一类拟线性方程弱解的存在性

Existence of Weak Solutions for a Class of Quasilinear Equations

研究一类拟线性椭圆方程-Δu-uΔu^(2)=λ|u|^(q-2)u(inΩ)弱解的存在性,其中Ω⊂R^(N)是N>3的光滑有界区域,1<q<2,λ>0是一个实参数,且在边界处u=0.利用变量替换将拟线性问题转换到在零点处是奇异的和在无穷远处是超线性的半线性问题,并利用上下解和比较原理的方法证明拟线性方程弱解的存在性.

In this paper,we consider existence of weak solutions for a class of quasilinear elliptic equation-Δu-uΔu^(2)=λ|u|^(q-2)u(inΩ).WhereΩ⊂R^(N) with N>3 is a smooth bounded region,1<q<2,λ>0 is a real parameter,and u=0 at the boundary.The quasilinear problem is transformed through variable substitution to a semilinear problem which remains singular at zero and behaves superlinearly at infinity,and the existence of weak solutions of quasilinear equations is proved by upper and lower solutions and comparison principle.