一类含渐近线性的奇异椭圆边值问题正解的存在性

利用临界点理论,研究了一类含有渐近线性项和奇异项的半线性椭圆方程的边值问题.首先,利用椭圆算子特征值的性质,结合函数f(u)的渐近线性,证明了椭圆边值所对应的泛函J在凸闭集Γε={u∈C10(-Ω)|u≥εφ1}上满足PS条件.其次,利用Banach空间中的常微分方程理论,证明了对任意的a∈R+,J在Γε上具有收缩性,并利用Schauder型条件,证明了Γε是泛函J的一个下降流不变集.最后,对于u∈Γε,证明了J(u)是下方有界的.从而得到了奇异椭圆方程的边值问题至少存在一个正解的结论.

Existence of Positive Solution to a Class of Problems of Singular Elliptic Boundary Value with Asymptotical Linearity

According to the critical point theory, a class of problems of elliptic boundary value with an asymptotically linear term and singular term is studied. It is proved that the functional J corresponding to the elliptic boundary value satisfies PS condition on the convex closed set ΓΕ = {u ∈ C0¹ (Ω¯)|u > Εφ1} by the property of elliptic operator eigenvalue in combination with the asymptotical linearity of the function f(u). Then it is also proved that J is retractable to a ∈ R⁺ on ΓΕ by the ordinary differential equation theory in Banach space. Furthermore, ΓΕ is proved an invariant set of decent flow of J by Schauder condition, and J(u) is proved lower bounded for u ∈ΓΕ. A conclusion is therefore reasoned out that there is a positive solution at least to the problems of singular elliptic boundary value.