Construction of bubbling solutions of the Brezis–Nirenberg problem in general bounded domains (I): The dimensions 4 and 5

In this paper, we consider the Brezis–Nirenberg problem (Formula presented.) where (Formula presented.), (Formula presented.) is a bounded domain with smooth boundary (Formula presented.) and (Formula presented.). We prove that every eigenvalue of the Laplacian operator (Formula presented.) with the Dirichlet boundary is a concentration value of the Brezis–Nirenberg problem in dimensions (Formula presented.) and (Formula presented.) by constructing bubbling solutions with precisely asymptotic profiles via the Lyapunov–Schmidt reduction arguments. Our results suggest that the bubbling phenomenon of the Brezis–Nirenberg problem in dimensions (Formula presented.) and (Formula presented.) as the parameter (Formula presented.) is close to the eigenvalues are governed by crucial functions related to the eigenfunctions, which has not been observed yet in the literature to our best knowledge. Moreover, as the parameter (Formula presented.) is close to the eigenvalues, there are arbitrary number of multibump bubbling solutions in dimension (Formula presented.) while, there are only finitely many number of multibump bubbling solutions in dimension (Formula presented.), which are also new findings to our best knowledge.