Search

We are concerned with the existence of blowing-up solutions to the following boundary value problem - Δ u = λ V (x) e u - 4 π N δ 0 in B 1 , u = 0 on ∂ B 1 , where B 1 is the unit ball in R 2 centered at the origin, V(x) is a positive smooth potential, N is a positive integer ( N ≥ 1 ). Here δ 0 defines the Dirac measure with pole at 0, and λ > 0 is a small parameter. We assume that N = 1 and, under some suitable assumptions on the derivatives of the potential V at 0, we find a solution which exhibits a non-simple blow-up profile as λ → 0 + . [ABSTRACT FROM AUTHOR]

We consider the Euler equations in R 3 expressed in vorticity form ω → t + (u · ∇) ω → = (ω → · ∇) u u = curl ψ → , - Δ ψ → = ω →. A classical question that goes back to Helmholtz is to describe the evolution of solutions with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called binormal curvature flow. Existence of true solutions concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings. We provide what appears to be the first rigorous construction of helical filaments, associated to a translating-rotating helix. The solution is defined at all times and does not change form with time. The result generalizes to multiple polygonal helical filaments travelling and rotating together. [ABSTRACT FROM AUTHOR]

We study the Bose–Einstein condensates (BEC) in two or three dimensions with attractive interactions, described by L 2 constraint Gross-Pitaevskii energy functional. First, we give the precise description of the chemical potential of the condensate μ and the attractive interaction a. Next, for a class of degenerate trapping potential with non-isolated critical points, we obtain the existence and the local uniqueness of the excited states by accurately analyzing the location of the concentrated points and the Lagrange multiplier. Our results on degenerate trapping potential with non-isolated critical points are new for ground states of BEC and singularly perturbed nonlinear Schrödinger equations. [ABSTRACT FROM AUTHOR]

We consider the nonlinear half-Laplacian heat equation u t + (- Δ) 1 2 u - | u | p - 1 u = 0 , R n × (0 , T). We prove that all blows-up are type I, provided that n ≤ 4 and 1 < p < p ∗ (n) where p ∗ (n) is an explicit exponent which is below n + 1 n - 1 , the critical Sobolev exponent. Central to our proof is a Giga-Kohn type monotonicity formula for half-Laplacian and a Liouville type theorem for self-similar nonlinear heat equation. This is the first instance of a monotonicity formula at the level of the nonlocal equation, without invoking the extension to the half-space. [ABSTRACT FROM AUTHOR]

We construct infinite time blow-up solution to the following heat equation with Sobolev critical exponent and drift terms u t = Δ u + ∇ b (x) · ∇ u + u n + 2 n - 2 in R n × (0 , + ∞) , u (· , 0) = u 0 in R n , where b(x) is a smooth bounded function in R n with n ≥ 5 and the initial datum u 0 is positive and smooth. Let q j ∈ R n , j = 1 , … , k , be distinct nondegenerate local minimum points of b(x). Assume that an eigenvalue condition (1.6) is satisfied. We prove the existence of a positive smooth solution u(x, t) which blows up at infinite time near those points with the form u (x , t) ≈ ∑ j = 1 k α n μ j (t) μ j (t) 2 + | x - ξ j (t) | 2 n - 2 2 , as t → + ∞. Here ξ j (t) → q j and 0 < μ j (t) → 0 exponentially as t → + ∞ . [ABSTRACT FROM AUTHOR]

Let $$\Omega $$ be a bounded domain in $$\mathbb {R}^n$$ , $$n\ge 3$$ with smooth boundary $$\partial \Omega $$ and a small hole. We give the first example of sign-changing bubbling solutions to the nonlinear elliptic problem where $$\varepsilon $$ is a small positive parameter. The basic cell in the construction is the sign-changing nodal solution to the critical Yamabe problem which has large number (3 n) of kernels. [ABSTRACT FROM AUTHOR]

We study the following generalized SU(3) Toda System where $$\mu >-2$$ . We prove the existence of radial solutions bifurcating from the radial solution $$(\log \frac{64}{(2+\mu ) (8+|x|^2)^2}, \log \frac{64}{(2+\mu ) (8+|x|^2)^2})$$ at the values $$\mu =\mu _n=2\frac{2-n-n^2}{2+n+n^2},\ n\in \mathbbm {N}$$ . [ABSTRACT FROM AUTHOR]

We consider the standing-wave problem for a nonlinear Schrödinger equation, corresponding to the semilinear elliptic problem where $$V(x)$$ is a uniformly positive potential and $$p>1$$ . Assuming that for instance if $$p>2$$ , $$m>2$$ and $$\sigma >1$$ we prove the existence of infinitely many positive solutions. If $$V(x)$$ is radially symmetric, this result was proved in []. The proof without symmetries is much more difficult, and for that we develop a new intermediate Lyapunov-Schmidt reduction method, which is a compromise between the finite and infinite dimensional versions of it. [ABSTRACT FROM AUTHOR]

In this paper, we construct a wealth of bounded, entire solutions of the Allen-Cahn equation in the plane. The asymptotic behavior at infinity of these solutions is determined by $$2L$$ half affine lines, in the sense that, along each of these half affine lines, the solution is close to a suitable translated and rotated copy of a one dimensional heteroclinic solution. The solutions we construct belong to a smooth $$2L$$ -dimensional family of bounded, entire solutions of the Allen-Cahn equation, in agreement with the result of del Pino (Trans Am Math Soc 365(2):721-766, ) and, in some sense, they provide a description of a collar neighborhood of part of the compactification of the moduli space of $$2L$$ -ended solutions for the Allen-Cahn equation. Our construction is inspired by a construction of minimal surfaces by Traizet [Ann. Inst. Fourier (Grenoble) 46(5), 1385-1442, ]. [ABSTRACT FROM AUTHOR]

We consider the following nonlinear Schrödinger equation where $$V$$ is a continuous potential and $$ f(u)$$ is a nonlinearity satisfying some decay condition and some non-degeneracy condition, respectively. Using localized energy method, we prove that there exists a $$\delta _0$$ such that for $$0<\delta <\delta _0$$ , the above problem has infinitely many positive solutions. This generalizes and gives a new proof of the results by Cerami et al. (Comm. Pure Appl. Math. 66, 372-413, ). The new techniques allow us to establish the existence of infinitely many positive bound states for elliptic systems. [ABSTRACT FROM AUTHOR]

Studies of near periodic patterns in many self-organizing physical and biological systems give rise to a nonlocal geometric problem in the entire space involving the mean curvature and the Newtonian potential. One looks for a set in space of the prescribed volume such that on the boundary of the set the sum of the mean curvature of the boundary and the Newtonian potential of the set, multiplied by a parameter, is constant. Despite its simple form, the problem has a rich set of solutions and its corresponding energy functional has a complex landscape. When the parameter is sufficiently large, there exists a solution that consists of two tori: a larger torus and a smaller torus. Due to the axisymmetry, the problem is formulated on a half plane. A variant of the Lyapunov-Schmidt procedure is developed to reduce the problem to minimizing the energy of the set of two exact tori, as an approximate solution, with respect to their radii. A re-parameterization argument shows that the double tori so obtained indeed solves the equation of mean curvature and Newtonian potential. One also obtains the asymptotic formulae for the radii of the tori in terms of the parameter. This double tori set is the first known disconnected solution. [ABSTRACT FROM AUTHOR]

We investigate nonlinear Schrödinger-Poisson systems in the 3-sphere. We prove existence results for these systems and discuss the question of the stability of the systems with respect to their phases. While, in the subcritical case, we prove that all phases are stable, we prove in the critical case that there exists a sharp explicit threshold below which all phases are stable and above which resonant frequencies and multi-spikes blowing-up solutions can be constructed. Solutions of the Schrödinger-Poisson systems are standing waves solutions of the electrostatic Maxwell-Schrödinger system. Stable phases imply the existence of a priori bounds on the amplitudes of standing waves solutions. Unstable phases give rise to resonant states. [ABSTRACT FROM AUTHOR]

Let ( M, g) be a compact Riemannian manifold of dimension N ≥ 5 and Q be its Q curvature. The prescribed Q curvature problem is concerned with finding metric of constant Q curvature in the conformal class of g. This amounts to finding a positive solution to $${P_{g}(u)=c u^{\frac{N+4}{N-4}},\quad u > 0 \quad {\rm on} \; M}$$ where P is the Paneitz operator. We show that for dimensions N ≥ 25, the set of all positive solutions to the prescribed Q curvature problem is non-compact. [ABSTRACT FROM AUTHOR]

We study the existence of 2 π-periodic positive solutions of the equation where a( θ) is a positive smooth 2 π-periodic function. A priori estimates and sufficient conditions for the existence of solutions of the equation are established. [ABSTRACT FROM AUTHOR]

For the Neumann sinh-Gordon equation on the unit ball $${B \subset \mathbb {R}^2}$$ we construct sequence of solutions which exhibit a multiple blow up at the origin, where λ ± are positive parameters. It answers partially an open problem formulated in Jost et al. [Calc Var Partial Diff Equ 31(2):263–276]. [ABSTRACT FROM AUTHOR]

We consider the following Liouville equation in $${\mathbb{R}}^{4}$$ For each fixed $$x^0 \in {\mathbb{R}}^{4}, 1 \le k \le 4, \alpha \in (1-\frac{k}{4}, 2)$$ and a j > 0 for 1 ≤ j ≤ k, we construct a solution to the above equation with the following asymptotic behavior: [ABSTRACT FROM AUTHOR]

A droplet solution characterizes the lamellar phase of a diblock copolymer when the two composing monomers maintain a skewed ratio. We study the threshold case where the free energy of a droplet solution is comparable to the free energy of the constant solution. Using a Lyapunov-Schmidt reduction approach, adapted to calculus of variations, we prove the existence of a free energy local minimizer with a given number of droplets. Also determined are the free energy, the droplet location, and the droplet size. [ABSTRACT FROM AUTHOR]

Examines a singularly perturbed semilinear elliptic problem. Asymptotic expansion; Single boundary spike; Higher-order expansion; Applications of the expansion. Boundary mean curvature function.

We prove the existence of smooth non-radial entire solution to Δ 2 u + u - q = 0 in R 3 , u > 0 , for q > 1 . This answers an open question raised by McKenna and Reichel (Electron J Differ Equ 37:1–3, 2003). [ABSTRACT FROM AUTHOR]

We consider the following quasilinear Schrödinger equations of the form ▵ u - ε V (x) u + u ▵ u 2 + u p = 0 , u > 0 in R N and lim | x | → ∞ u (x) = 0 , where N ≥ 3 , p > N + 2 N - 2 , ε > 0 and V(x) is a positive function. By imposing appropriate conditions on V(x), we prove that, for ε = 1 , the existence of infinity many positive solutions with slow decaying O (| x | - 2 p - 1 ) at infinity if p > N + 2 N - 2 and, for ε sufficiently small, a positive solution with fast decaying O (| x | 2 - N) if N + 2 N - 2 < p < 3 N + 2 N - 2. The proofs are based on perturbative approach. To this aim, we also analyze the structure of positive solutions for the zero mass problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of the existence of positive weak solutions of (-Δ)su=upinΩu=0onRn\Ωhaving prescribed isolated interior singularities. We prove that if nn-2s