In the present paper, we consider the problem
$\begin{equation} \left\{\begin{array}{ll}\label{0001} -\Delta u=u^{\beta_1}|\nabla u|^{\beta_2}, &\ \ { in} \ \Omega,\\ u=0,&\ \ { on} \ \partial{\Omega},\\ u>0,&\ \ { in} \ {\Omega},\\ \end{array}\right. \end{equation}$ $ \ \ \ \ \ $ (0.1)
where $\beta_1,\beta_2>0$ and $\beta_1+\beta_2<1$, and $\Omega $ is a convex domain in $ \mathbb{R}^{n} $. The existence, uniqueness, regularity and $\frac{2-\beta_{2}}{1-\beta_1-\beta_2}$-concavity of the positive solutions of the problem (0.1) are proven.
Bo Chen
,
Zhengmao Chen
,
Junhui Xie
. PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION*[J]. Acta mathematica scientia, Series B, 2023
, 43(3)
: 1161
-1174
.
DOI: 10.1007/s10473-023-0310-5
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