Sign-changing solutions of the Brezis-Nirenberg problem : asymptotics and existence results

Abstract

In this PhD thesis we show some recent results about sign-changing solutions for the Brezis{ Nirenberg problem _􀀀_u = _u + juj2_􀀀1u in u = 0; on @; (0.1) where is a bounded smooth domain of RN, N _ 3, _ is a positive parameter, and 2_ = 2N N􀀀2 is the critical Sobolev exponent for the embedding of H1 0 () into Lp(). In the _rst part we analyze the asymptotic behavior of least-energy radial sign-changing solu-tions in the ball for N _ 7, as _ ! 0, and prove that their positive and negative part concentrate and blow up (with di_erent concentration speeds) at the same point, which is the center of the ball. This provides the _rst existence result of sign-changing bubble-tower solutions for the Brezis{Nirenberg problem. For the lower dimensions N = 4; 5; 6 we analyze the asymptotic behavior of radial sign- changing solutions (with two nodal regions) as _ goes to some strictly positive limit value ob- tained by studying the associated ordinary di_erential equation. We prove that the positive part concentrate and blows-up at the center of the ball, and its limit pro_le is that of a standard bubble in RN. On the contrary, the negative part converges to zero, when N = 4; 5, and it converges to the unique positive radial solution of (0.1) in the ball, for _ = _0, when N = 6, where _0 2 (0; _1), being _1 the _rst eigenvalue of 􀀀_. In view of the results obtained in the radial case for N _ 7, by applying a variant of the Lyapunov-Schmidt reduction method, we prove that such sign-changing bubble tower solutions exist in symmetric bounded domains, as _ ! 0. On the other hand, for the low dimensions N = 4; 5; 6, by applying the Pohozaev's identity and _ne estimates, we prove that such solutions cannot exist for _ close to zero.