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Asymptotic analysis for a singularly perturbed Dirichlet problem
http://hdl.handle.net/2307/600
<Title>Asymptotic analysis for a singularly perturbed Dirichlet problem</Title>
<Authors>Petralla, Maristella</Authors>
<Issue Date>2010-05-10</Issue Date>
<Abstract>Let us consider the problem −∆u + λV (x)u = up in Ω, u = 0 on ∂ Ω, where Ω is a smooth
bounded domain, p > 1, V is a positive potential and λ > 0. We are interested in the regime λ → +∞, which is equivalent to a singularly perturbed Dirichlet problem. It is known that
solutions u must blow up as λ → +∞, and we address here the asymptotic description of such
a blow up behavior. When the ”energy” is uniformly bounded, the behavior is well understood
and the solutions can develop just a ﬁnite number of sharp peaks. When V is not constant, the
blow up points must be c.p.’s of the potential V. The situation is more involved when V = 1,
and the crucial role is played by the mutual distances between the blow-up points as well as the
boundary distances. The construction of these blowing-up solutions has also been addressed.
The ﬁrst part in the thesis is devoted to strengthen such an analysis when just a Morse index
information is available. A posteriori, we obtain an equivalence in the form of a double-side
bound between Morse index and ”energy” with essentially optimal constants. This result can be
seen as a sort of Rozenblyum-Lieb-Cwikel inequality, where the number of negative eigenvalues
of a Schrodinger operator −∆ + V can be estimated in terms of a suitable Lebesgue norm of the
negative part V− . Thanks to the speciﬁcity of our problem, we improve it by getting the correct
Lebesgue exponent (in view of the double-side bound) as well as the sharp constants. We then
turn to the question of concentration on manifolds of positive dimensions. The problem is well
understood by a constructive approach but the asymptotic analysis is in general missing. Let
us notice that on the annulus the radial ground state solution has Morse index and ”energy”
which blow up as λ → +∞. Nonetheless, the radial Morse index is one which has allowed
Esposito-Mancini-Santra-Srikanth to develop a ﬁne asymptotic analysis to localize the limiting
concentration radii. They are c.p.’s of a modiﬁed potential, whose role had been already
clariﬁed by the constructive results. The second part part of the thesis is devoted to develop an
asymptotic analyis for solutions on the annulus which have partial symmetries. In particular,
we consider the three-dimensional annulus and solutions which are invariant under rotations
around the z-axis. Assuming an uniform bound on the reduced invariant Morse index, we obtain
a localization of the limiting concentration circles in terms of a suitable modiﬁed potential. The
main difficulty here is related to the presence of ﬁxed points w.r.t. the group action (the z-axis)
and the aim is to exhibit potentials V for which the concentration circles (for example, for the
ground state solution) do not degenerate to points on the z-axis.</Abstract>Sun, 09 May 2010 22:00:00 GMThttp://hdl.handle.net/2307/6002010-05-09T22:00:00Z