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Solutions of semilinear elliptic equations in tubes

Abstract : Given a smooth compact k-dimensional manifold \Lambda embedded in $\mathbb {R}^m$, with m\geq 2 and 1\leq k\leq m-1, and given \epsilon>0, we define B_\epsilon (\Lambda) to be the geodesic tubular neighborhood of radius \epsilon about \Lambda. In this paper, we construct positive solutions of the semilinear elliptic equation \Delta u + u^p = 0 in B_\epsilon (\Lambda) with u = 0 on \partial B_\epsilon (\Lambda), when the parameter \epsilon is chosen small enough. In this equation, the exponent p satisfies either p > 1 when n:=m-k \leq 2 or p\in (1, \frac{n+2}{n-2}) when n>2. In particular p can be critical or supercritical in dimension m\geq 3. As \epsilon tends to zero, the solutions we construct have Morse index tending to infinity. Moreover, using a Pohozaev type argument, we prove that our result is sharp in the sense that there are no positive solutions for p>\frac{n+2}{n-2}, n\geq 3, if \epsilon is sufficiently small.
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Contributor : Carole Juppin <>
Submitted on : Thursday, August 15, 2013 - 12:24:48 AM
Last modification on : Thursday, March 5, 2020 - 6:26:00 PM

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Frank Pacard, Filomena Pacella, Berardino Sciunzi. Solutions of semilinear elliptic equations in tubes. 2013. ⟨hal-00851589⟩



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