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Nodal bubble-tower solutions to radial elliptic problems near criticality

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  • We describe as $\varepsilon \to 0$ radially symmetric sign-changing solutions to the problem

    $ -\Delta u =|u|^{\frac 4{N-2} -\varepsilon} u \quad \text{in } B $

    where $B$ is the unit ball in $\R^N$, $N\ge 3$, under zero Dirichlet boundary conditions. We construct radial solutions with $k$ nodal regions which resemble a superposition of "bubbles'' of different signs and blow-up orders, concentrating around the origin. A dual phenomenon is described for the slightly supercritical problem

    $ -\Delta u =|u|^{\frac 4{N-2} +\varepsilon} u \quad \text{in } \R^N \setminus B $

    under Dirichlet and fast vanishing-at-infinity conditions.

    Mathematics Subject Classification: Primary: 35J25, 35J20; Secondary: 35B33.


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