Li Yongxiang, Ma Weifeng
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China.
Entropy (Basel). 2022 Dec 15;24(12):1829. doi: 10.3390/e24121829.
This paper deals with the existence of solutions of the elliptic equation with nonlinear gradient term -Δu=f(x,u,∇u) on Ω restricted by the boundary condition u|∂Ω=0, where Ω is a bounded domain in RN with sufficiently smooth boundary ∂Ω, N≥2, and f:Ω¯×R×RN→R is continuous. The existence results of classical solutions and positive solutions are obtained under some inequality conditions on the nonlinearity f(x,ξ,η) when |(ξ,η)| is small or large enough.
本文研究在边界条件(u|\partial\Omega = 0)限制下，(\Omega)上具有非线性梯度项(-\Delta u = f(x, u, \nabla u))的椭圆方程解的存在性，其中(\Omega)是(R^N)中边界(\partial\Omega)足够光滑的有界区域，(N\geq2)，且(f:\overline{\Omega}\times R\times R^N\rightarrow R)连续。当(|(\xi, \eta)|)足够小或足够大时，在关于非线性项(f(x, \xi, \eta))的一些不等式条件下，得到了经典解和正解的存在性结果。
