ C 2 (Ω) ∩ C 1 (Ω) of (1.1), by arguments in the proof of Theorem 1.3 of [14], and v ≤ cd(x), where d(x) is the Euclidean distance from x to ∂Ω. However, in our paper, we prove that there exists a bounded super-solution v ∈ C 1 (R N )∩C 2 (R N \{0}) under weaker conditions.…”

“…And Zhang [18] showed that (1.1) has a solution provided that f < 0 and lim s→0 f (s) = ∞, g(x) ≡ 0. Goncalves, Santos [13] showed that (1.1) is solvable under the conditions:…”

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

“…in Ω) and g, f : (0, ∞) → [0, ∞) be locally Lipschitz continuous functions, and furthermore, g or f (or both of them) be singular at zero. Goncalves and Santos [32] showed both the existence, uniqueness and boundary behavior of classical solutions and the nonexistence of weak solution to problem (1.1). Most recently, when g satisfies (g 1 )-(g 2 ), f satisfies (f 1 ), b, a satisfy (H 1 ), by establishing a local comparison principle of solutions near the boundary, Zhang et al [33] studied the first order expansion of classical solutions to problem (1.1), and they also showed the existence and nonexistence of classical solutions.…”

Existence and multiplicity of weak solutions for a singular semilinear elliptic equation

“…w(x) → ∞ as x → ∂Ω. In the last years, there are a large number of papers involving the above problems, see for example [3,4,11,13,14,16,21,23,25,27].…”