Asymptotic analysis for a singularly perturbed Dirichlet problem

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.