The existence of multiple solutions of p-Laplacian elliptic equation

This paper considers the quasilinear elliptic equation (1)(lambda) {-Delta (p) u = \u \ (m-1) u + lambda \u \ (q-1)u, x is an element of Omega {u is an element of W-0(1.p)(Omega), where - Delta (p)u = -div(\ delu \ (p-2)delu), and 0 < m < p-1 < q < +infinity, Omega is a bounded domain in R-N (N greater than or equal to 3). A is a positive number. Object is to estimate exactly the magnitute of lambda* such that (1)(lambda) has at least one positive solution if lambda is an element of (0, lambda*) and no positive solutions if lambda > lambda*. Furthermore, (1)(lambda) has at least one positive solution when lambda = lambda* and at least two positive solutions when lambda is an element of (0, lambda*) and q less than or equal to N-p/N-p - 1. Finally, the authors obtain a multiplicity result with positive energy of (1)(lambda) when 0 < m < p-1 < q = N-p/N-p -1.