一类奇异拟线性椭圆方程正解的多重性

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λ0是参数,1pN(N3),1rpgp*=0a(N—p)/p,p*=Np/{N~pd),aa+l,d=a+l-60,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.

The Existence of Solutions for a Singular Quasilinear Elliptic Equation

In this paper,we study the existence of solutions for the singular quasilinear elliptic problem{-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,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 ≤b0.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.