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Solutions for semilinear elliptic equations with critical exponents and Hardy potential. (English) Zbl 1154.35346

The authors deal with the following semilinear elliptic problem: \[ -\Delta u-\mu\frac{u}{|x|^2}=\lambda u+|u|^{2^*-2}\cdot u\quad\text{in}\quad\Omega\quad u=0\quad\text{on}\quad\partial\Omega,\tag{1} \] where \(\Omega \subset\mathbb R^N\) \((N\geq 5)\) is a bounded domain with smooth boundary \(\partial\Omega\), \(0\in\text{int}\,\Omega\), \(2^*= \frac{2N}{N-2}\), \(\lambda>0\) and \(0\leq\mu<\overline\mu=(\frac{N-2}{2})^2\). Under some additional assumption on \(\mu\), the authors show for all \(\lambda >0\) the existence of a nontrivial solution and study its critical level thereby giving an affirmative answer to an open problem in [A. Ferrero and F. Gazzola, J. Differ. Equ. 177, No. 2, 494–522 (2001; Zbl 0997.35017)]. To this end, the authors employ Palais-Smale arguments.

MSC:

35J60 Nonlinear elliptic equations
35B33 Critical exponents in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
47J30 Variational methods involving nonlinear operators

Citations:

Zbl 0997.35017
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References:

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