具有凹凸非线性项和变号位势函数拟线性椭圆系统解的多重结果

Multiple Results for Quasilinear Elliptic Systems Involving Concave-Convex Nonlinearities and Sign-Changing Weight Functions

研究拟线性椭圆系统(?)的非平凡非负解或正解的多重性,这里Ω(?)R^N是具有光滑边界(?)Ω的有界域,1≤q<p,△_pω=div(|▽ω、|^(p-2)▽ω)代表p-Laplacian算子,F∈C^1((?)×(R^+)~2,R^+)满足次临界增长条件,λ为正参数,α(x),b(x)∈L^r(Ω)在Ω上可能变号,r>p~*/p~*-q,其中当N≤p时,p~*=+∞,而当1<p<N时,p~*=Np/N-p.利用Ekeland变分原理和山路引理,证明了这一拟线性椭圆系统非平凡非负解或正解的多重性.

The multiplicity of nontrivial nonnegative or positive solutions to the following quasilinear elliptic system is studied,whereΩ R^N is a bounded domain with smooth boundary Ω,1≤qp,Δ_pω=div（｜▽_ω｜^（p-2）▽ω）denotes the p-Laplacian operator,F∈C^1×（R^＋）^2,R^＋）satisfies the subcritical growth condition;λ0 is a positive parameter;a（x）,b（x）∈L^r（Ω）are allowed to change sign,r（p^＊）/（p^＊-q）such that p^＊=＋∞if N≤p and p^＊=（N_p/N-p）if 1pN.The multiple results of the weak solutions to the above quasilinear elliptic system are obtained by using the Ekeland＇s variational principle and the mountain pass theorem.