Pointwise Lower Bounds for Solutions of Semilinear Elliptic Equations and Applications

Abstract

AbstractWe consider the semilinear elliptic problem −Δu = f (x, u), posed in a smooth bounded domain Ω of ℝNwith Dirichiel data u|∂Ω = 0, where f : Ω × [0, αf) → ℝ+(0 < αf≤ +∞) is a function of appropriate regularity which blows up at αf. We give pointwise lower bounds for the supersolutions under some appropriate conditions on f , and apply them to eigenvalue problem −Δu = λ f (x, u), by giving upper and lower bounds for the extremal parameter λ∗ and the extremal solution u∗. To demonstrate the sharpness of our results, we consider the eigenvalue problem −Δu = λ f (up) (p ≥ 1) with Dirichlet boundary condition, and show that for every increasing, convex and superlinear C2function f: ℝ+→ℝ+with, where ψΩ is the maximum of the torsion function of Ω. Also, we consider the eigenvalue problem −Δu = λρ(x) f (u), where f is either a regular singularity such as f (u) = eu, or a singular one such asand give explicit estimates on λ∗ and u∗, that improve and extend several results in the literature, by Payne[17], Sperb [21], Brezis-Vasquez [3], Guo-Pan-Ward [11], Ghoussoub-Guo [10], Cowan-Ghoussoub [6], and others.