Positive Solutions of a Nonlocal and Nonvariational Elliptic Problem

In this paper, we will study the nonlocal and nonvariational elliptic problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {\matrix{{ - (1 + a\|u\|_q^{\alpha q})\Delta u = |u{|^{p - 1}}u + h(x,u,\nabla u)\,{\rm{in}}\,\,\,\Omega ,} \hfill \cr {u = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,on\,\,\,\partial \Omega ,} \hfill \cr } } \right.$$\end{document} where a > 0, α > 0, 1 < q < 2*, p ∈ (0, 2* − 1) {1} and Ω is a bounded smooth domain in ℝN (N ≥ 2). Under suitable assumptions about h(x, u, ∇u), we obtain a priori estimates of positive solutions for the problem (0.1). Furthermore, we establish the existence of positive solutions by making use of these estimates and of the method of continuity.