NORMALIZED SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH COMBINED NONLINEARITIES: THE SOBOLEV CRITICAL CASE

In this paper, we study the Kirchhoff equation with Sobolev critical exponent -(a + b integral(R3) |del u|(2)) Delta u = lambda u + mu|u|(q-2) mu + |u|(4) u in R-3 under the normalized constraint integral(R3) u(2) = c(2), where a, b, c > 0 are constants, lambda, mu is an element of R and 2 < q < 6. The number 2 + 8/3 behaves as the L-2 -critical exponent for the above problem. When mu > 0, we distinguish the problem into four cases: 2 < q < 2 + 4/3, q = 2 + 4/3, 2 + 4/3 < q < 2 + 8=3 and 2 + 8/3 <= q < 6, and prove the existence and multiplicity of normalized solutions under suitable assumptions on mu and c. The solution obtained is either a minimum (local or global) or a mountain pass solution. When mu <= 0, we establish the nonexistence of nonnegative normalized solutions. Finally, we investigate the asymptotic behavior of normalized solutions obtained above as mu -> 0(+) and as b -> 0(+) respectively.