一类半正椭圆方程径向正解的存在性

Existence of Radial Positive Solutions for a Class ofSemipositone Elliptic Equations

首先,用变分法理论讨论带有Dirichlet边界条件的半正椭圆方程-Δu=λk(x)f(u),x∈Ω径向正解的存在性问题,结果表明:当λ充分小时,方程不存在非负解;当λ充分大时,方程存在径向正解.其次,证明该方程每个解处的线性化算子均有非负的第一特征值.其中ΩℝN(N≥2)是一个球或环,参数λ>0,f∈C([0,∞),ℝ)且f(0)<0(半正),k:[a,b]→[0,∞)且k(x)不恒为0.此外,当Ω为球时,k为线性映射;当Ω为环时,k为单调增函数.

Firstly,the author discussed the existence of radial positive solutions for a semipositone elliptic equation-Δu=λk(x)f(u),x∈Ωwith the Dirichlet boundary condition by using the variational method,the results show that when theλis sufficiently small,the equation has no nonnegative solution;when theλis sufficiently large,the equation has a radial positive solution.Secondly,the author prove that the linearized operator at each solution of the equation has the nonnegative first eigenvalue.WhereΩR N(N≥2)is a ball or an annulus,the parameterλ>0,f∈C([0,∞),R)and f(0)<0(semipositone),k:[a,b]→[0,∞)and k(x)is not always 0.Furthermore,when theΩis a ball,the k is a linear map;whenΩis an annulus,the k is a monotone increasing function.