EXISTENCE AND MULTIPLICITY OF POSITIVE SOLUTIONS OF A CRITICAL KIRCHHOFF TYPE ELLIPTIC PROBLEM IN DIMENSION FOUR

In this paper, we study a Kirchhoff type elliptic problem, {-(1+alpha integral(Omega)vertical bar del u vertical bar(2)dx) Delta u = lambda u(q) + u(3), u > 0 in Omega, u = 0 on partial derivative Omega, where Omega subset of R-4 is a bounded domain with smooth boundary partial derivative Omega and we assume alpha, lambda > 0 and 1 <= q < 3. For q = 1, we prove the existence of possibly multiple solutions for alpha > 0 and lambda >= lambda(1) in suitable intervals, where lambda(1) > 0 is the first eigenvalue of -Delta on Omega. On the other hand for q is an element of (1, 3), we show the existence of a solution for all small alpha > 0 and all lambda > 0. We establish the results by the method of the Nehari manifold and the concentration compactness analysis for Palais-Smale sequences.