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Uniqueness of positive solutions of a nonlinear elliptic equation involving the critical exponent. (English) Zbl 0797.35048
We consider the problem $(P_ \varepsilon)\qquad \qquad -\Delta u = N(N-2)u^ p + \varepsilon u,\;u>0 \text{ in } \Omega,\;u = 0 \text{ on }\partial \Omega,$ where $$\Omega$$ is a bounded domain in $$\mathbb{R}^ N$$ (with $$N\geq 5)$$ with a smooth boundary $$\partial\Omega$$, $$\varepsilon>0$$ is a parameter which will be small and $$p$$ the critical exponent, that is $$p=(N+2)/(N-2)$$. We are interested in the limiting behavior of solutions $$u_ \varepsilon$$ of problem $$(P_ \varepsilon)$$ as $$\varepsilon\to 0$$ when problem $$(P_ 0)$$ has no solution.

##### 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 35B40 Asymptotic behavior of solutions to PDEs
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##### References:
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