Hilbert空间方法在半线性方程上的应用

运用Sobolev嵌入定律和Schauder不动点定理,在一个比文2]更宽松的条件下,建立了Hilbert空间方法,并有效地解决了如下方程解的存在性Lu-C(u,0)u=f(u).其中L是线性自伴算子,C(u,0)是非线性的并满足与L的谱有关的一系列条件,同时突破了文1]中非共振条件中两个常数p,q的限制及文2]中类似的限制条件,从而推广了文1,2]的结论.

Hilbert Space Method and Application to Semilinear Equation

In this paper,by the Sobolev imbedding theorem a nd Schauder fixed point theorem, we establish Hilbert space method in a more gener al sence than 2], and obtain effectively the existence of solution to t he equation of the form Lu-C(u,0)u=f(u). Here L is a linear and selfadjoint ope ra tor while C(u,o)is nonlinear and satisfies certain conditions relating to the sp ectrum of L. Moreover, the limit of two constats p,q in1]is being taken out , the same to2].Thus the main theorem of 1,2]are corollaries.