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On a variational problem with lack of compactness: The topological effect of the critical points at infinity. (English) Zbl 0814.35032
Summary: We study the subcritical problems \((P_ \varepsilon)-\Delta u=u^{p- \varepsilon}\), \(u>0\) on \(\Omega\); \(u=0\) on \(\partial\Omega\), \(\Omega\) being a smooth and bounded domain in \(\mathbb{R}^ N\), \(N \geq 3\), \(p + 1 = 2N/(N-2)\) the critical Sobolev exponent and \(\varepsilon>0\) going to zero – in order to compute the difference of topology that the critical points at infinity induce between the level sets of the functional corresponding to the limit case \((P_ 0)\).

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35J20 Variational methods for second-order elliptic equations
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