NORMALIZED SOLUTIONS FOR KIRCHHOFF EQUATIONS WITH CHOQUARD NONLINEARITY

This paper is concerned with the qualitative analysis of solutions for the following Kirchhoff equation with mixed nonlinearities -(a + b integral(3)(R) |del u|(2)dx) Delta u -lambda u = mu|u|(q-2) + (I-infinity * |u|(p))|u|(p-2)u, x is an element of R-3 with prescribed mass integral(3)(R) |u|(2)dx = c(2), where a; b; c > 0, mu is an element of R, alpha is an element of ( 5/3, 3), 2 < q < 10/3, 14/3 < p <= 3 + alpha. We prove several existence results to the above problem, where mu is a positive parameter. Here, 3 + ff is the HardyLittlewood-Sobolev upper critical exponent, which can be seen as the Sobolev critical exponent 2*. The proof of the Palais-Smale condition is a challenge when p = 3+ alpha. So we present a different method of dealing with compactness compared to the general way handling the Sobolev critical term. Finally, while for the defocusing situation mu < 0, we prove an existence result by constructing a minimax characterization for the energy functional.