EXISTENCE OF MULTIPLE SOLUTIONS AND ESTIMATES OF EXTREMAL VALUES FOR A KIRCHHOFF TYPE PROBLEM WITH FAST INCREASING WEIGHT AND CRITICAL NONLINEARITY

In this article, we study the Kirchhoff type problem - (a +epsilon integral K-R3(x)vertical bar del u vertical bar(2)dx) div( K(x)del u) = lambda K(x) f (x) vertical bar u vertical bar(q-2)u + K(x)vertical bar u vertical bar(4)u, where x is an element of R-3, 1 < q < 2, K(x) = exp(vertical bar x vertical bar(alpha)/4) with alpha > 2, epsilon > 0 is small enough, and the parameters a, lambda > 0. Under some assumptions on f(x), we establish the existence of two nonnegative nontrivial solutions and obtain uniform lower estimates for extremal values of the problem via variational methods.