Normalized solutions for Kirchhoff equations with Sobolev critical exponent and mixed nonlinearities

This paper focuses on the existence of normalized solutions for the following Kirchhoff equation: { -(a + b integral(3 )(R)|del u|(2)dx) Delta u + lambda u = u(5 )+ mu|u|(q-2)u, x is an element of R-3, integral(R)u(3 )(2)dx = c, where a, b, c > 0, mu is an element of R and 2 < q < 6, lambda is an element of R will arise as a Lagrange multiplier that is not a priori given. By using new analytical techniques, the paper establishes several existence results for the case mu > 0: (1) The existence of two solutions, one being a local minimizer and the other of mountain-pass type, under explicit conditions on c when 2 < q < 10/3. (2) The existence of a mountain-pass type solution under explicit conditions on c when 10/3 <= q < 14/3. (3) The existence of a ground state solution for all c > 0 when 14/3 <= q < 6. Furthermore, the paper presents the first non-existence result for the case mu <= 0 and 2 < q < 6. In particular, refined estimates of energy levels are proposed, suggesting a new threshold of compactness in the L-2-constraint. This study addresses an open problem for 2 < q < 10/3 and fills a gap in the case 10/3 <= q < 14/3. We believe that our approach can be applied to a broader range of nonlinear terms with Sobolev critical growth, and the underlying ideas have potential for future development and applicability.