Narrow Results

Given (M,g) a smooth compact Riemannian manifold of dimension n ≥ 5, we investigate compactness for fourth order critical equations like P_gu = u^2♯-1, where is a Paneitz–Branson operator with constant coefficients b and c, u is required to be positive, and is critical from the Sobolev viewpoint. We prove that such equations are compact on locally conformally flat manifolds, unless b lies in some closed interval associated to the spectrum of the smooth symmetric (2,0)-tensor field involved in the definition of the geometric Paneitz–Branson operator.

For n ≥ 6, using the Lyapunov–Schmidt reduction method, we describe how to construct (scalar curvature) functions on Sⁿ, so that each of them enables the conformal scalar curvature equation to have an infinite number of positive solutions, which form a blow-up sequence. The prescribed scalar curvature function is shown to have C^{n - 1,β} smoothness. We present the argument in two parts. In this first part, we discuss the uniform cancellation property in the Lyapunov–Schmidt reduction method for the scalar curvature equation. We also explore relation between the Kazdan–Warner condition and the first-order derivatives of the reduced functional, and symmetry in the second-order derivatives of the reduced functional.

An ℓ-groupG is an abelian group equipped with a translation invariant lattice-order. Baker and Beynon proved that G is finitely generated projective if and only if it is finitely presented. A unital ℓ-group is an ℓ-group G with a distinguished order unit, i.e. an element 0 ≤ u ∈ G whose positive integer multiples eventually dominate every element of G. Unital ℓ-homomorphisms between unital ℓ-groups are group homomorphisms that also preserve the order unit and the lattice structure. A unital ℓ-group (G, u) is projective if whenever ψ : (A, a) → (B, b) is a surjective unital ℓ-homomorphism and ϕ : (G, u) → (B, b) is a unital ℓ-homomorphism, there is a unital ℓ-homomorphism θ : (G, u) → (A, a) such that ϕ = ψ ◦ θ. While every finitely generated projective unital ℓ-group is finitely presented, the converse does not hold in general. Classical algebraic topology (à la Whitehead) is combined in this paper with the Włodarczyk–Morelli solution of the weak Oda conjecture for toric varieties, to describe finitely generated projective unital ℓ-groups.

In this paper, we investigate the localization of solutions of the Cauchy problem to a doubly degenerate parabolic equation with a strongly nonlinear source

Let $m\ge 2$ be an integer. For any open domain $\mathrm{\Omega }\subset {ℝ}^{2m}$, non-positive function $\phi \in C^{\infty }(\mathrm{\Omega })$ such that ${\mathrm{\Delta }}^m\phi \equiv 0$, and bounded sequence $(V_k)\subset L^{\infty }(\mathrm{\Omega })$ we prove the existence of a sequence of functions $(u_k)\subset C^{2m-1}(\mathrm{\Omega })$ solving the Liouville equation of order $2m$

By using the Lyapunov–Schmidt reduction method without perturbation, we consider existence results for the conformal scalar curvature on $S^n$ ($n\ge 3$) when the prescribed function (after being projected to ${\mathrm{\text{I}}\mathrm{\text{R}}}^n$) has two close critical points, which have the same value (positive), equal “flatness” (“twin”; flatness $<n-2$), and exhibit maximal behavior in certain directions (“pseudo-peaks”). The proof relies on a balance between the two main contributions to the reduced functional — one from the critical points and the other from the interaction of the two bubbles.

We study the fractional Laplacian problem

In this work, we study a system of Schrödinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in terms of the ground state solutions associated with the corresponding elliptic system, which in turn are obtained by applying variational methods. By using the concentration-compactness method we also investigate the nonlinear stability/instability of the ground states.

In this paper, we prove the existence of an extremal function for the Adams–Moser–Trudinger inequality on the Sobolev space $H_0^m(\mathrm{\Omega })$, where $\mathrm{\Omega }$ is any bounded, smooth, open subset of ${ℝ}^{2m}$, $m\ge 1$. Moreover, we extend this result to improved versions of Adams’ inequality of Adimurthi-Druet type. Our strategy is based on blow-up analysis for sequences of subcritical extremals and introduces several new techniques and constructions. The most important one is a new procedure for obtaining capacity-type estimates on annular regions.

We investigate the well-posedness in the generalized Hartree equation $i{u}_t+\mathrm{\Delta }u+(|x{|}^{-(N-\gamma )}\ast |u{|}^p)|u{|}^{p-2}u=0$, $x\in {ℝ}^N$, $0<\gamma <N$, for low powers of nonlinearity, $p<2$. We establish the local well-posedness for a class of data in weighted Sobolev spaces, following ideas of Cazenave and Naumkin, Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Comm. Contemp. Math. 19(2) (2017) 1650038. This crucially relies on the boundedness of the Riesz transform in weighted Lebesgue spaces. As a consequence, we obtain a class of data that exists globally, moreover, scatters in positive time. Furthermore, in the focusing case in the $L^2$-supercritical setting we obtain a subset of locally well-posed data with positive energy, which blows up in finite time.