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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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