一类p(x)-Laplacian问题弱解的存在性

Existence of weak solutions for class of p(x)-Laplacian problem

在变指数Lebesgue空间Lp(x)(Ω)和变指数Sobolev空间Wk,p(x)(Ω)基本理论体系上,研究了下面的p(x)-Laplacian问题:-div[(d+|u|2)p(2x)-1u]=-λ|u|p(x)-2u+f(x,u),x∈Ω{u=0,x∈Ω其中Ω是瓗N中的具有光滑边界的有界区域,0<d<∞,λ>0为常数.利用山路引理证明了上述的主部为-div[(d+|u|2)p(2x)-1u]一类p(x)-Laplacian问题在超线性的情况下解的存在性.

Based on the theory of the variable exponent Lebesgue spaces Lp（x）（Ω） and Sobolev spaces Wk,p（x）（Ω） ,this paper studied the following p（x）-Laplacian problem：-div[（d＋｜u｜2）p（x）2-1u]=-λ｜u｜p（x）-2u＋f（x,u）,x∈Ω u=0,x∈Ω Where ΩN was a bounded domain with smooth boundary and 00 were constant.Applying the mountain pass lemma,the existence of solution for the above-mentioned p（x）Laplacian problem with principal parts-div[（d＋｜u｜2）p（x）2-1u] was proved and in the superlinear case.