3,356 research outputs found
Asymptotic properties of an optimal principal Dirichlet eigenvalue arising in population dynamics
No full textWe consider a shape optimization problem related to the persistence threshold for a biological species, the unknown shape corresponding to the zone of the habitat which is favorable to the population. Analytically, this translates in the minimization of a weighted eigenvalue of the Dirichlet Laplacian, with respect to a bang-bang indefinite weight. For such problem, we provide a full description of the singularly perturbed regime in which the volume of the favorable zone vanishes, with particular attention to the interplay between its location and shape. First, we show that the optimal favorable zone shrinks to a connected, nearly spherical set, in C1,1 sense, which aims at maximizing its distance from the lethal boundary. Secondly, we show that the spherical asymmetry of the optimal favorable zone decays exponentially, with respect to a negative power of its volume, in the C1,α sense, for every α<1. This latter property is based on sharp quantitative asymmetry estimates for the optimization of a weighted eigenvalue problem on the full space, of independent interest
Asymptotic properties of an optimal principal eigenvalue with spherical weight and Dirichlet boundary conditions
No full textWe consider a weighted eigenvalue problem for the Dirichlet laplacian in a smooth bounded domain Ω⊂RN, where the bang–bang weight equals a positive constant m ̄ on a ball B⊂Ω and a negative constant −m̲ on Ω∖B. The corresponding positive principal eigenvalue provides a threshold to detect persistence/extinction of a species whose evolution is described by the heterogeneous Fisher–KPP equation in population dynamics. In particular, we study the minimization of such eigenvalue with respect to the position of B in Ω. We provide sharp asymptotic expansions of the optimal eigenpair in the singularly perturbed regime in which the volume of B vanishes. We deduce that, up to subsequences, the optimal ball concentrates at a point maximizing the distance from ∂Ω
FRACTIONAL DIFFUSION WITH NEUMANN BOUNDARY CONDITIONS: THE LOGISTIC EQUATION
No full textMotivated by experimental studies on the anomalous diffusion of biological populations, we study the spectral square root of the Laplacian in bounded domains with Neumann homogeneous boundary conditions. Such op- erator arises in the continuous limit for long jumps random walks with reflecting barriers. Existence and uniqueness results for positive solutions are proved in the case of indefinite nonlinearities of logistic type by means of bifurcation theory
Normalized solutions to mass supercritical Schrödinger equations with negative potential
No full textWe study the existence of positive solutions with prescribed L2-norm for the mass supercritical Schrödinger equation −Δu+λu−V(x)u=|u|^(p−2)uu∈H^1(R^N),λ∈R, where V≥0, N≥1 and [Formula presented], [Formula presented] if N≥3 and 2⁎:=+∞ if N=1,2. We treat two cases. Firstly, under an explicit smallness assumption on V and no condition on the mass, we prove the existence of a mountain pass solution at positive energy level, and we exclude the existence of solutions with negative energy. Secondly, requiring that the mass is smaller than some explicit bound, depending on V, and that V is not too small in a suitable sense, we find two solutions: a local minimizer with negative energy, and a mountain pass solution with positive energy. Moreover, a nonexistence result is proved
Partially concentrating standing waves for weakly coupled Schrödinger systems
No full textWe study the existence of standing waves for the following weakly coupled system of two Schrodinger equations where V 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} and V 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} are radial potentials bounded from below. We address the case , m 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} constant, and prove the existence of a standing wave solution with both nontrivial components satisfying a prescribed asymptotic profile. In particular, the second component of such solution exhibits a concentrating behavior, while the first one keeps a quantum nature
Asymptotic spherical shapes in some spectral optimization problems
No full textWe study the optimization of the positive principal eigenvalue of an indefinite weighted problem, associated with the Neumann Laplacian in a box Ω⊂RN, which arises in the investigation of the survival threshold in population dynamics. When trying to minimize such eigenvalue with respect to the weight, one is led to consider a shape optimization problem, which is known to admit no spherical optimal shapes (despite some previously stated conjectures). We investigate whether spherical shapes can be recovered in some singular perturbation limit. More precisely we show that, whenever the negative part of the weight diverges, the above shape optimization problem approaches in the limit the so called spectral drop problem, which involves the minimization of the first eigenvalue of the mixed Dirichlet-Neumann Laplacian. Using α-symmetrization techniques on cones, we prove that, for suitable choices of the box Ω, the optimal shapes for this second problem are indeed spherical. Moreover, for general Ω, we show that small volume spectral drops are asymptotically spherical, centered near points of ∂Ω having largest mean curvature
Normalized solutions for Sobolev critical Schrödinger equations on bounded domains
No full textWe study the existence and multiplicity of positive solutions with prescribed L2norm for the Sobolev critical Schróodinger equation on a bounded domain \Omega \subset \BbbRN, N \geq 3,-\DeltaU = \lambdaU + U2\ast-1, U \in H01(\Omega), \int\Omega U2 dx = \rho2, where 2\ast = N2-N2 . First, we consider a general bounded domain \Omega in dimension N \geq 3, with a restriction, only in dimension N = 3, involving its inradius and first Dirichlet eigenvalue. In this general case, we show the existence of a mountain pass solution on the L2-sphere for \rho belonging to a subset of positive measure of the interval (0, \rho\ast\ast) and for a suitable threshold \rho\ast\ast > 0. Next, assuming that \Omega is star-shaped, we extend the previous result to all values \rho \in (0, \rho\ast\ast). With respect to that of local minimizers, already known in the literature, the existence of mountain pass solutions in the Sobolev critical case is much more elusive. In particular, our proofs are based on the sharp analysis of the bounded Palais-Smale sequences, provided by a nonstandard adaptation of the Struwe monotonicity trick that we develop
Local minimizers in absence of ground states for the critical NLS energy on metric graphs
No full textWe consider the mass-critical non-linear Schrödinger equation on non-compact metric graphs. A quite complete description of the structure of the ground states, which correspond to global minimizers of the energy functional under a mass constraint, is provided by Adami, Serra and Tilli in [R. Adami, E. Serra and P. Tilli. Negative energy ground states for the L2-critical NLSE on metric graphs. Comm. Math. Phys. 352 (2017), 387-406.], where it is proved that existence and properties of ground states depend in a crucial way on both the value of the mass, and the topological properties of the underlying graph. In this paper we address cases when ground states do not exist and show that, under suitable assumptions, constrained local minimizers of the energy do exist. This result paves the way to the existence of stable solutions in the time-dependent equation in cases where the ground state energy level is not achieved
Multiplicity of solutions on a Nehari set in an invariant cone
No full textFor 1 < p < 2 and q large, we prove the existence of two positive, nonconstant, radial and radially nondecreasing solutions of the supercritical equation −∆pu + u^p−1 = u^q−1 under Neumann boundary conditions, in the unit ball of R^N. We use a variational approach in an invariant cone. We distinguish the two solutions upon their energy: one is a ground state inside a Nehari-type subset of the cone, the other is obtained via a mountain pass argument inside the Nehari set. As a byproduct of our proofs, we detect the limit profile of the low energy solution as q → ∞ and show that the constant solution 1 is a local minimizer on the Nehari set. This marks a strong difference with the case p ≥ 2
Ergodic mean field games: existence of local minimizers up to the Sobolev critical case
No full textWe investigate the existence of solutions to viscous ergodic Mean Field Games systems in bounded domains with Neumann boundary conditions and local, possibly aggregative couplings. In particular we exploit the associated variational structure and search for constrained minimizers of a suitable functional. Depending on the growth of the coupling, we detect the existence of global minimizers in the mass subcritical and critical case, and of local minimizers in the mass supercritical case, notably up to the Sobolev critical case
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