Qualitative Properties of Saddle-Shaped Solutions to Bistable Diffusion Equations

We consider the elliptic equation ? Δu = f(u) in the whole ?^2m , where f is of bistable type. It is known that there exists a saddle-shaped solution in ?^2m . This is a solution which changes sign in ?^2m and vanishes only on the Simons cone 𝒞 = {(x ¹, x ²) ∈ ? ^m × ? ^m : |x ¹| = |x ²|}. It is also known that these solutions are unstable in dimensions 2 and 4.

In this article we establish that when 2m = 6 every saddle-shaped solution is unstable outside of every compact set and, as a consequence has infinite Morse index. For this we establish the asymptotic behavior of saddle-shaped solutions at infinity. Moreover we prove the existence of a minimal and a maximal saddle-shaped solutions and derive monotonicity properties for the maximal solution.

These results are relevant in connection with a conjecture of De Giorgi on 1D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1D solutions, to be global minimizers in high dimensions, a property not yet established.     

Families of Periodic Solutions of Resonant PDEs

Summary. We construct some families of small amplitude periodic solutions close to a completely resonant equilibrium point of a semilinear reversible partial differential equation. To this end, we construct, using averaging methods, a suitable map from the configuration space to itself. We prove that to each nondegenerate zero of such a map there corresponds a family of small amplitude periodic solutions of the system. The proof is based on Lyapunov-Schmidt decomposition. This establishes a relation between Lyapunov-Schmidt decomposition and averaging theory that could be interesting in itself. As an application, we construct countable many families of periodic solutions of the nonlinear string equation u _tt -u _xx ± u ³ =0 (and of its perturbations) with Dirichlet boundary conditions. We also prove that the fundamental periods of solutions belonging to the n ^th family converge to 2π/n when the amplitude tends to zero. Received August 8, 2000; accepted November 21, 2000 Online publication February 26, 2001     

Finite Morse Index Implies Finite Ends

We prove that finite Morse index solutions to the Allen-Cahn equation in ℝ² have finitely many ends and linear energy growth. The main tool is a curvature decay estimate on level sets of these finite Morse index solutions, which in turn is reduced to a problem on the uniform second-order regularity of clustering interfaces for the singularly perturbed Allen-Cahn equation. Using an indirect blowup technique, in the spirit of the classical Colding-Minicozzi theory in minimal surfaces, we show that the obstruction to the uniform second-order regularity of clustering interfaces in ℝⁿ is associated to the existence of nontrivial entire solutions to a (finite or infinite) Toda system in ℝ^n–1. For finite Morse index solutions in ℝ², we show that this obstruction does not exist by using information on stable solutions of the Toda system. © 2019 Wiley Periodicals, Inc.     

Divergence of Infinite-Variance Nonradial Solutions to the 3D NLS Equation

We consider solutions u(t) to the 3d NLS equation i? _t u + Δu + |u|² u = 0 such that ‖xu(t)‖ _{L ²}  = ∞ and u(t) is nonradial. Denoting by M[u] and E[u], the mass and energy, respectively, of a solution u, and by Q(x) the ground state solution to ?Q + ΔQ + |Q|² Q = 0, we prove the following: if M[u]E[u] < M[Q]E[Q] and ‖u ₀‖ _{L ²} ‖?u ₀‖ _{L ²}  > ‖Q‖ _{L ²} ‖?Q‖ _{L ²} , then either u(t) blows-up in finite positive time or u(t) exists globally for all positive time and there exists a sequence of times t _n  → + ∞ such that ‖?u(t _n )‖ _{L ²}  → ∞. Similar statements hold for negative time.     

Nonsmooth Neumann-Type Problems Involving the p-Laplacian

This paper deals with the problem ? Δ _p u + α(x)|u| ^p?2 u = β(x)f(|u|) in Ω, subjected to the zero Neumann boundary condition, where p > 1, Ω ? ? ^N is bounded with smooth boundary, α, β ? L ^∞(Ω), essinf_Ωβ > 0, and f:[0,+ ∞) → ? is a not necessarily continuous nonlinearity that oscillates either at the origin or at the infinity. By using nonsmooth variational methods, we establish in both cases the existence of infinitely many distinct non-negative solutions of the Neumann problem. In our framework, α:Ω → ? may be a sign-changing or even a nonpositive potential, which is not permitted usually in earlier works.     

Catenoidal Layers for the Allen-Cahn Equation in Bounded Domains

This paper presents a new family of solutions to the singularly perturbed Allen-Cahn equation α~2Δu + u(1- u~2) = 0 in a smooth bounded domain Ω R~3, with Neumann boundary condition and α 0 a small parameter. These solutions have the property that as α→ 0, their level sets collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature intersecting ?Ω orthogonally and that is non-degenerate respect to ?Ω. The authors provide explicit examples of surfaces to which the result applies.     

Morse index of layered solutions to the heterogeneous Allen-Cahn equation

Let u? be a single layered radially symmetric unstable solution of the Allen-Cahn equation −?²Δu=u(u−a(|x|))(1−u) over the unit ball with Neumann boundary conditions. We estimate the small eigenvalues of the linearized eigenvalue problem at u? when ? is small. As a consequence, we prove that the Morse index of u? is asymptotically given by [μ^∗+o(1)]?−(N−1)/2 with μ^∗ a certain positive constant expressed in terms of parameters determined by the Allen-Cahn equation. Our estimates on the small eigenvalues have many other applications. For example, they may be used in the search of other non-radially symmetric solutions, which will be considered in forthcoming papers.     

Convergence to Equilibrium for a Thin-Film Equation on a Cylindrical Surface

The degenerate parabolic equation u _t + ? _x [u ³(u _xxx  + u _x  ? sin x)] = 0 models the evolution of a thin liquid film on a stationary horizontal cylinder. It is shown here that for each mass there is a unique steady state, given by a droplet hanging from the bottom of the cylinder that meets the dry region with zero contact angle. The droplet minimizes the associated energy functional and attracts all strong solutions that satisfy certain energy and entropy inequalities, including all positive solutions. The distance of solutions from the steady state cannot decay faster than a power law.     

Partial Regularity for Stationary Solutions to Liouville-Type Equation in Dimension 3

In dimension n = 3, we prove that the singular set of any stationary solution to the Liouville equation ? Δ u = e ^u , which belongs to W ^1,2, has Hausdorff dimension at most 1.     

Hydrodynamic Limit of the Gross-Pitaevskii Equation

We study dynamics of vortices in solutions of the Gross-Pitaevskii equation i? _t u = Δu + ?^?2 u(1 ? |u|²) on ?² with nonzero degree at infinity. We prove that vortices move according to the classical Kirchhoff-Onsager ODE for a small but finite coupling parameter ?. By carefully tracking errors we allow for asymptotically large numbers of vortices, and this lets us connect the Gross-Pitaevskii equation on the plane to two dimensional incompressible Euler equations through the work of Schochet [19 Schochet , S. ( 1996 ). The point vortex method for periodic weak solutions of the 2D Euler equations . Comm. Pure Appl. Math. 49 : 911 – 965 .[Crossref], [Web of Science ®] , [Google Scholar]].     

The heterogeneous Allen-Cahn equation in a ball: Solutions with layers and spikes

Let u? be a single layered radially symmetric unstable solution of the Allen-Cahn equation −?²Δu=u(u−a(|x|))(1−u) over the unit ball with Neumann boundary conditions. Based on our estimate of the small eigenvalues of the linearized eigenvalue problem at u? when ? is small, we construct solutions of the form u?+v?, with v? non-radially symmetric and close to zero in the unit ball except near one point x₀ such that |x₀| is close to a nondegenerate critical point of a(r). Such a solution has a sharp layer as well as a spike.     

KAM tori for higher dimensional beam equation with a fixed constant potential

In this paper, we consider the higher dimensional nonlinear beam equation:utt + △2u + σu + f(u)=0 with periodic boundary conditions, where the nonlinearity f(u) is a real-analytic function of the form f(u)=u3+ h.o.t near u=0 and σ is a positive constant. It is proved that for any fixed σ>0, the above equation admits a family of small-amplitude, linearly stable quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system.     

Existence and asymptotic behavior of entire blow-up solutions for quasilinear elliptic equation

In this article, we discuss the blow-up problem of entire solutions of a class of second-order quasilinear elliptic equation Δ _p u ≡ div(|?u| ^p?2?u) = ρ(x)f(u), x ∈ R ^N . No monotonicity condition is assumed upon f(u). Our method used to get the existence of the solution is based on sub-and supersolutions techniques.     

Classification of radial solutions to the Emden–Fowler equation on the hyperbolic space

We study the Emden–Fowler equation ?Δu = |u| ^p?1 u on the hyperbolic space ${{\mathbb H}^n}$ . We are interested in radial solutions, namely solutions depending only on the geodesic distance from a given point. The critical exponent for such equation is p = (n + 2)/(n ? 2) as in the Euclidean setting, but the properties of the solutions show striking differences with the Euclidean case. While the papers (Bhakta and Sandeep, Poincaré Sobolev equations in the hyperbolic space, 2011; Mancini and Sandeep, Ann Sci Norm Sup Pisa Cl Sci 7(5):635–671, 2008) consider finite energy solutions, we shall deal here with infinite energy solutions and we determine the exact asymptotic behavior of wide classes of finite and infinite energy solutions.     

Bifurcation for Quasilinear Elliptic Singular BVP

For a continuous function g ≥ 0 on (0, + ∞) (which may be singular at zero), we confront a quasilinear elliptic differential operator with natural growth in ?u, ? Δu + g(u)|?u|², with a power type nonlinearity, λu ^p  + f ₀(x). The range of values of the parameter λ for which the associated homogeneous Dirichlet boundary value problem admits positive solutions depends on the behavior of g and on the exponent p. Using bifurcations techniques we deduce sufficient conditions for the boundedness or unboundedness of the cited range.     

Many Solutions of Elliptic Problems on R n of Irrational Slope

ABSTRACT

We consider the problem ? Δ u + F _u (x, u) = 0 on R ⁿ , where F is a smooth function periodic of period 1 in all its variables. We show that, under suitable hypotheses on F, this problem has a family of non-self-intersecting solutions u _D , which are at finite distance from a plane of slope (ω,0,…,0) with ω irrational. These solutions depend on a real parameter D; if D ≠ D ^′, then the closures of the integer translates of u _D and of u _{D ^′} do not intersect.     

Periodic solutions of the Navier–Stokes equations in a perturbed half‐space and an aperture domain

We shall construct a periodic strong solution of the Navier–Stokes equations for some periodic external force in a perturbed half‐space and an aperture domain of the dimension n?3. Our proof is based on L^p–L^q estimates of the Stokes semigroup. We apply L^p–L^q estimates to the integral equation which is transformed from the original equation. As a result, we obtain the existence and uniqueness of periodic strong solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Spectral Analysis for Stationary Solutions of the Cahn–Hilliard Equation in ℝ d

We consider the spectrum associated with three types of bounded stationary solutions for the Cahn–Hilliard equation on ? ^d , d ≥ 2: radial solutions, saddle solutions (only for d = 2), and planar periodic solutions. In particular, we establish spectral instability for each type of solution. The important case of multiply periodic solutions does not fit into the framework of our approach, and we do not consider it here.