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Existence of solutions for singular critical growth semilinear elliptic equations. (English) Zbl 0997.35017

Summary: A semilinear elliptic problem \[ \begin{matrix} {\l}\quad & {\l}\\ -\Delta u-\mu\;\frac{u}{|x|^2}=\lambda u+|u|^{2^*-2}u & \text{in }\Omega\\ u=0 & \text{on }\partial\Omega,\end{matrix} \] where \(\Omega\subset \mathbb{R}^n\) \((n\geq 3)\) is an open bounded domain with smooth boundary \(\partial\Omega\) and containing the origin 0, \(2^*=\frac{2n}{n-2}\) is the critical Sobolev exponent, \(0\leq \mu<\overline\mu=(n-2)^2/4\) and \(\lambda>0\), containing both a singularity and a critical growth term is considered. Existence results are obtained by variational methods. The solvability of the problem depends on the space dimension \(n\) and on the coefficient of the singularity; the results obtained describe the behavior of critical dimensions and nonresonant dimensions when the Brezis-Nirenberg problem is modified with a singular term.

MSC:

35J60 Nonlinear elliptic equations
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B35 Stability in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
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