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The role played by space dimension in elliptic critical problems. (English) Zbl 0938.35058

Weak \(H_{0}^{1}(\Omega)\) solutions of the nonlinear critical second order elliptic problem \[ \begin{cases}-\Delta u-\mu u/|x|^{2}=u^{2^{\ast }-1}+\lambda u & \text{in \(\Omega \)},\\ u>0 & \text{in }\Omega,\\ u=0 & \text{on }\partial \Omega\end{cases} \] are considered, where \(\Omega \) is a smooth bounded domain in \(\mathbb{R}^{N}\), \(N\geq 3\), such that \(0\in \Omega \). Here \(2^{\ast }=2N/(N-2)\) is the so-called critical exponent, \(\mu \) is a real parameter and \(\lambda >0.\) One can say that a dimension \(N\) is critical for the problem (1) if there exists a smooth domain in \(\mathbb{R}^{N}\) in which (1) has no solutions for some \(\lambda \in (0,\lambda _{1}),\) where \(\lambda _{1}\) is the first eigenvalue of \(-\Delta \) in \(\Omega \) [see for instance P. Pucci and J. Serrin, J. Math. Pures Appl. 69, 55-83 (1990; Zbl 0717.35032)]. In H. Brezis and L. Nirenberg [Commun. Pure Appl. Math. 36, 437-477 (1983; Zbl 0541.35029)], was already proved that the dimension \(N\geq 4\) is not critical, and \(N=3\) is critical for (1) with \(\mu =0.\) In the paper, these results are generalized for the case \(\mu \neq 0.\) Moreover, it is shown that any fixed dimension \(N\geq 3\) may be critical or not, as follows: if \(\mu \leq \overline{\mu }-1\) then \(N\) is not critical and if \(\overline{\mu }-1<\mu <\overline{\mu }\) then \(N\) is critical, where \(\overline{\mu }=(N-2)^{2}/4.\)

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
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