Volume 17, Issue 1

We prove several existence results for some nonlinear elliptic Kirchhoff equations.

We consider magnetic Tonelli Hamiltonian systems on the cotangent bundle of the 2-sphere, where the magnetic form is not necessarily exact. It is known that, on very low and on high energy levels, these systems may have only finitely many periodic orbits. Our main result asserts that almost all energy levels in a precisely characterized intermediate range (e0,e1)${(e_{0},e_{1})}$ possess infinitely many periodic orbits. Such a range of energies is non-empty, for instance, in the physically relevant case where the Tonelli Lagrangian is a kinetic energy and the magnetic form is oscillating (in which case, e0=0${e_{0}=0}$ is the minimal energy of the system).

We consider u⁢(x,t)${u(x,t)}$, a solution of ∂t⁡u=Δ⁢u+|u|p-1⁢u${\partial_{t}u=\Delta u+|u|^{p-1}u}$ which blows up at some time T>0${T>0}$, where u:ℝN×[0,T)→ℝ${u:\mathbb{R}^{N}\times[0,T)\to\mathbb{R}}$, p>1${p>1}$ and (N-2)⁢p<N+2${(N-2)p<N+2}$. Define S⊂ℝN${S\subset\mathbb{R}^{N}}$ to be the blow-up set of u , that is, the set of all blow-up points. Under suitable non-degeneracy conditions, we show that if S contains an (N-ℓ)${(N-\ell)}$-dimensional continuum for some ℓ∈{1,…,N-1}${\ell\in\{1,\dots,N-1\}}$, then S is in fact a 𝒞2${\mathcal{C}^{2}}$ manifold. The crucial step is to make a refined study of the asymptotic behavior of u near blow-up. In order to make such a refined study, we have to abandon the explicit profile function as a first-order approximation and take a non-explicit function as a first-order description of the singular behavior. This way we escape logarithmic scales of the variable (T-t)${(T-t)}$ and reach significant small terms in the polynomial order (T-t)μ${(T-t)^{\mu}}$ for some μ>0${\mu>0}$. Knowing the refined asymptotic behavior yields geometric constraints of the blow-up set, leading to more regularity on S .

The paper is concerned with the slightly subcritical elliptic problem with Hardy-critical term { - Δ ⁢ u - μ ⁢ u | x | 2 = | u | 2 ∗ - 2 - ε ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω $\left\{\begin{aligned} \displaystyle-\Delta u-\mu\frac{u}{|x|^{2}}&% \displaystyle=|u|^{2^{\ast}-2-\varepsilon}u&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega\end{% aligned}\right.$ in a bounded domain Ω⊂ℝN${\Omega\subset\mathbb{R}^{N}}$ with 0∈Ω${0\in\Omega}$, in dimensions N≥7${N\geq 7}$. We investigate the possible blow-up behavior of solutions as μ,ε→0${\mu,\varepsilon\to 0}$. In particular, we prove the existence of nodal solutions that blow up positively at the origin and negatively at a different point as μ,ε→0+${\mu,\varepsilon\to 0^{+}}$. The location of the negative blow-up point is determined by the geometry of Ω. Moreover, the asymptotic shape of the solutions is described in detail. An interesting new consequence of our results is that the type of blow-up solutions considered here exists for μ=O⁢(εα)${\mu=O(\varepsilon^{\alpha})}$ with α>N-4N-2${\alpha>\frac{N-4}{N-2}}$. The bound N-4N-2${\frac{N-4}{N-2}}$ is sharp.

We establish the existence of nodal solutions to the supercritical problem - Δ ⁢ u = | u | p - 2 ⁢ u   in ⁢ Ω , u = 0   on ⁢ ∂ ⁡ Ω , $-\Delta u=|u|^{p-2}u\quad\text{in }\Omega,\qquad u=0\quad\text{on }\partial\Omega,$ in a symmetric bounded smooth domain Ω of ℝN${\mathbb{R}^{N}}$, N≥3${N\geq 3}$, for p>2N∗:=2⁢NN-2${p>2_{N}^{\ast}:=\frac{2N}{N-2}}$, up to some range which depends on the symmetries, and we study their asymptotic behavior as p→2N∗${p\rightarrow 2_{N}^{\ast}}$. We exhibit solutions up${u_{p}}$ to this problem in symmetric domains with a shrinking hole, which concentrate at a single point as the hole shrinks and p approaches 2N∗${2_{N}^{\ast}}$ from above, and whose limit profile is a rescaling of a nonradial sign-changing solution to the limit problem - Δ ⁢ u = | u | 2 N ∗ - 2 ⁢ u , u ∈ D 1 , 2 ⁢ ( ℝ N ) . $-\Delta u=|u|^{2_{N}^{\ast}-2}u,\quad u\in D^{1,2}(\mathbb{R}^{N}).$

Given (M,g)${(M,g)}$, a smooth compact Riemannian n -manifold, we prove that for a generic Riemannian metric g the critical points of the function 𝒲g⁢(ξ):=|Weylg⁢(ξ)|g2${\mathcal{W}_{g}(\xi):=\lvert\mathrm{Weyl}_{g}(\xi)\rvert^{2}_{g}}$ with 𝒲g⁢(ξ)≠0${\mathcal{W}_{g}(\xi)\not=0}$ are nondegenerate.

We consider some singular Liouville equations and systems motivated by uniformization problems in a non-smooth setting, as well as from models in mathematical physics. We will study the existence of solutions from a variational point of view, using suitable improvements of the Moser–Trudinger inequality. These reduce the problem to a topological one by studying the concentration property of conformal volume, which will be constrained by the functional inequalities of geometric flavour. We will mainly describe some common strategies from the papers [11, 12, 20] in simple situations to give an idea to the non-expert reader about the general methods we use.

We consider the following Liouville-type equation on domains of ℝ2${\mathbb{R}^{2}}$ under Dirichlet boundary conditions: { - Δ ⁢ u = ϱ ⁢ K ⁢ e u ∫ Ω K ⁢ e u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , $\left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=\varrho\frac{Ke^{u% }}{\int_{\Omega}Ke^{u}}&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right.$ where ϱ∈ℝ${\varrho\in\mathbb{R}}$ and K is a smooth nonnegative function having N zeros q1,…,qN${q_{1},\ldots,q_{N}}$, which takes in a neighborhood of a zero qj${q_{j}}$ the following form: K ⁢ ( x ) = K j ⁢ ( x ) ⁢ | x - q j | 2 ⁢ γ j   with ⁢ K j ⁢ ( x ) > 0 ⁢ and ⁢ γ i ∈ ℝ ⁢ such that ⁢ 0 < γ j := γ j ⁢ ( q j ) ∉ ℕ . $K(x)=K_{j}(x)\lvert x-q_{j}\rvert^{2\gamma_{j}}\quad\text{with }K_{j}(x)>0% \text{ and }\gamma_{i}\in\mathbb{R}\text{ such that }0<\gamma_{j}:=\gamma_{j}(% q_{j})\notin\mathbb{N}.$ Using some dynamical and topological tools from the “critical point theory at infinity” of Bahri, we study the critical points at infinity of the related variational problem. Then we derive from our analysis some existence results in the so-called resonant case, that is, when the parameter ϱ is of the form ∑i=1σ8⁢π⁢(1+γi)+∑i=σ+1m8⁢π${\sum_{i=1}^{\sigma}8\pi(1+\gamma_{i})+\sum_{i=\sigma+1}^{m}8\pi}$ for a subset (qi1,…,qiσ)${(q_{i_{1}},\ldots,q_{i_{\sigma}})}$ of Σ:={q1,…,qN}${\Sigma:=\{q_{1},\ldots,q_{N}\}}$. In particular, we provide an Euler–Poincaré-type criterium for existence of solutions.

In this article, we prove that data assimilation by feedback nudging can be achieved for the three-dimensional quasi-geostrophic equation in a simplified scenario using only large spatial scale observables on the dynamical boundary. On this boundary, a scalar unknown (buoyancy or surface temperature of the fluid) satisfies the surface quasi-geostrophic equation. The feedback nudging is done on this two-dimensional model, yet ultimately synchronizes the streamfunction of the three-dimensional flow. The main analytical difficulties are due to the presence of a nonlocal dissipative operator in the surface quasi-geostrophic equation. This is overcome by exploiting a suitable partition of unity, the modulus of continuity characterization of Sobolev space norms, and the Littlewood–Paley decomposition to ultimately establish various boundedness and approximation-of-identity properties for the observation operators.

This work is an adaptation of one of the methods based on the variational critical points at infinity theory of Abbas Bahri [1, 3, 2, 4, 5, 6, 7, 8] to the Cauchy–Riemann settings. Following Gamara [24], we give new existence results for the Kazdan–Warner problem on Cauchy–Riemann spheres, whereas multiplicity results will be discussed in a forthcoming paper. Using algebraic topological methods and the theory of critical points at infinity, we provide a variety of functions which can be realized as the scalar curvature for these manifolds.