Nodal solutions for singularly perturbed equations with critical exponential growth

The existence and concentration behavior of nodal solutions are established for the equation −?²Δu+V(z)u=f(u) in Ω, where Ω is a domain in R², not necessarily bounded, V is a positive Hölder continuous function and f∈C¹ is an odd function having critical exponential growth.     

Blow-up rates of radially symmetric large solutions

This paper adapts a technical device going back to [J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations 224 (2006) 385-439] to ascertain the blow-up rate of the (unique) radially symmetric large solution given through the main theorem of [J. López-Gómez, Uniqueness of radially symmetric large solutions, Discrete Contin. Dyn. Syst., Supplement dedicated to the 6th AIMS Conference, Poitiers, France, 2007, pp. 677-686]. The requested underlying estimates are based upon the main theorem of [S. Cano-Casanova, J. López-Gómez, Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line, J. Differential Equations 244 (2008) 3180-3203]. Precisely, we show that if Ω is a ball, or an annulus, f∈C[0,∞) is positive and non-decreasing, V∈C[0,∞)∩C²(0,∞) satisfies V(0)=0, V^′(u)>0, V^″(u)?0, for every u>0, and V(u)∼Hup−1 as u↑∞, for some H>0 and p>1, then, for each λ?0,

−Δu=λu−f(dist(x,∂Ω))V(u)u     

Existence and asymptotic behavior for elliptic equations with singular anisotropic potentials

We study the equation Δu+u|u|^p−1+V(x)u+f(x)=0 in Rn, where n?3 and p>n/(n−2). The forcing term f and the potential V can be singular at zero, change sign and decay polynomially at infinity. We can consider anisotropic potentials of form h(x)|x|⁻² where h is not purely angular. We obtain solutions u which blow up at the origin and do not belong to any Lebesgue space Lr. Also, u is positive and radial, in case f and V are. Asymptotic stability properties of solutions, their behavior near the singularity, and decay are addressed.     

Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control

We study the Schrödinger equation i∂_tu+Δu+V₀u+V₁u=0 on R³×(0,T), where V₀(x,t)=|x-a(t)|^-1, with a∈W^2,1(0,T;R³), is a coulombian potential, singular at finite distance, and V₁ is an electric potential, possibly unbounded. The initial condition u₀∈H²(R³) is such that . The potential V₁ is also real valued and may depend on space and time variables. We prove that if V₁ is regular enough and at most quadratic at infinity, this problem is well-posed and the regularity of the initial data is conserved for the solution. We also give an application to the bilinear optimal control of the solution through the electric potential.     

A Fourier approach for nonlinear equations with singular data

For 0 < m < n, p a positive integer and p > n/(n ? m), we study the inhomogeneous equation L _u +u ^p + V (x)u + f(x) = 0 in ? ⁿ with singular data f and V. The symbol σ of the operator L is bounded from below by |ξ| ^m . Examples of L are Laplacian, biharmonic and fractional order operators. Here f and V can have infinite singular points, change sign, oscillate at infinity, and be measures. Also, f and V can blow up on an unbounded (n?1)-manifold. The solution u can change sign, be nonradial and singular. If σ, f and V are radial, then u is radial. The assumptions on f and V are in terms of their Fourier transforms and we provide some examples.     

On singular perturbations of superlinear elliptic systems

We study the existence, multiplicity and shape of positive solutions of the system −ε²Δu+V(x)u=K(x)g(v), −ε²Δv+V(x)v=H(x)f(u) in RN, as ε→0. The functions f and g are power-like nonlinearities with superlinear and subcritical growth at infinity, and V, H, K are positive and locally Hölder continuous.     

Multiple solutions for a superlinear and periodic elliptic system on \mathbbRN{\mathbb{R}^N}

This paper is concerned with the following periodic Hamiltonian elliptic system

Sparse connectivity certificates via MA orderings in graphs

For an undirected multigraph G=(V,E), let α be a positive integer weight function on V. For a positive integer k, G is called (k,α)-connected if any two vertices u,v∈V remain connected after removal of any pair (Z,E^′) of a vertex subset Z⊆V-{u,v} and an edge subset E^′⊆E such that ∑_v∈Zα(v)+|E^′|<k. The (k,α)-connectivity is an extension of several common generalizations of edge-connectivity and vertex-connectivity. Given a (k,α)-connected graph G, we show that a (k,α)-connected spanning subgraph of G with O(k|V|) edges can be found in linear time by using MA orderings. We also show that properties on removal cycles and preservation of minimum cuts can be extended in the (k,α)-connectivity.     

Hamilton cycles in digraphs of unitary matrices

A set S⊆V is called a q⁺-set (q^--set, respectively) if S has at least two vertices and, for every u∈S, there exists v∈S,v≠u such that N⁺(u)∩N⁺(v)≠∅ (N^-(u)∩N^-(v)≠∅, respectively). A digraph D is called s-quadrangular if, for every q⁺-set S, we have |∪{N⁺(u)∩N⁺(v):u≠v,u,v∈S}|?|S| and, for every q^--set S, we have |∪{N^-(u)∩N^-(v):u,v∈S)}?|S|. We conjecture that every strong s-quadrangular digraph has a Hamilton cycle and provide some support for this conjecture.     

Some results on Hu's conjecture concerning binary trees

Given n weights, w₁, w₂,…, w_n, such that 0?w₁?w₂???w₁, we examine a property of permutation π¹, where π¹=(w₁, w_n, w₂, w_n?1,…), concerning alphabetical binary trees.For each permutation π of these n weights, there is an optimal alphabetical binary tree corresponding to π, we denote it's cost by V(π). There is also an optimal almost uniform alphabetical binary tree, corresponding to π, we denote it's cost by V_u(π).This paper asserts that V_u(π¹)?V_u(π)?V(π) for all π. This is a preliminary result concerning the conjecture of T.C. Hu. Hu's conjecture is V(π¹)?V(π) for all π.     

Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations II

We construct spike layered solutions for the semilinear elliptic equation −ε²Δu+V(x)u=K(x)up−1 on a domain Ω⊂RN which may be bounded or unbounded. The solutions concentrate simultaneously on a finite number of m-dimensional spheres in Ω. These spheres accumulate as ε→0 at a prescribed sphere in Ω whose location is determined by the potential functions V,K.     

Multiplicity of solutions for a class of semilinear Schrödinger equations with sign-changing potential

In this paper, we study the existence of infinitely many nontrivial solutions for a class of semilinear Schrödinger equations −Δu+V(x)u=f(x,u), x∈RN, where the primitive of the nonlinearity f is of superquadratic growth near infinity in u and the potential V is allowed to be sign-changing.     

Existence and concentration behavior of sign-changing solutions for quasilinear Schrödinger equations

We consider the quasilinear Schrödinger equations of the form ?ε²Δu + V(x)u ? ε²Δ(u²)u = g(u), x∈ R^N, where ε > 0 is a small parameter, the nonlinearity g(u) ∈ C¹(R) is an odd function with subcritical growth and V(x) is a positive Hölder continuous function which is bounded from below, away from zero, and inf_ΛV(x) < inf?_ΛV(x) for some open bounded subset Λ of R^N. We prove that there is an ε₀ > 0 such that for all ε ∈ (0, ε₀], the above mentioned problem possesses a sign-changing solution uε which exhibits concentration profile around the local minimum point of V(x) as ε → 0⁺.     

Steiner intervals, geodesic intervals, and betweenness

The concept of the k-Steiner interval is a natural generalization of the geodesic (binary) interval. It is defined as a mapping S:V×?×V?2^V such that S(u₁,…,u_k) consists of all vertices in G that lie on some Steiner tree with respect to a multiset W={u₁,…,u_k} of vertices from G. In this paper we obtain, for each k, a characterization of the class of graphs in which every k-Steiner interval S has the so-called union property, which says that S(u₁,…,u_k) coincides with the union of geodesic intervals I(u_i,u_j) between all pairs from W. It turns out that, as soon as k>3, this class coincides with the class of graphs in which the k-Steiner interval enjoys the monotone axiom (m), respectively (b2) axiom, the conditions from betweenness theory. Notably, S satisfies (m), if x₁,…,x_k∈S(u₁,…,u_k) implies S(x₁,…,x_k)⊆S(u₁,…,u_k), and S satisfies (b2) if x∈S(u₁,u₂,…,u_k) implies S(x,u₂,…,u_k)⊆S(u₁,…,u_k). In the case k=3, these three classes are different, and we give structural characterizations of graphs for which their Steiner interval S satisfies the union property as well as the monotone axiom (m). We also prove several partial observations on the class of graphs in which the 3-Steiner interval satisfies (b2), which lead to the conjecture that these are precisely the graphs in which every block is a geodetic graph with diameter at most two.     

Standing waves for supercritical nonlinear Schrödinger equations

Let V(x) be a non-negative, bounded potential in RN, N?3 and p supercritical, . We look for positive solutions of the standing-wave nonlinear Schrödinger equation Δu−V(x)u+up=0 in RN, with u(x)→0 as |x|→+∞. We prove that if V(x)=o(⁻²|x|) as |x|→+∞, then for N?4 and this problem admits a continuum of solutions. If in addition we have, for instance, V(x)=O(|x|^−μ) with μ>N, then this result still holds provided that N?3 and . Other conditions for solvability, involving behavior of V at ∞, are also provided.     

Invariant tori of nonlinear Schrödinger equation

In this paper, we consider one-dimensional nonlinear Schrödinger equation iut−uxx+V(x)u+f(²|u|)u=0 on [0,π]×R under the boundary conditions a₁u(t,0)−b₁ux(t,0)=0, a₂u(t,π)+b₂ux(t,π)=0, , for i=1,2. It is proved that for a prescribed and analytic positive potential V(x), the above equation admits small-amplitude quasi-periodic solutions corresponding to d-dimensional invariant tori of the associated infinite-dimensional dynamical system.     

The critical Fujita exponent for a diffusion equation with a potential term

We study the initial-boundary-value problem of the diffusion equation u _t = Δu ^m ? V (x)u ^m + u ^p in a conelike domain D = [1,∞) × Ω, where V (x) ~ ω ₂ |x| ^?2 with ω ₂ > 0. Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace–Beltrami operator on Ω, and let l denote the positive root of l ² + (n ? 2)l = ω ₁ + ω ₂. We prove that if m < p ≤ m + 2/(n + l), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if p > m + 2/(n + l), then the problem has global solutions for some \( {u_0}\gneq 0 \) .     

Self-adjointness of Schrödinger-type operators with locally integrable potentials on manifolds of bounded geometry

We consider a Schrödinger-type differential expression , where ∇ is a C^∞-bounded Hermitian connection on a Hermitian vector bundle E of bounded geometry over a manifold of bounded geometry (M,g) with metric g and positive C^∞-bounded measure dμ, and V∈L_loc¹(EndE) is a linear self-adjoint bundle map. We define the maximal operator H_V,max associated to H_V as an operator in L²(E) given by H_V,maxu=H_Vu for all , where ∇∗∇u in is understood in distributional sense. We give a sufficient condition for the self-adjointness of H_V,max. The proof adopts Kato's technique to our setting, but it requires a more general version of Kato's inequality for Bochner Laplacian operator as well as a result on the positivity of u∈L²(M) satisfying the equation (Δ_M+b)u=ν, where Δ_M is the scalar Laplacian on M, b>0 is a constant and ν?0 is a positive distribution on M. For local estimates, we use a family of cut-off functions constructed with the help of regularized distance on manifolds of bounded geometry.     

Existence of Gevrey approximate solutions for certain systems of linear vector fields applied to involutive systems of first-order nonlinear pdes

Given a Gs-involutive structure, (M,V), a Gevrey submanifold X⊂M which is maximally real and a Gevrey function u₀ on X we construct a Gevrey function u which extends u₀ and is a Gevrey approximate solution for V. We then use our construction to study Gevrey micro-local regularity of solutions, u∈C²(RN), of a system of nonlinear pdes of the form

     

Hyper- and reverse-Wiener indices of F-sums of graphs

The Wiener index W(G)=∑_{u,v}⊂V(G)d(u,v), the hyper-Wiener index and the reverse-Wiener index , where d(u,v) is the distance of two vertices u,v in G, d²(u,v)=d(u,v)², n=|V(G)| and D is the diameter of G. In [M. Eliasi, B. Taeri, Four new sums of graphs and their Wiener indices, Discrete Appl. Math. 157 (2009) 794-803], Eliasi and Taeri introduced the F-sums of two connected graphs. In this paper, we determine the hyper- and reverse-Wiener indices of the F-sum graphs and, subject to some condition, we present some exact expressions of the reverse-Wiener indices of the F-sum graphs.