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.
Lingjun LIU
,
Feilin SHI
. POSITIVE SOLUTIONS OF A NONLOCAL AND NONVARIATIONAL ELLIPTIC PROBLEM[J]. Acta mathematica scientia, Series B, 2021
, 41(5)
: 1764
-1776
.
DOI: 10.1007/s10473-021-0522-5
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