一类奇异拟线性椭圆方程正解的多重性

The Existence of Solutions for a Singular Quasilinear Elliptic Equation

研究奇异拟线性椭圆型方程{-div(|x|^(-ap)|▽u|^(p-2)▽u) + f(x)|u|^(p-2) = g(x)\u|^(q-2)u + λh(x)|u|^(r-2),x R^N,u(x) > 0,x∈ R^N,其中λ>0是参数,1<p<N(N>3),1<r<p<g<p*=0<a<(N—p)/p,p*=Np/{N^pd),a<&<a+l,d=a+l-6>0,权函数f(x),g(x),h(x)满足一定的条件.利用山路引理和Ekeland变分原理证明了问题至少有两个非平凡的弱解.

In this paper, we study the existence of solutions for the singular quasilinear elliptic problem where λ 〉 0 is a real parameter and 1 〈 p 〈 N（N ≥ 3）,1 〈 r 〈 p 〈 q 〈 p＊,0 ≤ a 〈 （N - p）/p,p＊ = Np/（N - pd）,a 〈 b 〈 a＋ 1,d = a＋ 1 -b 〉 O. The weight functions f（x）, g（x）, h（x） satisfy some suitable conditions. We will prove the problem has at least two nontrivial weak solutions by Mountain Pass Theorem and Ekeland＇s variational principle.