1,721,006 research outputs found
Uniqueness results for higher order Lane-Emden systems
No full textIn this paper we develop a Gidas–Ni–Nirenberg technique for polyharmonic equations and systems of Lane-Emden type. As far as we are concerned with Dirichlet boundary conditions, we prove uniqueness of solutions up to eighth order equations, namely which involve the fourth iteration of the Laplace operator. Then, we can extend the result to arbitrary polyharmonic operators of any order, provided some natural boundary conditions are satisfied but not for Dirichlet's: the obstruction is apparently a new phenomenon and seems due to some loss of information. When the polyharmonic operator turns out to be a power of the Laplacian, and this is the case of Navier's boundary conditions, as byproduct uniqueness of solutions holds in a fairly general context. New existence results for systems are also established
Blow-up phenomena and asymptotic profiles passing from h1-critical to super-critical quasilinear Schrödinger equations
Get PDFWe study the asymptotic profile, as h → 0, of positive solutions to where γ≥0 is a parameter with relevant physical interpretations, V and K are given potentials and the dimension N is greater than or equal to 5, as we look for finite L2-energy solutions. We investigate the concentrating behavior of solutions when γ>0 and, differently from the case γ=0 where the leading potential is V, the concentration is here localized by the source potential K. Moreover, surprisingly for γ>0 we find a different concentration behavior of solutions in the case p=2NN-2 and when 2NN-24 NN-2. This phenomenon does not occur when γ=0
Global vs Blow-Up Solutions and Optimal Threshold for Hyperbolic ODEs with Possibly Singular Nonlinearities
Get PDFWe consider a hyperbolic ordinary differential equation perturbed by a nonlinearity which can be singular at a point and in particular this includes MEMS type equations. We first study qualitative properties of the solution to the stationary problem. Then, for small value of the perturbation parameter as well as initial value, we establish the existence of a global solution by means of the Lyapunov function and we show that the omega limit set consists of a solution to the stationary problem. For strong perturbations or large initial values, we show that the solution blows up. Finally, we discuss the relationship between upper bounds of the perturbation parameter for the existence of time-dependent and stationary solutions, for which we establish an optimal threshold
Asymptotic Behavior of Ground States and Local Uniqueness for Fractional Schrodinger Equations with Nearly Critical Growth
Get PDFWe study quantitative aspects and concentration phenomena for ground states of the following nonlocal Schrodinger equation(-Delta)(s)u + V (x)u = u(2* s -1-epsilon) in R-N,where epsilon > 0, s is an element of (0, 1), 2*(s) := 2N/N-2s and N > 4s, as we deal with finite energy solutions. We show that the ground state u blows u(epsilon) and precisely with the following rate parallel to u(epsilon)parallel to(L infinity (RN)) similar to epsilon-(N-2s/4s), as epsilon -> 0(+). We also localize the concentration points and, in the case of radial potentials V, we prove local uniqueness of sequences of ground states which exhibit a concentrating behavior
Schrödinger–Newton equations in dimension two via a Pohozaev–Trudinger log-weighted inequality
Get PDFWe study the following Choquard type equation in the whole plane (C)-Δu+V(x)u=(I2∗F(x,u))f(x,u),x∈R2where I2 is the Newton logarithmic kernel, V is a bounded Schrödinger potential and the nonlinearity f(x, u), whose primitive in u vanishing at zero is F(x, u), exhibits the highest possible growth which is of exponential type. The competition between the logarithmic kernel and the exponential nonlinearity demands for new tools. A proper function space setting is provided by a new weighted version of the Pohozaev–Trudinger inequality which enables us to prove the existence of variational, in particular finite energy solutions to (C)
Hardy-Rellich inequalities with boundary remainder terms and applications
No full textWe prove a family of Hardy-Rellich inequalities with optimal constants and additional boundary terms. These inequalities are used to study the behavior of extremal solutions to biharmonic Gelfand-type equations under Steklov boundary conditions
Bose fluids and positive solutions to weakly coupled systems with critical growth in dimension two
No full textWe prove, using variational methods, the existence in dimension two of positive vector ground states solutions for the Bose-Einstein type systems. The nonlinear interaction between two Bose fluids is assumed to be of critical exponential type in the sense of J. Moser. For ‘small’ solutions the system is asymptotically equivalent to the corresponding one in higher dimensions with power-like nonlinearities
Maximum principle for higher order elliptic operators with inertia in general domains and any dimension
No full textIt is well known how the Maximum Principle (MP) in general fails to hold for uniformly elliptic operators of order higher than two, even in smooth convex domains. In D. Cassani and A. Tarsia (2022) it was shown in dimension N = 2, 3, by establishing a new Harnack type inequality, that the validity of the positivity preserving property can be restored when lower order derivatives are taken into account as a perturbation of the higher order differential operator. The restriction to the dimension was due to regularity issues which we develop here, extending the validity of the MP to any dimension and fairly general domains. Moreover, we show that the presence of inertial terms affects the range of the perturbation parameter, providing a balance between the positivity restoring effect of lower order derivatives and the mass energy. The method provided here is flexible with respect to the form of differential operators involved and thus suitable to be further extended to other classes of operators than just elliptic
Nonlocal Schrödinger-Poisson systems in RN: the fractional Sobolev limiting case
Get PDFWe study the existence of positive solutions for nonlocal systems in gradient form and set in the whole RN. A quasilinear fractional Schr ̈odinger equation, where the leading operator is the N s - fractional Laplacian, is coupled with a higher-order and possibly fractional Poisson equation. For both operators the dimension N ≥ 2 corresponds to the limiting case of the Sobolev embedding, hence we consider nonlinearities with exponential growth. Since standard variational tools cannot be applied due to the sign-changing logarithmic Riesz kernel of the Poisson equation, we employ a variational approximating procedure for an auxiliary Choquard equation, where the Riesz kernel is uniformly approximated by polynomial kernels. Qualitative properties of solutions such as symmetry, regularity and decay are also established. Our results extend and complete the analysis carried out in the planar case in [13
Su alcune nuove prove speditive per rilevare le condizioni di superficie della pista P1 dell’aeroporto di Malpensa
No full textQuaderni Tecnico-scientifici del Laboratorio Stradale del Dipartimento di Sistemi di Trasporto e Movimentazione, Politecnico di Milano, 2/99, Ital
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