POSITIVE SOLUTIONS FOR p-LAPLACIAN EQUATIONS OF KIRCHHOFF TYPE PROBLEM WITH A PARAMETER

In this article, we consider the existence and non-existence of positive solutions for the Kirchhoff type equation - (a + lambda M (integral(Omega) vertical bar del u vertical bar(p) dx)) Delta(p)u = f(u), in Omega, u = 0, on partial derivative Omega, where Omega subset of R-N is a bounded domain with a smooth boundary partial derivative Omega, a is a positive constant, N >= 3, lambda >= 0, 2 <= p < N, M and f are positive continuous functions. Under some weak assumptions on f, we show that the above problem has at least one positive solution when lambda is small and has no nonzero solution when lambda is large. Our argument is based on iterative technique and variational methods.