## 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

### Citations:

Zbl 0717.35032; Zbl 0541.35029
Full Text:

### References:

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