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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
##### Keywords:
critical Sobolev exponent
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##### References:
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