Existence and multiplicity of positive solutions for classes of singular elliptic PDEs

Abstract

We consider the boundary value problemwhere Ω ⊂ R N is a bounded domain, φ is a nonnegative function in L ∞ (Ω) such that φ > 0 on some subset of Ω of positive measure, and g : [0, ∞) → R is continuous. We establish the existence of three positive solutions when g(0) > 0 (positone), the graph of s α+1 g(s) is roughly S-shaped, and α > 0. We also prove that there exists at least one positive solution when g(0) < 0 (semipositone), g(s) is eventually positive for s > 0, and 0 < α < 1.We employ the method of sub-super solutions to prove our results.

Multiplicity Results for Classes of Infinite Positone Problems

“…However, our multiplicity result is restricted two positive solutions. In [1], the author studied this singular problem (2) when p = 2 by treating it as a limit problem of the class of non-singular problems defined by −∆u ǫ = λ e αu α+u (u+ǫ) β in Ω and u ǫ = 0 on ∂Ω. Here we establish the our results for all p > 1 directly by method of sub-and supersolutions associated with such singular problems.…”

Positive solution to extremal Pucci's equations with singular and gradient nonlinearity

“…(c) In case of M + λ,Λ (D 2 u) = ∆u, the existence of positive classical solution to (1) has been established in [22,28,30,48]. In this continuation, we also mention the work [14], where authors establish the existence of positive solution to (1) with M + λ,Λ (D 2 u) = ∆u, H ≡ 0 and f (0) < 0. So our works generalize these works to fully nonlinear elliptic equations in the framework of viscosity solution.…”

“…For the sake of completeness, we also discuss the existence of a positive solution to (1) in case f (0) < 0. The existence result for the case f (0) < 0 is motivated by the study of similar problems but in the context of Laplace equation, see [14]. The article is organized as follows.…”

Existence of Multiple Positive Radial Solutions to Elliptic Equations in an Annulus

“…It has been observed that very few papers exist in the literature on the existence of at least two positive solutions of (1.1) and (1.2). Maya and Robinson [22] used supersolution and subsolution method to obtain sufficient conditions for the existence of at least three positive solutions of the equation…”