221 research outputs found
Radial Positive Solutions for p-Laplacian Supercritical Neumann Problems
No full textThis paper deals with existence and multiplicity of positive solutions for a quasilinear problem with Neumann boundary conditions. The problem is set in a ball and admits at least one constant non-zero solution; moreover, it involves a nonlinearity that can be supercritical in the sense of Sobolev embeddings. The main tools used are variational techniques and the shooting method for ODE's. These results are contained in A. Boscaggin, F. Colasuonno, B. Noris. Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions. ESAIM Control Optim. Calc. Var., DOI: 10.1051/cocv/2016064 (2017; F. Colasuonno, B. Noris. A p-Laplacian supercritical Neumann problem. Discrete Contin. Dyn. Syst., 37 (2017) 3025-3057
On the Aharonov–Bohm Operators with Varying Poles: The Boundary Behavior of Eigenvalues
No full textWe consider a magnetic Schrödinger operator with magnetic field concentrated at one point (the pole) of a domain and half integer circulation, and we focus on the behavior of Dirichlet eigenvalues as functions of the pole. Although the magnetic field vanishes almost everywhere, it is well known that it affects the operator at the spectral level (the Aharonov–Bohm effect, Phys Rev (2) 115:485–491, 1959). Moreover, the numerical computations performed in (Bonnaillie-Noël et al. Anal PDE 7(6):1365–1395, 2014; Noris and Terracini, Indiana Univ Math J 59(4):1361–1403, 2010) show a rather complex behavior of the eigenvalues as the pole varies in a planar domain. In this paper, in continuation of the analysis started in (Bonnaillie-Noël et al. Anal PDE 7(6):1365–1395, 2014; Noris and Terracini, Indiana Univ Math J 59(4):1361–1403, 2010), we analyze the relation between the variation of the eigenvalue and the nodal structure of the associated eigenfunctions. We deal with planar domains with Dirichlet boundary conditions and we focus on the case when the singular pole approaches the boundary of the domain: then, the operator loses its singular character and the k-th magnetic eigenvalue converges to that of the standard Laplacian. We can predict both the rate of convergence and whether the convergence happens from above or from below, in relation with the number of nodal lines of the k-th eigenfunction of the Laplacian. The proof relies on the variational characterization of eigenvalues, together with a detailed asymptotic analysis of the eigenfunctions, based on an Almgren-type frequency formula for magnetic eigenfunctions and on the blow-up technique.SCOPUS: ar.jinfo:eu-repo/semantics/publishe
Positive constrained minimizers for supercritical problems in the ball
No full textWe provide a sufficient condition for the existence of a positive solution to -Delta u + V(vertical bar x vertical bar)u = u(p) in B-1, when p is large enough. Here B-1 is the unit ball of R-n, n >= 2, and we deal with both Neumann and Dirichlet homogeneous boundary conditions. The solution turns out to be a constrained minimum of the associated energy functional. As an application we show that in case V(vertical bar x vertical bar) >= 0, V not equivalent to 0 is smooth and p is sufficiently large, and the Neumann problem always admits a solution
Nodal sets of magnetic Schroedinger operators of Aharonov-Bohm type and energy minimizing partitions
No full textA p-Laplacian supercritical Neumann problem
No full textFor p > 2, we consider the quasilinear equation -Δpu+SCOPUS: ar.jinfo:eu-repo/semantics/publishe
Asymptotics for a high-energy solution of a supercritical problem
No full textIn this paper we deal with the equation
for , under Neumann boundary conditions in the unit ball of
. We focus on the three positive, radial, and radially
non-decreasing solutions, whose existence for large is proved in [13]. We
detect the limit profile as of the higher energy solution and show
that, unlike the minimal energy one, it converges to the constant . The
proof requires several tools borrowed from the theory of minimization problems
and accurate a priori estimates of the solutions, which are of independent
interest.Comment: 14 pages, revised versio
Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition
No full textFor the positive solutions of the competitive Gross-Pitaevskii system of two equations, we prove that L^\infty boundedness implies uniform H\"older boundedness as the competition parameter goes to infinity. Moreover we prove that the limiting profile is Lipschitz continuous. The proof relies upon the blow-up technique and the monotonicity formulae by Almgren and Alt-Caffarelli-Friedman. This system arises in the Hartree-Fock approximation theory for binary mixtures of Bose-Einstein condensates in different hyperfine states. Extensions to systems with more than two densities are given
A priori bounds and multiplicity of positive solutions for p-Laplacian Neumann problems with sub-critical growth
No full textExistence and orbital stability of the ground states with prescribed mass for the L2-critical and supercritical NLS on bounded domains
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