Positive solutions to the p-laplacian with singular weights

By the Mountain Pass Theorem, we study existence and multiplicity of positive solutions of p-laplacian equation of the form −Δpu=λf(x,u), the nonlinearity f(x, u) grows as uσ at infinity with a singular coefficient, where σ∈(p−1,p*−1). To manage the asymptotic behavior of its positive solutions with respect to λ, we establish a new Liouville-type theorem for the p-Laplacian operator.