Abstract
We study the following Brezis–Nirenberg type critical exponent equation which is related to the Yamabe problem:
where Ω is a smooth bounded domain in \({{\mathbb R}^N(N\ge3)}\) and 2* is the critical Sobolev exponent. We show that, if N ≥ 5, this problem has at least \({\lceil\frac{N+1}{2}\rceil}\) pairs of nontrivial solutions for each fixed λ ≥ λ1, where λ1 is the first eigenvalue of −Δ with the Dirichlet boundary condition. For N ≥ 3, we give energy estimates from below for ground state solutions.
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Z. Chen and W. Zou are partially supported by NSFC (10871109); N. Shioji is partially supported by the Grant-in-Aid for Scientific Research (C) (No. 21540214) from Japan Society for the Promotion of Science.
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Chen, Z., Shioji, N. & Zou, W. Ground state and multiple solutions for a critical exponent problem. Nonlinear Differ. Equ. Appl. 19, 253–277 (2012). https://doi.org/10.1007/s00030-011-0127-0
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DOI: https://doi.org/10.1007/s00030-011-0127-0