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Sign changing solutions to a Bahri-Coron's problem in pierced domains

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  • We consider the problem

    $-\Delta u= |u|^{\frac4{N-2}} u \mbox{ in } \Omega \setminus \{B(\xi_1,\varepsilon)\cup B(\xi_2,\varepsilon)\},$
    $ u = 0 \mbox{ on } \partial( \Omega \setminus \{B(\xi_1,\varepsilon)\cup B(\xi_2,\varepsilon)\}),$

    where $\Omega$ is a smooth bounded domain in $R^N$, $N\ge 3,$ $\xi_1,$ $\xi_2$ are different points in $\Omega$ and ε is a small positive parameter. We show that, for ε small enough, the equation has at least one pair of sign changing solutions, whose positive and negative parts concentrate at $\xi_1$ and $\xi_2$ as ε goes to zero.

    Mathematics Subject Classification: Primary: 35J60; Secondary: 35J25.


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