BOUNDED STATE SOLUTIONS OF KIRCHHOFF TYPE PROBLEMS WITH A CRITICAL EXPONENT IN HIGH DIMENSION

In the present paper, we consider the following Kirchhoff type problem {-(a+lambda integral(N)(R)vertical bar del u vertical bar(2)dx)Delta u + V(x)u = vertical bar u vertical bar(2)*(-2)u in R-N, u is an element of D-1,D-2(R-N), where a is a positive constant, lambda is a positive parameter, V is an element of L-N/2 (R-N) is a given nonnegative function and 2* is the critical exponent. The existence of bounded state solutions for Kirchhoff type problem with critical exponents in the whole RN (N >= 5) has never been considered so far. We obtain sufficient conditions on the existence of bounded state solutions in high dimension N >= 4, and especially it is the fist time to consider the case when N >= 5 in the literature.