Existence and iteration of positive solutions for a singular two-point boundary value problem with a <Emphasis Type="Italic">p</Emphasis>-Laplacian operator

In the paper, we obtain the existence of symmetric or monotone positive solutions and establish a corresponding iterative scheme for the equation (ϕ _p (u′))′+q(t)f(u) = 0, 0 < t < 1, where ϕ _p (s):= |s| ^p−2 s, p > 1, subject to nonlinear boundary condition. The main tool is the monotone iterative technique. Here, the coefficient q(t) may be singular at t = 0; 1.     

Characterization of the absence of a spectral gapfor a class of periodic Jacobi operators

Let q ∈ {2, 3} and let 0 = s₀ < s₁ < … < s_q = T be integers. For m, n ∈ Z, we put ¯m,n = {j ∈ Z| m? j ? n}. We set l_j = s_j − s_j−1 for j ∈ 1, q. Given (p₁,, p_q) ∈ R^q, let b: Z → R be a periodic function of period T such that b(·) = p_j on s_j−1 + 1, s_j for each j ∈ 1, q. We study the spectral gaps of the Jacobi operator (Ju)(n) = u(n + 1) + u(n − 1) + b(n)u(n) acting on l²(Z). By [λ_2j , λ_2j−1] we denote the jth band of the spectrum of J counted from above for j ∈ 1, T. Suppose that p_m ≠ p_n for m ≠ n. We prove that the statements (i) and (ii) below are equivalent for λ ∈ R and i ∈ 1, T − 1.     

A complete classification of bifurcation diagrams of a p -Laplacian Dirichlet problem

We study the bifurcation diagrams of (classical) positive solutions u with |u |_∞ ∈ (0, ∞) of the p -Laplacian Dirichlet problem (φ_p (u ′(x)))′ + λf_q (u (x))) = 0, –1 ≤ x ≤ 1, u (–1) = 0 = u (1), where p > 1, φ_p (y) = |y |^{p –2} y, (φ_p (u ′))′ is the one-dimensional p -Laplacian, λ > 0 is a bifurcation parameter, and the nonlinearity f_q (u) = |1 – u |^q is defined on [0, ∞) with constant q > 0. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global attractors for p-Laplacian equation

The existence of a -global attractor is proved for the p-Laplacian equation ut−div(|∇u|^p−2∇u)+f(u)=g on a bounded domain Ω⊂Rn(n?3) with Dirichlet boundary condition, where p?2. The nonlinear term f is supposed to satisfy the polynomial growth condition of arbitrary order c₁q|u|−k?f(u)u?c₂q|u|+k and f^′(u)?−l, where q?2 is arbitrary. There is no other restriction on p and q. The asymptotic compactness of the corresponding semigroup is proved by using a new a priori estimate method, called asymptotic a priori estimate.     

A local smoothing estimate for the Schrödinger equation

We show that the Schrödinger operator eitΔ is bounded from Wα,q(Rn) to Lq(Rn×[0,1]) for all α>2n(1/2−1/q)−2/q and q?2+4/(n+1). This is almost sharp with respect to the Sobolev index. We also show that the Schrödinger maximal operator sup0<t<1|eitΔf| is bounded from Hs(Rn) to when s>s₀ if and only if it is bounded from Hs(Rn) to L²(Rn) when s>2s₀. A corollary is that sup0<t<1|eitΔf| is bounded from Hs(R²) to L²(R²) when s>3/4.     

Explosive solutions of elliptic equations with absorption and nonlinear gradient term

Letf be a non-decreasing C¹-function such that andF(t)/f ² a(t)→ 0 ast → ∞, whereF(t)=∫ ₀ ^t f(s) ds anda ∈ (0, 2]. We prove the existence of positive large solutions to the equationΔu +q(x)|Δu| ^a =p(x)f(u) in a smooth bounded domain Ω ⊂R^N, provided thatp, q are non-negative continuous functions so that any zero ofp is surrounded by a surface strictly included in Ω on whichp is positive. Under additional hypotheses onp we deduce the existence of solutions if Ω is unbounded.     

Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation u_t = (uⁿ)_xx + lf(u)/(ò_-1¹ f(u)dx)²{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s ^n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}.     

Coincidence sets in quasilinear elliptic problems of monostable type

This paper concerns the formation of a coincidence set for the positive solution of the boundary value problem: −εΔpu=uq−1f(a(x)−u) in Ω with u=0 on ∂Ω, where ε is a positive parameter, Δpu=div(|∇u|^p−2∇u), 1<q?p<∞, f(s)∼|s|^θ−1s(s→0) for some θ>0 and a(x) is a positive smooth function satisfying Δpa=0 in Ω with infΩ|∇a|>0. It is proved in this paper that if 0<θ<1 the coincidence set Oε={x∈Ω:uε(x)=a(x)} has a positive measure for small ε and converges to Ω with order O(ε1/p) as ε→0. Moreover, it is also shown that if θ?1, then Oε is empty for any ε>0. The proofs rely on comparison theorems and the energy method for obtaining local comparison functions.     

A uniform finite element method for a conservative singularly perturbed problem

We examine the problem ϵ(p(x)u′) + (q(x)u)′ − r(x)u = f(x) for 0 < x < 1, p>0, q>0, r ⩾ 0; p, q, r and f in C²[0, 1], ϵ in (0, 1], u(0) and u(1) given. Existence of a unique solution u and bounds on u and its derivatives are obtained. Using finite elements on an equidistant mesh of width h we generate a tridiagonal difference scheme which is shown to be uniformly second order accurate for this problem (i.e., the nodal errors are bounded by Ch², where C is independent of h and ϵ). With a natural choice of trial functions, uniform first order accuracy is obtained in the L^∞(0, 1) norm. Using trial functions which interpolate linearly between the nodal values generated by the difference scheme gives uniform first order accuracy in the L¹(0, 1) norm.     

Multiple positive solutions of nonhomogeneous elliptic systems with strongly indefinite structure and critical Sobolev exponents

In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems −Δv=λu+u^p+εf(x), −Δu=μv+v^q+δg(x) in Ω;u,v>0 in Ω; u=v=0 on ∂Ω, where Ω is a bounded domain in R^N (N?3); 0?f, g∈L∞(Ω); 1/(p+1)+1/(q+1)=(N−2)/N, p,q>1; λ,μ>0. Using sub- and supersolution method and based on an adaptation of the dual variational approach, we prove the existence of at least two nontrivial positive solutions for all λ,μ∈(0,λ₁) and ε,δ∈(0,δ₀), where λ₁ is the first eigenvalue of the Laplace operator −Δ with zero Dirichlet boundary conditions and δ₀ is a positive number.     

Semilinear elliptic problems near resonance with a nonprincipal eigenvalue

We consider the Dirichlet problem for the equation −Δu=λu±f(x,u)+h(x) in a bounded domain, where f has a sublinear growth and h∈L². We find suitable conditions on f and h in order to have at least two solutions for λ near to an eigenvalue of −Δ. A typical example to which our results apply is when f(x,u) behaves at infinity like a(x)|u|^q−2u, with M>a(x)>δ>0, and 1<q<2.     

Labeling bipartite permutation graphs with a condition at distance two

An L(p,q)-labeling of a graph G is an assignment f from vertices of G to the set of non-negative integers {0,1,…,λ} such that |f(u)−f(v)|≥p if u and v are adjacent, and |f(u)−f(v)|≥q if u and v are at distance 2 apart. The minimum value of λ for which G has L(p,q)-labeling is denoted by λ_p,q(G). The L(p,q)-labeling problem is related to the channel assignment problem for wireless networks.In this paper, we present a polynomial time algorithm for computing L(p,q)-labeling of a bipartite permutation graph G such that the largest label is at most (2p−1)+q(bc(G)−2), where bc(G) is the biclique number of G. Since λ_p,q(G)≥p+q(bc(G)−2) for any bipartite graph G, the upper bound is at most p−1 far from optimal.     

Nonexistence of solutions of the dirichlet problem for some quasilinear elliptic equations in a half-space

We prove the monotonicity of nonnegative bounded solutions of the Dirichlet problem for the quasilinear elliptic equation ?Δ_pu = f(u), p ≥ 3, in a half-space. This assertion implies new results on the nonexistence of solutions for the case in which f(u) = u^q with appropriate values of q.     

Labeling graphs with two distance constraints

Given a graph G and integers p,q,d₁ and d₂, with p>q, d₂>d₁?1, an L(d₁,d₂;p,q)-labeling of G is a function f:V(G)→{0,1,2,…,n} such that |f(u)−f(v)|?p if d_G(u,v)?d₁ and |f(u)−f(v)|?q if d_G(u,v)?d₂. A k-L(d₁,d₂;p,q)-labeling is an L(d₁,d₂;p,q)-labeling f such that max_v∈V(G)f(v)?k. The L(d₁,d₂;p,q)-labeling number ofG, denoted by , is the smallest number k such that G has a k-L(d₁,d₂;p,q)-labeling. In this paper, we give upper bounds and lower bounds of the L(d₁,d₂;p,q)-labeling number for general graphs and some special graphs. We also discuss the L(d₁,d₂;p,q)-labeling number of G, when G is a path, a power of a path, or Cartesian product of two paths.     

Positive solutions for the one-dimensional p-Laplacian

In this paper, the criterion for the existence of at least one positive solution of the one-dimensional p-Laplacian (b(t)Φ(u)′ + c(t)f(u) = 0, are obtained, where Φ(u) = |u|^p−1u, p > 0 is a constant, and b(t) > 0 for t > 0. The method used in this paper is shooting method.     

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.     

A method for finding coefficients of a quasilinear hyperbolic equation

The inverse problem of finding the coefficients q(s) and p(s) in the equation u _tt = a ² u _xx + q(u)u _t ? p(u)u _x is investigated. As overdetermination required in the inverse setting, two additional conditions are set: a boundary condition and a condition with a fixed value of the timelike variable. An iteration method for solving the inverse problem is proposed based on an equivalent system of integral equations of the second kind. A uniqueness theorem and an existence theorem in a small domain are proved for the inverse problem to substantiate the convergence of the algorithm.     

Localization on the Boundary of Blow-up for Reaction–Diffusion Equations with Nonlinear Boundary Conditions

Abstract

In this work we analyze the existence of solutions that blow-up in finite time for a reaction–diffusion equation u _t  ? Δu = f(x, u) in a smooth domain Ω with nonlinear boundary conditions ?u/?n = g(x, u). We show that, if locally around some point of the boundary, we have f(x, u) = ?βu ^p , β ≥ 0, and g(x, u) = u ^q then, blow-up in finite time occurs if 2q > p + 1 or if 2q = p + 1 and β < q. Moreover, if we denote by T _b the blow-up time, we show that a proper continuation of the blowing up solutions are pinned to the value infinity for some time interval [T, τ] with T _b  ≤ T < τ. On the other hand, for the case f(x, u) = ?βu ^p , for all x and u, with β > 0 and p > 1, we show that blow-up occurs only on the boundary.     

Uniqueness and Flat Core of Positive Solutions for Quasilinear Elliptic Eigenvalue Problems in General Smooth Domains

The structure of positive solutions to the quasilinear elliptic problems –div(|Du|^p–2Du = λf(u) in Ω, u = 0 on ∂Ω, p > 1, Ω ⊂ R^Na bounded smooth domain, is precisely studied when λ is sufficiently large, for a class of logistic‐type nonlinearities f(u) satisfying that f(0) = f(a) = 0, a > 0, f(u) > 0 for u ∈ (0,a), , while u = a is a zero point of f with order ω. It is shown that if ω ≥ p – 1, the problem has a unique positive solution u_λ with sup _Ω u_λ < a, which develops a boundary layer near ∂Ω. It is shown that if 0 < ω < p – 1, the problem also has a unique positive solution u _λ, but the flat core {x ∈ Ω : u_λ(x) = a} ≠ ∅︁ exists. Moreover, the asymptotic behaviour of the flat core is studied as λ → ∞.     

Local existence of solutions to some degenerate parabolic equation associated with the p-Laplacian

The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|^q−2u(x,t)=f(x,t), (x,t)∈Ω×(0,T), where 2?p<q<+∞, Ω is a bounded domain in RN, is given and Δp denotes the so-called p-Laplacian defined by Δpu:=∇⋅(|∇u|^p−2∇u), with initial data u₀∈Lr(Ω) is proved under r>N(q−p)/p without imposing any smallness on u₀ and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, Lr-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0,T₀] in which the problem admits a solution. More precisely, T₀ depends only on Lr|u₀| and f.