Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.57 (2025), 1st Issue

We provide a detailed proof that the Nonlinear Fermi Golden Rule coefficient that appears in our recent proof of the asymptotic stability of ground states for the pure power Nonlinear Schr¨odinger equations in R with exponent 0 < |p − 3| ≪ 1 is nonzero.

Let ΔG be a sublaplacian on a Carnot group, and let μ be a local measure on the open set Ω ⊂ G. If u ∈ L1l oc(Ω) is such that −ΔGu = μ, u ≥ 0 on Ω, then μc ≥ 0, where μc is the concentrated component of μ with respect to the G-capacity. This extends to the Carnot group setting a result contained in [9].

The paper concerns universal a priori estimates for positive solutions to a large class of elliptic quasilinear equations and systems involving the p-Laplacian operator on arbitrary domains of RN and a convective term in the reaction. Our main theorems, new even for the Laplacian operator, extend previous estimates by Pol´aˇcik, Quitter and Souplet in [38] to very general nonlinearities admitting solely a lower bound, yielding a curious dichotomy. The main ingredients are a key doubling property, a rescaling argument, different from the classical blow-up technique of Gidas and Spruck, and Liouville-type theorems for inequalities. A discussion on the sharpness of the exponent in the power type term is also included.

Let D ⊆ Rn, n ≥ 3, be a bounded open set and let x0 ∈ D. Assume that the Newtonian potential of D is proportional outside D to the Newtonian potential of a mass concentrated at {x0}. Then D is a Euclidean ball centered at x0. This Theorem, proved by Aharonov, Schiffer and Zalcman in 1981, was extended to the caloric setting by Suzuki and Watson in 2001. In this note, we show that Suzuki–Watson Theorem is a particular case of a more general rigidity result related to a class of Kolmogorov-type PDEs.

For the third order Galerkin approximation of the Navier- Stokes equations under Navier boundary conditions in a cube we prove global existence and qualitative behaviour of the solution. By modifying properly the signs of the resulting ODEs system and using the test function technique developed by Mitidieri-Pohoˇzaev we prove, instead, finite time blow-up.

We consider a class of perturbed (p, q)-eigenvalue problems. Using the Nehari method, we show that for all small values of the parameter λ > 0, the problem has at least three nontrivial bounded solutions all with sign information (positive, negative and nodal).

This research project considers the d-dimensional MagnetoHydroDynamics (MHD) system defined on a sufficiently smooth bounded domain, d = 2, 3 with homogeneous boundary conditions, and subject to external sources assumed to cause instability. The initial conditions for both fluid and magnetic equations are taken of low regularity. We then seek to uniformly stabilize such MHD system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of explicitly constructed, static, feedback controls, which are localized on an arbitrarily small interior subdomain. In addition, the actuators will be minimal in number. The resulting space of well-posedness and stabilization is a suitable product space eB2−2/p q,p (Ω) × eB2−2/p q,p (Ω), 1 < p < 2q 2q−1 , q > d, of tight Besov spaces for the fluid velocity component and the magnetic field component (each “close” to L3(Ω) for d = 3). It is known that such Besov space does not recognize compatibility conditions at the boundary, yet it provides a “minimal” level of regularity necessary to handle the nonlinear terms. In this paper we provide a solution of the first step: uniform stabilization of the linearized MHD. Showing maximal Lp-regularity up to T = ∞ for the feedback stabilized linearized system is critical for the analysis of well-posedness and stabilization of the feedback nonlinear problem. The solution of the nonlinear stabilization problem is to be given in a successive paper [29].

Via the global bifurcation theorem due to Rabinowitz, the paper shows bifurcation properties of the solutions of the following nonlinear Dirichlet problem, involving a double phase operator, that is ( −Δap u − νΔmu = λa(x)|u|m−2u + f(x, u) in Ω, u = 0 on ∂Ω, where 1 < m < p < N, p/m < 1 + 1/N and λ, ν ∈ R.

We classify completely the equivalence groups and the classical Lie point symmetry groups of generalized radial Liouville-Bratu- Gelfand problems.

We give a brief and concise guide for the analysis of the local behavior of the elements of local and nonlocal homogeneous De Giorgi classes: local boundedness, local H¨older continuity and Harnacktype inequalities. In the local case, we promote a simplified itinerary in the classic theory, propaedeutic for the successive part; while in the nonlocal case, we gather recent new developments into an unitary and concise framework. Employing a suitable definition of De Giorgi classes, we show a new proof of the Harnack inequality, way easier than in the local case, that bypasses any sort of Krylov-Safonov argument or cube decomposition.

We 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].

We establish the existence of multiple positive weak solutions of the quasilinear elliptic Neumann problem driven by the mean curvature operator ( −div 􀀀 ∇u/ p 1 + |∇u|2 _ = λw(x) |u|p−2u in Ω, −∇u ν/ p 1 + |∇u|2 = 0 on ∂Ω. Here, Ω is a bounded regular domain in RN, with N ≥ 2, p ∈ (1, 1∗), w is a sign-changing weight function, and λ > 0 is a parameter. Our findings provide the existence, for sufficiently small λ, of two positive solutions, the smaller in C1(Ω), the larger in BV (Ω), which respectively bifurcate from (λ, u) = (0, 0) and from (λ, u) = (0,+∞). This way we extend to a genuine PDE setting some results obtained in [22, 23] for the corresponding one-dimensional problem.

In this paper, we prove existence results for quasilinear parabolic bilateral variational inequalities of the form: Find u ∈ K ⊂ X with u(・, 0) = 0 satisfying 0 ∈ ut − Δpu + aF(u) + ∂IK(u) in X∗ in the unbounded cylindrical domain Q = RN × (0, τ ), where Δp is the p-Laplacian acting on X = Lp(0, τ ;D1,p(RN)) with its dual space X∗, and with D1,p(RN) denoting the Beppo-Levi space (or homogeneous Sobolev space). The bilateral constraint is represented by the closed convex set K ⊂ X given by K = {v ∈ X : ϕ(x, t) ≤ v(x, t) ≤ ψ(x, t) for a.a. (x, t) ∈ Q} and IK is the indicator function related to K with ∂IK denoting its subdifferential in the sense of convex analysis. The main goal and the novelty of this paper is to prove existence and directedness results without assuming coercivity conditions on the operator −Δp + aF : X → X∗, and without supposing the existence of sub- and supersolutions. Moreover, additional difficulties we are faced with arise due to the lack of compact embedding of D1,p(RN) into Lebesgue spaces Lσ(RN), and the fact that the domain K of ∂IK has empty interior, which prevents us to use recent results on evolutionary variational inequality. Instead our approach is based on an appropriately designed penalty technique and the use of weighted Lebesgue spaces as well as pseudomontone operator theory.

We prove advanced regularity results for solutions to a sixth order equation arising in the mechanical Kirchhoff-Love’s type model of the static equilibrium of a nanoplate in bending. Such regularity properties play a crucial role in the treatment, among others, of the inverse problem consisting in the determination of the Winkler coefficient of a nanoplate.

By using mountain pass arguments and a novel grouptheoretical approach, in this paper we study the existence of multiple sequences of nodal solutions with prescribed different symmetries for a wide class of (sub)critical elliptic problems settled on the unit sphere (Sd, h), of dimension d ≥ 5, whose simple prototype is given by the celebrated Yamabe equation on the sphere.

The first aim of this paper is to discuss some of the contents of Hutson et al. [15] versus the contents of a well known paper of Y. Lou, [20], as many experts are attributing, incorrectly, to Lou [20] some of the pioneering findings of Hutson et al. [15], published 11 years before. The second aim is contextualize the most pioneering results versus the most recent ones by the team of the author. Finally, some new multiplicity and uniqueness results are given for a symmetric diffusive competition model.