Abstract
We deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by−Δpu=fuγ+guqinΩ,u=0on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left \\{\\begin {array}{ll} \\displaystyle -{\\Delta }_{p} u= \\frac {f}{u^{\\gamma }} + g u^{q} & \ ext { in } {\\Omega }, \\\\ u = 0 & \ ext {on } \\partial {\\Omega }, \\end {array}\\right . $$\\end{document} where Ω is an open bounded subset of mathbb {R}^{N} where Ω is an open bounded subset of mathbb {R}^{N}, Δpu := ÷(|∇u|p− 2∇u) is the usual p-Laplacian operator, γ ≥ 0 and 0 ≤ q ≤ p − 1; f and g are nonnegative functions belonging to suitable Lebesgue spaces.
Highlights
In this paper we deal with an elliptic problem which simplest model is ⎧ ⎪⎪⎨− ⎪⎪⎩uu > = 0 0 in, on ∂, (1.1)R
Oliva where is an open bounded subset of RN, pu := div(|∇u|p−2∇u) is the p-Laplacian operator (1 < p < N ), γ, q ≥ 0 are such that q < p − 1 or q = p − 1, which correspond to the sublinear and to the linear behaviour in case p = 2; here f, g are nonnegative functions belonging to suitable Lebesgue spaces
The Dirichlet problem (1.1) is singular since the request that the solution is zero on the boundary of the set implies that the right hand side blows up
Summary
Let us mention that in [35], for p > 1, the authors show existence and uniqueness of finite energy solutions to (1.1) under suitable assumptions on f, g. It is worth mentioning that (1.3) allows to deal with the case q ≤ p − 1, at least for positive f if one considers the model case given by (1.1) This result is presented as the comparison principle given by Theorem 2.2 which, as a simple corollary, takes to uniqueness of finite energy solutions. We are interested to instances of finite energy solutions to (1.2); this is done both in the mild and in the strongly singular case by means of approximation arguments firstly if q < p − 1; we give an existence result in case q = p − 1.
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