POSITIVE SOLUTIONS OF A NONLOCAL AND NONVARIATIONAL ELLIPTIC PROBLEM

doi:10.1007/s10473-021-0522-5

Abstract

Abstract: In this paper, we will study the nonlocal and nonvariational elliptic problem

$\begin{equation} \left\{\begin{array}{ll}\label{eq0.1} -(1+a||u||_q^{\alpha q})\Delta u=|u|^{p-1}u+h(x,u,\nabla u) & \mbox{in}\ \ \Omega,\\ u=0 & \mbox{on}\ \ \partial\Omega,\\ \end{array} (0.1)\right. \end{equation}$ where

$a>0, \alpha>0, 1< q< 2^*, p\in(0,2^*-1)\setminus\{1\}$ and

$\Omega$ is a bounded smooth domain in

$\mathbb{R}^N$

$(N\geq 2)$ . Under suitable assumptions about

$h(x,u,\nabla u)$ , we obtain \emph{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.

Key words: positive solutions, nonvariational elliptic problem, a priori estimates

CLC Number:

35A05

Lingjun LIU, Feilin SHI. POSITIVE SOLUTIONS OF A NONLOCAL AND NONVARIATIONAL ELLIPTIC PROBLEM[J].Acta mathematica scientia,Series B, 2021, 41(5): 1764-1776.

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