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In this paper, the fractional problem of the Ambrosetti-Prodi type involving the critical Sobolev exponent is taken into account in a bounded domain of R N { A α u = u + 2 α ⁎ − 1 + λ u − s ¯ φ 1 , u > 0 , in Ω , u = 0 , on ∂ Ω , where A α is the spectral fractional operator, λ and s ¯ are real numbers, Ω ⊂ R N is bounded, 2 α ⁎ = 2 N N − 2 α is a critical exponent, 0 < α < 1 , φ 1 is the first eigenfunction of −Δ with zero Dirichlet boundary condition. We will construct bubbling solutions when the parameter is large enough, and the location of the bubbling point is near the boundary of the domain. [ABSTRACT FROM AUTHOR]

In this paper, we consider the multi-species nonlinear Schrödinger systems in R N : { − Δ u j + V j (x) u j = μ j u j 3 + ∑ i = 1 ; i ≠ j d β i , j u i 2 u j in R N , u j (x) > 0 in R N , u j (x) → 0 as | x | → + ∞ , j = 1 , 2 , ⋯ , d , where N = 2 , 3 , μ j > 0 are constants, β i , j = β j , i ≠ 0 are coupling parameters, d ≥ 2 and V j (x) are potentials. By Ljapunov-Schmidt reduction arguments, we construct infinitely many nonradial positive solutions of the above system under some mild assumptions on potentials V j (x) and coupling parameters { β i , j } , without any symmetric assumptions on the limit case of the above system. Our result, giving a positive answer to the conjecture in Pistoia and Viara [50] and extending the results in [50,52] , reveals new phenomenon in the case of N = 2 and d = 2 and is almost optimal for the coupling parameters { β i , j }. [ABSTRACT FROM AUTHOR]

This paper aims to give an affirmative answer to a conjecture raised by Soave (2020) [25] and considers the qualitative properties of normalized solutions to Sobolev critical/subcritical Schrödinger equations with combined nonlinearities. Precisely, we establish the mass threshold a ¯ such that the mountain pass type normalized solution exists for the Sobolev critical/subcritical Schrödinger equation with combined mass critical and mass supercritical nonlinearities. We then show that a ¯ is also a threshold of the limit behavior of the mountain pass type normalized solution of the Schrödinger equation with combined nonlinearities as the exponent of lower order term tending to the mass critical exponent. Among which, the results in the case that the mass small than the threshold a ¯ give an affirmative answer to the conjecture raised by Soave. [ABSTRACT FROM AUTHOR]

In this paper we give elementary proofs of energy conservation for weak solutions to the Euler and Navier-Stokes equations in the class of Hölder continuous functions, relaxing some of the assumptions on the time variable (both integrability and regularity at initial time) and presenting them in a unified way. Then, in the final section we prove (for the Navier-Stokes equations) a result of energy conservation in presence of a solid boundary and with Dirichlet boundary conditions. This result seems the first one –in the viscous case– with Hölder type hypotheses, but without additional assumptions on the pressure. [ABSTRACT FROM AUTHOR]

We consider nonautonomous master equations of finite-state, continuous-time Markovian jump processes with uniformly continuous and bounded transition matrix functions. The Earnshaw–Keener conjecture states that if the omega-limit set of the transition matrix function contains at least one matrix which is neither decomposable nor splitting, then the difference of any two probability distribution solutions tends to zero at infinity. The conjecture has been confirmed under the additional assumption that the transition matrix function is almost-automorphic. In this paper, we prove the conjecture in its full generality. [ABSTRACT FROM AUTHOR]

Bernstein problem for affine maximal type equation (0.1) u i j D i j w = 0 , w ≡ det ⁡ D 2 u − θ , ∀ x ∈ Ω ⊂ R N has been a core problem in affine geometry. A conjecture proposed firstly by Chern (1977) [6] for entire graph and then extended by Trudinger-Wang (2000) [14] to its fully generality asserts that any Euclidean complete, affine maximal type, locally uniformly convex C 4 -hypersurface in R N + 1 must be an elliptic paraboloid. At the same time, this conjecture was solved completely by Trudinger-Wang for dimension N = 2 and θ = 3 / 4 , and later extended by Jia-Li (2009) [12] to N = 2 , θ ∈ (3 / 4 , 1 (see also Zhou (2012) [16] for a different proof). On the past twenty years, many efforts were done toward higher dimensional issues but not really successful yet, even for the case of dimension N = 3. In this paper, we will construct non-quadratic affine maximal type hypersurfaces which are Euclidean compete for N ≥ 3 , θ ∈ (1 / 2 , (N − 1) / N). [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to prove Artés–Llibre–Valls's conjectures on the uniqueness of limit cycles for the Higgins–Selkov system and the Selkov system. In order to apply the limit cycle theory for Liénard systems, we change the Higgins–Selkov and the Selkov systems into Liénard systems first. Then, we present two theorems on the nonexistence of limit cycles of Liénard systems. At last, the conjectures can be proven by these theorems and some techniques applied for Liénard systems. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we prove that Wright's equation y ′ (t) = − α y (t − 1) { 1 + y (t) } has a unique slowly oscillating periodic solution for parameter values α ∈ (π 2 , 1.9 ] , up to time translation. This result proves Jones' Conjecture formulated in 1962, that there is a unique slowly oscillating periodic orbit for all α > π 2. Furthermore, there are no isolas of periodic solutions to Wright's equation; all periodic orbits arise from Hopf bifurcations. [ABSTRACT FROM AUTHOR]

We consider initial-boundary value problems for a quasi linear parabolic equation, k t = k 2 ( k θ θ + k ) , with zero Dirichlet boundary conditions and positive initial data. It has known that each of solutions blows up at a finite time with the rate faster than ( T − t ) − 1 . In this paper, it is proved that sup θ ⁡ k ( θ , t ) ≈ ( T − t ) − 1 log ⁡ log ⁡ ( T − t ) − 1 as t ↗ T under some assumptions. Our strategy is based on analysis for curve shortening flows that with self-crossing brought by S.B. Angenent and J.J.L. Velázquez. In addition, we prove some of numerical conjectures by Watterson which are keys to provide the blow-up rate. [ABSTRACT FROM AUTHOR]

This paper settles a conjecture by Gazzola and Pavani [10] regarding solutions to the fourth order ODE w ( 4 ) + k w ″ + f ( w ) = 0 which arises in models of traveling waves in suspension bridges when k > 0 . Under suitable assumptions on the nonlinearity f and initial data, we demonstrate blow-up in finite time. The case k ≤ 0 was first investigated by Gazzola et al., and it is also handled here with a proof that requires less differentiability on f . Our approach is inspired by Gazzola et al. and exhibits the oscillatory mechanism underlying the finite-time blow-up. This blow-up is nonmonotone, with solutions oscillating to higher amplitudes over shrinking time intervals. In the context of bridge dynamics this phenomenon appears to be a consequence of mutually-amplifying interactions between vertical displacements and torsional oscillations. [ABSTRACT FROM AUTHOR]

Abstract: We consider the Hardy–Hénon equation with and and we are concerned in particular with the Liouville property, i.e. the nonexistence of positive solutions in the whole space . It has been conjectured that this property is true if (and only if) , where is the Hardy–Sobolev exponent, given by . However, when , the conjecture had up to now been proved only for . Indeed the case seems more difficult, due to . In this paper, we prove the conjecture for in dimension , in the case of bounded solutions. Next, for the conjecture in the case , and for related estimates near isolated singularities and at infinity, we give new proofs – based in particular on doubling-rescaling arguments – and we provide some extensions of these estimates. These proofs are significantly simpler than the previously known ones. Finally, we clarify some of the previous results on a priori estimates for the related Dirichlet problem. [Copyright &y& Elsevier]

Abstract: Based on geometric singular perturbation theory we prove the existence of classical Liénard equations of degree 6 having 4 limit cycles. It implies the existence of classical Liénard equations of degree , having at least limit cycles. This contradicts the conjecture from Lins, de Melo and Pugh formulated in 1976, where an upperbound of limit cycles was predicted. This paper improves the counterexample from Dumortier, Panazzolo and Roussarie (2007) by supplying one additional limit cycle from degree 7 on, and by finding a counterexample of degree 6. We also give a precise system of degree 6 for which we provide strong numerical evidence that it has at least 3 limit cycles. [ABSTRACT FROM AUTHOR]

Abstract: Let be the Cauchy–Riemann operator and f be a real-valued function in a neighborhood of 0 in in which for all . In such cases, is known as a Loewner vector field due to its connection with Loewner''s conjecture that the index of such a vector field is bounded above by n. The case of Loewner''s conjecture implies Carathéodory''s conjecture that any -immersion of into must have at least two umbilics. Recent work of F. Xavier produced a formula for computing the index of Loewner vector fields when using data about the Hessian of f. In this paper, we extend this result and establish an index formula for for all . Structurally, our index formula provides a defect term, which contains geometric data extracted from Hessian-like objects associated with higher order derivatives of f. [Copyright &y& Elsevier]

Abstract: In the paper I study the gradient field of a harmonic function f in in a neighborhood of a critical point 0. I show that the flow of ∇f, as a mapping between level sets of f, is a stratified mapping – that gives, in our case, an answer to the problem of stratifying the space of orbits of the field ∇f posed by R. Thom. I also show that the trajectories of ∇f having 0 as a limit point satisfy the finiteness conjecture and have generalized tangents at 0. [Copyright &y& Elsevier]