Critical exponents and critical dimensions for nonlinear elliptic problems with singular coefficients

Abstract

Let B1 ⊂ ℝN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients: { u|∂ B1 = 0,−div(|∇u|p−2∇u)=|x|s|u|p*(s)−2u+λ|x|t|u|p−2u, x ∈ B1,where t,s > −p, 2 ≤p < N, p*(s)=(N+s)pN−p and λ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N > p(p−1)t + p(p2−p+1) and λ ∈ (0, λ1, t), where λ1, t is the first eigenvalue of -Δp with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤ (ps+p)min{1,p+tp+s}+p2p−(p−1)min{1,p+tp+s} λ > 0 is small.