Radial and non radial solutions for Hardy–Hénon type elliptic systems

We discuss existence and non-existence of positive solutions for the following system of Hardy and Hénon type: $$\left\{\begin{array}{ll} {-\Delta v=|x|^{\alpha}u^{p},\,-\Delta u=|x|^{\beta}v^{q} \,\,{\rm in}\, \Omega,}\\ {u=v=0 \quad\quad\quad\quad\quad\quad\quad\quad\quad{\rm on}\, \partial \Omega}, \end{array}\right.$$ where ${\Omega\ni 0}$ is a bounded domain in ${\mathbb{R}^{N}}$ , N ≥ 3, p, q > 1, and α, β > ?N. We also study symmetry breaking for ground states when Ω is the unit ball in ${\mathbb{R}^{N}}$ .     

Radial symmetry of standing waves for nonlinear fractional Hardy–Schrödinger equation

In this paper, by applying the method of moving planes, we conclude the conclusions for the radial symmetry of standing waves for a nonlinear Schrödinger equation involving the fractional Laplacian and Hardy potential. First, we prove the radial symmetry of solution under the condition of decay near infinity. Based upon that, under the condition of no decay, by the Kelvin transform, we establish the results for the non-existence and radial symmetry of solution.     

Multi-peak solutions for the Hénon equation with slightly subcritical growth

We study the Dirichlet problem for the Hénon equation

Non radial positive solutions for the Hénon equation with critical growth

We study the Dirichlet problem in a ball for the Hénon equation with critical growth and we establish, under some conditions, the existence of a positive, non radial solution. The solution is obtained as a minimizer of the quotient functional associated to the problem restricted to appropriate subspaces of H₀¹ invariant for the action of a subgroup of . Analysis of compactness properties of minimizing sequences and careful level estimates are the main ingredients of the proof. Received: 18 October 2003, Accepted: 5 July 2004, Published online: 3 September 2004 Mathematics Subject Classification (2000): 35J20, 35B33 This research was supported by MIUR, Project "Variational Methods and Nonlinear Differential Equations".     

Multiplicity of solutions to the Dirichlet problem for generalized Hénon equation

For the boundary value problem

Asymptotic behavior on the Hénon equation with supercritical exponent

The main purpose of this paper is to analyze the asymptotic behavior of the radial solution of Hénon equation −Δu = |x| ^α u ^p−1, u > 0, x ∈ B _R (0) ⊂ ℝ ⁿ (n ⩾ 3), u = 0, x ∈ ∂B _R (0), where $ p \to p(\alpha ) = \frac{{2(n + \alpha )}} {{n - 2}} $ p \to p(\alpha ) = \frac{{2(n + \alpha )}} {{n - 2}} from left side, α > 0.     

Non-radial least energy solutions of the generalized Hénon equation

In this paper, we study the generalized Hénon equation with a radial coefficient function in the unit ball and show the existence of a positive non-radial solution. Our result is applicable to a wide class of coefficient functions. Our theorem ensures that if the ratio of the density of the coefficient function in $|x|<a$ to that in $a<|x|<1$ is small enough and a is sufficiently close to 1, then a least energy solution is not radially symmetric.     

Complicated behavior in cubic Hénon maps

Theoretical and Mathematical Physics - We study a generalized Hénon map in two-dimensional space. We find a region of the phase space where the nonwandering set exists, specify parameter...     

Hölder regularity for the time fractional Schrödinger equation

In this paper, we investigate the Hölder regularity of solutions to the time fractional Schrödinger equation of order 1<α<2, which interpolates between the Schrödinger and wave equations. This is inspired by Hirata and Miao's work which studied the fractional diffusion-wave equation. First, we give the asymptotic behavior for the oscillatory distributional kernels and their Bessel potentials by using Fourier analytic techniques. Then, the space regularity is derived by employing some results on singular Fourier multipliers. Using the asymptotic behavior for the above kernels, we prove the time regularity. Finally, we use mismatch estimates to prove the pointwise convergence to the initial data in Hölder spaces. In addition, we also prove Hölder regularity result for the Schrödinger equation.     

Collision and escape orbits in a generalized Hénon–Heiles model

The motion of a material point of unit mass in a field determined by a generalized Hénon–Heiles potential $U=A{q}_1^2+B{q}_2^2+C{q}_1^2{q}_2+D{q}_2^3$, with $(q_1,q_2)=$ standard Cartesian coordinates and $(A,B,C,D)\in {(0,\infty )}^2\times R^2$, is addressed for two limit situations: collision and escape. Using McGehee-type transformations, the corresponding collision and infinity boundary manifolds pasted on the phase space are determined. These are fictitious manifolds, but, due to the continuity with respect to initial data, their flow determines the near by orbit behaviour.The dynamics on the collision and infinity manifolds is fully described. The topology of the flow on the collision manifold is independent of the parameters. In the full phase space, while spiraling collision orbits are present, most of the orbits avoid collision. The topology of the flow on the infinity manifold changes as the ratio between $C$ and $D$ varies. More precisely, there are two symmetric pitchfork bifurcations along the line $2C?3D=0$, due to the reshaping of the potential along the bifurcation line. Besides rectilinear and spiraling orbits, the near-escape dynamics includes oscillatory orbits, for which angular momentum alternates sign.     

Least energy solutions of the generalized Hénon equation in reflectionally symmetric or point symmetric domains

We study the generalized Hénon equation in reflectionally symmetric or point symmetric domains and prove that a least energy solution is neither reflectionally symmetric nor even. Moreover, we prove the existence of a positive solution with prescribed exact symmetry.     

Ray Hölder-continuity for fractional Sobolev spaces in infinite dimensions and applications

We prove H?lder-continuity on rays in the direction of vectors in the (generalized) Cameron-Martin space for functions in Sobolev spaces in L ^p of fractional order α∈ (, 1) over infinite dimensional linear spaces. The underlying measures are required to satisfy some easy standard structural assumptions only. Apart from Wiener measure they include Gibbs measures on a lattice and Euclidean interacting quantum fields in infinite volume. A number of applications, e.g., to the two-dimensional polymer measure, are presented. In particular, irreducibility of the Dirichlet form associated with the latter measure is proved without restrictions on the coupling constant. Received: 9 November 1998 / Published online: 30 March 2000     

On fractional Schrödinger equation in α-dimensional fractional space

The Schrödinger equation is solved in $\alpha$-dimensional fractional space with a Coulomb potential proportional to $\frac{1}{r^{\beta ?2}}$, $2\le \beta \le 4$. The wave functions are studied in terms of spatial dimensionality $\alpha$ and $\beta$ and the results for $\beta =3$ are compared with those obtained in the literature.     

Monotone method for initial value problem for fractional diffusion equation

Using the method of upper and lower solutions and its associated monotone iterative, consider the existence and uniqueness of solution of an initial value problem for the nonlinear fractional diffusion equation.     

Elliptic solutions in the Hénon–Heiles model

Equations of motion corresponding to the Hénon–Heiles Hamiltonian are considered. A method enabling one to find all elliptic solutions of an autonomous ordinary differential equation or a system of autonomous ordinary differential equations is described. New families of elliptic solutions of a fourth-order equation related to the Hénon–Heiles system are obtained. A classification of elliptic solutions up to the sixth order inclusively is presented.     

Singularity and blow-up estimates via Liouville-type theorems for Hardy–Hénon parabolic equations

We consider the Hardy–Hénon parabolic equation ${u_t-\Delta u =|x|^a |u|^{p-1}u}$ with p > 1 and ${a\in \mathbb{R}}$ . We establish the space-time singularity and decay estimates, and Liouville-type theorems for radial and nonradial solutions. As applications, we study universal and a priori bound of global solutions as well as the blow-up estimates for the corresponding initial-boundary value problem.