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On the Brezis-Nirenberg problem in a ball. (English) Zbl 1265.35072

Summary: We study the following Brezis-Nirenberg type critical exponent problem: \(-\Delta u=\lambda u^q+u^{2^{*}-1}\) in \(B_R\), \(u>0\) in \(B_R\), \(u=0\) on \(\partial B_R\), where \(B_R\) is a ball with radius \(R\) in \(\mathbb {R}\) (\(N\geq 3\)), \(\lambda > 0\), \(1\leq q <2^{*}-1\) and \(2^{*}\) is the critical Sobolev exponent. We prove the uniqueness results of the least-energy solution when \(3\leq N\leq 5\) and \(1\leq q <2^{*}-1\). We give extremely accurate energy estimates of the least-energy solutions as \(R\rightarrow 0\) for \(N\geq 4\) and \(q=1\).

MSC:

35J60 Nonlinear elliptic equations
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
58E30 Variational principles in infinite-dimensional spaces
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