We consider the Brezis-Nirenberg problem 1u = u + |u|p1u in , u = 0 on @, where is a smooth bounded domain in RN, N 3, p = NN+22 and > 0. We prove that, if is symmetric and N = 4, 5, there exists a sign-changing solution whose positive part concentrates and blowsup at the center of symmetry of the domain, while the negative part vanishes, as ! 1, where 1 = 1() denotes the first eigenvalue of 1 on , with zero Dirichlet boundary condition.
Sign-changing blowing-up solutions for the Brezis–Nirenberg problem in dimensions four and five
Iacopetti A.;
2018-01-01
Abstract
We consider the Brezis-Nirenberg problem 1u = u + |u|p1u in , u = 0 on @, where is a smooth bounded domain in RN, N 3, p = NN+22 and > 0. We prove that, if is symmetric and N = 4, 5, there exists a sign-changing solution whose positive part concentrates and blowsup at the center of symmetry of the domain, while the negative part vanishes, as ! 1, where 1 = 1() denotes the first eigenvalue of 1 on , with zero Dirichlet boundary condition.
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