一类含非局部源和非变分形式的椭圆型方程组正解的存在性(英文)

EXISTENCE OF POSITIVE SOLUTION FOR A NONLOCAL AND NONVARIATIONAL ELLIPTIC SYSTEMS

本文研究一类含非局部源的椭圆型方程组{-A(∫_Ω|u|~kdx)Δ_pu=λv^m∫_Ωu~αv~βdx,x∈Ω-B(∫_Ω|u|~sdx)Δ_qv=μu^n∫_Ωu~γv~δdx,x∈Ω(1)并且带有Dirichlet零边界条件的正解存在性．这里Ω是R^N,N≥1中的有界区域,边界(?)Ω光滑．为了得到它的解,我们先考虑与之相应的局部椭圆型方程组-Δ_pu=λv^m,-Δ_qv=μu^n inΩ;u=v=0,on (?)Ω(2)正解的存在性．我们将应用上下解方法得到问题(1)和(2)的解．

In this work, we consider the existence of positive solution for the following nonlocal elliptic system： {-A（∫Ω｜u｜^kdx）}△pu=λv^m∫Ωu^αv^βdx,x∈Ω, -B（∫Ω｜v｜^sdx）△qv=μu^n∫Ω^uγvδdx,x∈Ω （0.1） with zero Dirichlet boundary condition in a bounded domain Ω belong to R^N,N≥1.To obtain the solution, we establish the existence of positive solution for the corresponding stationary problem -△pu=λv^m,-△qv=μu^n,x∈Ω（0.2） with zero Dirichlet boundary condition . The method of sub- and super solution will be used for the existence of solutions for problems （0.1） and （0.2）.