Positive Periodic Solutions for Higher Order Functional Difference Systems Depending on a Parameter

In this paper, we consider a type of nth order functional difference systems depending on a parameter λ. By using the fixed point index theory, we prove that there exists a positive number ${\widetilde{\lambda}}$ separating ${(0,+ \infty) \backslash \widetilde{\lambda}}$ into two disjoint subintervals Λ₁ and Λ₂ such that the system has zero, at least one or at least two positive periodic solutions according to ${\lambda \in \Lambda_1, \lambda = \widetilde{\lambda}}$ , or ${\lambda \in \Lambda_2}$ . This work improves and extends some recent results in the literature for the first order systems. In addition, we obtain conditions for a general system to have nontrivial periodic solutions. The results are illustrated with an example.     

Geodesic Ricci solitons on unit tangent sphere bundles

In the paper, (Abbassi and Kowalski, Ann Glob Anal Geom, 38: 11–20, 2010) the authors study Einstein Riemannian $g$ natural metrics on unit tangent sphere bundles. In this study, we equip the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold $(M,g)$ with an arbitrary Riemannian $g$ natural metric $\tilde{G}$ and we show that if the geodesic flow $\tilde{\xi }$ is the potential vector field of a Ricci soliton $(\tilde{G},\tilde{\xi },\lambda )$ on $T_1M,$ then $(T_1M,\tilde{G})$ is Einstein. Moreover, we show that the Reeb vector field of a contact metric manifold is an infinitesimal harmonic transformation if and only if it is Killing. Thus, we consider a natural contact metric structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi }, \tilde{\xi })$ over $T_1 M$ and we show that the geodesic flow $\tilde{\xi }$ is an infinitesimal harmonic transformation if and only if the structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi },\tilde{\xi })$ is Sasaki $\eta $ -Einstein. Consequently, we get that $(\tilde{G},\tilde{\xi }, \lambda )$ is a Ricci soliton if and only if the structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi }, \tilde{\xi })$ is Sasaki-Einstein with $\lambda = 2(n-1) >0.$ This last result gives new examples of Sasaki–Einstein structures.     

A functional model associated with a generalized Nevanlinna pair

Let $ \mathcal{L} $ be a Hilbert space, and let $ \mathcal{H} $ be a Pontryagin space. For every self-adjoint linear relation $ \tilde{A} $ in $ \mathcal{H} \oplus \mathcal{L} $ , the pair $ \{ I + \lambda \psi (\lambda ),\,\psi (\lambda )\} $ where $ \psi (\lambda ) $ is the compressed resolvent of $ \tilde{A} $ , is a normalized generalized Nevanlinna pair. Conversely, every normalized generalized Nevanlinna pair is shown to be associated with some self-adjoint linear relation $ \tilde{A} $ in the above sense. A functional model for this selfadjoint linear relation $ \tilde{A} $ is constructed.     

Interface foliation for an inhomogeneous Allen–Cahn equation in Riemannian manifolds

Let ${(\mathcal{M}, \tilde{g})}$ be an N-dimensional smooth compact Riemannian manifold. We consider the problem ${\varepsilon^2 \triangle_{\tilde{g}} \tilde{u} + V(\tilde{z})\tilde{u}(1-\tilde{u}^2)=0\; {\rm in}\; \mathcal{M}}$ , where ${\varepsilon > 0}$ is a small parameter and V is a positive, smooth function in ${\mathcal{M}}$ . Let ${\kappa \subset \mathcal{M}}$ be an (N ? 1)-dimensional smooth submanifold that divides ${\mathcal{M}}$ into two disjoint components ${\mathcal{M}_{\pm}}$ . We assume κ is stationary and non-degenerate relative to the weighted area functional ${\int_{\kappa}V^{\frac{1}{2}}}$ . For each integer m ≥ 2, we prove the existence of a sequence ${\varepsilon = \varepsilon_\ell \rightarrow 0}$ , and two opposite directional solutions with m-transition layers near κ, whose mutual distance is ${{\rm O}(\varepsilon | \log \varepsilon | )}$ . Moreover, the interaction between neighboring layers is governed by a type of Jacobi–Toda system.     

Remarks on the existence of solutions to some quasilinear elliptic problems involving the Hardy-Leray potential

We study the solvability of the quasilinear problem $$\begin{aligned} -\Delta _p u =\frac{u^q }{|x|^p}+g(\lambda , x, u) \quad u>0 \quad \text{ in}\;\Omega , \end{aligned}$$ with $u=0$ on $\partial \Omega $ , where $-\Delta _p(\cdot )$ is the $p$ -Laplacian operator, $q>0, 1<p<N$ and $\Omega $ a smooth bounded domain in $\mathbb R ^N$ . We consider the following cases:

1. $g(\lambda ,x,u)\equiv 0$ ;
2. $g(\lambda ,x,u)=\lambda f(x)u^r$ , with $\lambda >0$ and $f(x) \gneq 0$ belonging to $L^{\infty }(\Omega )$ and $0 \le r<p-1$ .

In the case $(i)$ , the existence of solutions depends on the location of the origin in the domain, on the geometry of the domain and on the exponent $q$ . On the other hand, in the case $(ii)$ , the existence of solutions only depends on the position of the origin and on the coefficient $\lambda $ , but does not depend either on the exponent $q$ or on the geometry of $\Omega $ .     

Lower and upper bounds of Dirichlet eigenvalues for totally characteristic degenerate elliptic operators

Let λkbe the k-th Dirichlet eigenvalue of totally characteristic degenerate elliptic operator-ΔB defined on a stretched cone B0 ■ [0,1) × X with boundary on {x1 = 0}. More precisely,ΔB=(x1αx1)2+ α2x2+ + α2xnis also called the cone Laplacian. In this paper,by using Mellin-Fourier transform,we prove thatλk Cnk2 n for any k 1,where Cn=(nn+2)(2π)2(|B0|Bn)-2n,which gives the lower bounds of the Dirchlet eigenvalues of-ΔB. On the other hand,by using the Rayleigh-Ritz inequality,we deduce the upper bounds ofλk,i.e.,λk+1 1 +4n k2/nλ1. Combining the lower and upper bounds of λk,we can easily obtain the lower bound for the first Dirichlet eigenvalue λ1 Cn(1 +4n)-12n2.     

Bergman and Calderón Projectors for Dirac Operators

For a Dirac operator $D_{\bar{g}}$ over a spin compact Riemannian manifold with boundary $(\bar{X},\bar{g})$ , we give a new construction of the Calderón projector on $\partial\bar{X}$ and of the associated Bergman projector on the space of L ² harmonic spinors on $\bar{X}$ , and we analyze their Schwartz kernels. Our approach is based on the conformal covariance of $D_{\bar{g}}$ and the scattering theory for the Dirac operator associated with the complete conformal metric $g=\bar{g}/\rho^{2}$ where ρ is a smooth function on $\bar{X}$ which equals the distance to the boundary near $\partial\bar{X}$ . We show that $\frac{1}{2}(\operatorname{Id}+\tilde{S}(0))$ is the orthogonal Calderón projector, where $\tilde{S}(\lambda)$ is the holomorphic family in {?(λ)≥0} of normalized scattering operators constructed in Guillarmou et al. (Adv. Math., 225(5):2464–2516, 2010), which are classical pseudo-differential of order 2λ. Finally, we construct natural conformally covariant odd powers of the Dirac operator on any compact spin manifold.     

Translation invariant diffusions in the space of tempered distributions

In this paper we prove existence and pathwise uniqueness for a class of stochastic differential equations (with coefficients σ _ij , b _i and initial condition y in the space of tempered distributions) that may be viewed as a generalisation of Ito’s original equations with smooth coefficients. The solutions are characterized as the translates of a finite dimensional diffusion whose coefficients σ _ij $\tilde y$ , b _i $\tilde y$ are assumed to be locally Lipshitz.Here denotes convolution and $\tilde y$ is the distribution which on functions, is realised by the formula $\tilde y\left( r \right): = y\left( { - r} \right)$ . The expected value of the solution satisfies a non linear evolution equation which is related to the forward Kolmogorov equation associated with the above finite dimensional diffusion.     

A generalized min–max theorem for functionals of strongly indefinite sign

We consider non-linear Schrödinger equations of the following type: $$\begin{aligned} \left\{ \begin{array}{l} -\Delta u(x) + V(x)u(x)-q(x)|u(x)|^\sigma u(x) = \lambda u(x), \quad x\in \mathbb{R }^N \\ u\in H^1(\mathbb{R }^N)\setminus \{0\}, \end{array} \right. \end{aligned}$$ where $N\ge 1$ and $\sigma >0$ . We will concentrate on the case where both $V$ and $q$ are periodic, and we will analyse what happens for different values of $\lambda $ inside a spectral gap $]\lambda ^-,\lambda ^+[$ . We derive both the existence of multiple orbits of solutions and the bifurcation of solutions when $\lambda \nearrow \lambda ^+$ . Thereby we use the corresponding energy function ${I_\lambda }$ and we derive a new variational characterization of multiple critical levels for such functionals: in this way we get multiple orbits of solutions. One main advantage of our new view on some specific critical values $c_0(\lambda )\le c_1(\lambda )\le \cdots \le c_n(\lambda )\le \cdots $ is a multiplicity result telling us something about the number of critical points with energies below $c_n(\lambda )$ , even if for example two of these values $c_i(\lambda )$ and $c_j(\lambda )$ ( $0\le i<j\le n$ ) coincide. Let us close this summary by mentioning another main advantage of our variational characterization of critical levels: we present our result in an abstract setting that is suitable for other problems and we give some hints about such problems (like the case corresponding to a Coulomb potential $V$ ) at the end of the present paper.     

A smooth global branch of solutions for a semilinear elliptic equation on {\mathbb{R}^N}

The existence of a global branch of positive spherically symmetric solutions ${\{(\lambda,u(\lambda)):\lambda\in(0,\infty)\}}$ of the semilinear elliptic equation $$\Delta u - \lambda u + V(x)|u|^{p-1}u = 0 \quad \text{in}\,\mathbb{R}^N\,\text{with}\,N\geq3$$ is proved for ${1 < p < 1+\frac{4-2b}{N-2}}$ , where ${b\in(0,2)}$ is such that the radial function V vanishes at infinity like |x|^?b . V is allowed to be singular at the origin but not worse than |x|^?b . The mapping ${\lambda\mapsto u(\lambda)}$ is of class ${C^r((0,\infty),H^1(\mathbb{R}^N))}$ if ${V\in C^r(\mathbb{R}^N\setminus\{0\},\mathbb{R})}$ , for r = 0, 1. Further properties of regularity and decay at infinity of solutions are also established. This work is a natural continuation of previous results by Stuart and the author, concerning the existence of a local branch of solutions of the same equation for values of the bifurcation parameter λ in a right neighbourhood of λ = 0. The variational structure of the equation is deeply exploited and the global continuation is obtained via an implicit function theorem.     

Anderson’s Orthogonality Catastrophe for One-Dimensional Systems

The overlap, \({\mathcal{D}_N}\) , between the ground state of N free fermions and the ground state of N fermions in an external potential in one spatial dimension is given by a generalized Gram determinant. An upper bound is \({\mathcal{D}_N\leq\exp(-\mathcal{I}_N)}\) with the so-called Anderson integral \({\mathcal{I}_N}\) . We prove, provided the external potential satisfies some conditions, that in the thermodynamic limit \({\mathcal{I}_N = \gamma\ln N + O(1)}\) as \({N\to\infty}\) . The coefficient γ > 0 is given in terms of the transmission coefficient of the one-particle scattering matrix. We obtain a similar lower bound on \({\mathcal{D}_N}\) concluding that \({\tilde{C} N^{-\tilde{\gamma}} \leq \mathcal{D}_N \leq CN^{-\gamma}}\) with constants C, \({\tilde{C}}\) , and \({\tilde{\gamma}}\) . In particular, \({\mathcal{D}_N\to 0}\) as \({N\to\infty}\) which is known as Anderson’s orthogonality catastrophe.     

Existence of positive periodic solutions for second-order functional differential equations

In this paper, we show the existence of positive $T$ -periodic solutions of second-order functional differential equations $u^{\prime \prime }(t)-\rho ^2u(t)+\lambda g(t)f(u(t-\tau (t)))=0,\ \ t\in \mathbb R , $ where $\rho >0$ is a constant, $g\in C(\mathbb R ,[0,\infty ))$ , $\tau \in C(\mathbb R ,\mathbb R )$ are $T$ -periodic functions, $f\in C([0,\infty ),[0,\infty ))$ and $\lambda $ is a positive parameter. Our approach based on global bifurcation theorem.     

Classical Euclidean wormhole solutions in the Palatini f(\tilde R) cosmology

We study the classical Euclidean wormholes in the context of extended theories of gravity. Without loss of generality, we use the dynamical equivalence between f( $\tilde R$ ) gravity and scalar-tensor theories to construct a pointlike Lagrangian in the flat Friedmann-Robertson-Walker space-time. We first show the dynamical equivalence between the Palatini f( $\tilde R$ ) gravity and the Brans-Dicke theory with a self-interaction potential and then show the dynamical equivalence between the Brans-Dicke theory with a self-interaction potential and the minimally coupled O’Hanlon theory. We show the existence of new Euclidean wormhole solutions for this O’Hanlon theory; in a special case, we find the corresponding form of f( $\tilde R$ ) that has a wormhole solution. For small values of the Ricci scalar, this f( $\tilde R$ ) agrees with the wormhole solution obtained for the higher-order gravity theory $\tilde R + \varepsilon \tilde R^2 ,\varepsilon < 0$ .     

Existence and Multiplicity Results for Some Elliptic Systems in Unbounded Cylinders

We study the following nonlinear elliptic system of Lane–Emden type $$\left\{\begin{array}{ll} -\Delta u = {\rm sgn}(v) |v| ^{p-1} \qquad \qquad \qquad \; {\rm in} \; \Omega , \\ -\Delta v = - \lambda {\rm sgn} (u)|u| \frac{1}{p-1} + f(x, u)\; \; {\rm in}\; \Omega , \\ u = v = 0 \qquad \qquad \qquad \quad \quad \;\;\;\;\; {\rm on}\; \partial \Omega , \end{array}\right.$$ where ${\lambda \in \mathbb{R}}$ . If ${\lambda \geq 0}$ and ${\Omega}$ is an unbounded cylinder, i.e., ${\Omega = \tilde \Omega \times \mathbb{R}^{N-m} \subset \mathbb{R}^{N}}$ , ${N - m \geq 2, m \geq 1}$ , existence and multiplicity results are proved by means of the Principle of Symmetric Criticality and some compact imbeddings in partially spherically symmetric spaces. We are able to state existence and multiplicity results also if ${\lambda \in \mathbb{R}}$ and ${\Omega}$ is a bounded domain in ${\mathbb{R}^{N}, N \geq 3}$ . In particular, a good finite dimensional decomposition of the Banach space in which we work is given.     

Extinction profile of the logarithmic diffusion equation

Let ${N \geq 3}$ and u be the solution of u _t = Δ log u in ${\mathbb{R}^N \times (0, T)}$ with initial value u ₀ satisfying ${B_{k_1}(x, 0) \leq u_{0} \leq B_{k_2}(x, 0)}$ for some constants k ₁ > k ₂ > 0 where ${B_k(x, t) = 2(N - 2)(T - t)_{+}^{N/(N - 2)}/(k + (T - t)_{+}^{2/(N - 2)}|x|^{2})}$ is the Barenblatt solution for the equation and ${u_0 - B_{k_0} \in L^{1}(\mathbb{R}^{N})}$ for some constant k ₀ > 0 if ${N \geq 4}$ . We give a new different proof on the uniform convergence and ${L^1(\mathbb{R}^N)}$ convergence of the rescaled function ${\tilde{u}(x, s) = (T - t)^{-N/(N - 2)}u(x/(T - t)^{-1/(N - 2)}, t), s = -{\rm log}(T - t)}$ , on ${\mathbb{R}^N}$ to the rescaled Barenblatt solution ${\tilde{B}_{k_0}(x) = 2(N - 2)/(k_0 + |x|^{2})}$ for some k ₀ > 0 as ${s \rightarrow \infty}$ . When ${N \geq 4, 0 \leq u_0(x) \leq B_{k_0}(x, 0)}$ in ${\mathbb{R}^N}$ , and ${|u_0(x) - B_{k_0}(x, 0)| \leq f \in L^{1}(\mathbb{R}^{N})}$ for some constant k ₀ > 0 and some radially symmetric function f, we also prove uniform convergence and convergence in some weighted L ¹ space in ${\mathbb{R}^N}$ of the rescaled solution ${\tilde{u}(x, s)}$ to ${\tilde{B}_{k_0}(x)}$ as ${s \rightarrow \infty}$ .     

Biharmonic elliptic problems involving the 2nd Hessian operator

In this paper we will study the equation $$\begin{aligned} \Delta ^2 u=S_2(D^2u),\quad \Omega \subset \mathbb {R}^N, \end{aligned}$$ with \(N=3,\) where \( S_2(D^2u)(x)=\sum _{1\le i , being \(\lambda _i,\) the solutions to the equation $$\begin{aligned} \mathrm{det}\left( \lambda I-D^2u(x)\right) =0, \end{aligned}$$ \(i=1,\dots ,N,\) and \(\Omega \) is a bounded domain with smooth boundary. We deal with several boundary conditions looking for the appropriate framework to get existence and multiplicity of nontrivial solutions. This kind of equation is related to some models of growth, and for this reason it is natural to study the effect of zero order local reaction terms of the type \(F_{\lambda }(x,u)=\lambda |u|^{p-1}u\) , with \(\lambda \in \mathbb {R}\) , \(\lambda >0\) , and \(0 , and also the solvability of the boundary problems with a source term \(f\) satisfying some integrability hypotheses.