The Influence Of Domain Geometry In The Boundary Behavior Of Large Solutions To Semilinear Elliptic Problems

In this article, we are interested in the study of the first-term and the second-termexpansion of the solution to the semilinear elliptic problemnear (?)Ω, whereΩis a regular bounded domain in R^N with N≥2, b∈C⁰,μ(Ω|ˉ), forsomeμ∈(0,1), satisfies b > 0 inΩand b = 0 on (?)Ω.LetΛdenotes the set of all positive and non-decreasing functions k∈C¹(0,δ₀)(δ₀ > 0) which satisfyWe will assume the following conditions are satisfied(f1): f∈C1[0,∞) is non-negative and f(uu )is increasing on (0,∞);(f2): there exits p > 1 such that(b1): there exits k∈Λsuch thatBy a perturbation method and constructing comparison functions, we obtainTheorem 1.1. Let u∈C²(Ω) be a solution to problem (1.1) when f satisfy (f1), (f2)and b satisfy (b1). Then it satisfieswhere l is given byand h is given byBesides, problem (1.1) possesses a unique solution. In particular, if b(x) = d(x)α(α> 0)and f(u) = eu, the solution u(x) of problem (1.1) satisfies Considering the boundary influence in the explosion rate, we get the second-termexpansion of the solution near (?)Ω:Theorem 1.2. Let u∈C²(Ω) be a solution to problem (1.1) when b(x) = d(x)^α(α> 0)and f(u) = e^u. Then, on a su?ciently small neighborhood of (?)Ω:where x|ˉdenotes the unique point of the boundary such that d(x) = |x-x|ˉ| and H(x|ˉ)the mean curvature of the boundary at that point.