Existence and Nonexistence of Entire Positive Solutions of Semilinear Elliptic Systems

We show that entire positive solutions exist for the semilinear elliptic system Δu = p(x)v^α, Δv = q(x)u^β on R^N, N ≥ 3, for positive α and β, provided that the nonnegative functions p and q are continuous and satisfy appropriate decay conditions at infinity. We also show that entire solutions fail to exist if the functions p and q are of slow decay.     

Weighted Weak Behaviour of Fourier-Jacobi Series

Let w(x) = (1 - x)^α (1 + x)^β be a Jacobi weight on the interval [-1, 1] and 1 < p < ∞. If either α > ?1/2 or β > ?1/2 and p is an endpoint of the interval of mean convergence of the associated Fourier-Jacobi series, we show that the partial sum operators S_n are uniformly bounded from L^p,1 to L^p,∞, thus extending a previous result for the case that both α, β > ?1/2. For α, β > ?1/2, we study the weak and restricted weak (p, p)-type of the weighted operators f→uS_n(u^?1f), where u is also Jacobi weight.     

Multiple solutions of sublinear Lane-Emden elliptic equations

We study the sublinear elliptic equation, −Δ u = |u|^psgn u + f(x,u) in the bounded domain Ω under the zero Dirichlet boundary condition. We suppose that 0 < p < 1 and |f(x,u)| is small enough near u = 0 and do not suppose that f(x,u) is odd on u. Then we prove that this problem has infinitely many solutions. Supported in part by the Grant-in-Aid for Scientific Research (C) (No. 16540179), Ministry of Education, Culture, Sports, Science and Technology.     

The existence and nonexistence of entire positive solutions of semilinear elliptic systems with gradient term

We show the existence and nonexistence of entire positive solutions for semilinear elliptic system with gradient term Δu+|∇u|=p(|x|)f(u,v), Δv+|∇v|=q(|x|)g(u,v) on RN, N?3, provided that nonlinearities f and g are positive and continuous, the potentials p and q are continuous, c-positive and satisfy appropriate growth conditions at infinity. We find that entire large positive solutions fail to exist if f and g are sublinear and p and q have fast decay at infinity, while if f and g satisfy some growth conditions at infinity, and p, q are of slow decay or fast decay at infinity, then the system has infinitely many entire solutions, which are large or bounded.     

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.     

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.     

Fujita type conditions to heat equation with variable source

This paper studies heat equation with variable exponent u _t = Δu + u^p(x) + u ^q in ? ^N × (0, T), where p(x) is a nonnegative continuous, bounded function, 0 < p_? = inf p(x) ≤ p(x) ≤ sup p(x) = p₊. It is easy to understand for the problem that all nontrivial nonnegative solutions must be global if and only if max {p₊, q} ≤ 1. Based on the interaction between the two sources with fixed and variable exponents in the model, some Fujita type conditions are determined that that all nontrivial nonnegative solutions blow up in finite time if 0 < q ≤ 1 with p₊ > 1, or 1 < q < 1 + \(\frac{2}{N}\). In addition, if q > 1 + \(\frac{2}{N}\), then (i) all solutions blow up in finite time with 0 < p_? ≤ p₊ ≤ 1 + \(\frac{2}{N}\); (ii) there are both global and nonglobal solutions for p_? > 1 + \(\frac{2}{N}\); and (iii) there are functions p(x) such that all solutions blow up in finite time, and also functions p(x) such that the problem possesses global solutions when p_? < 1 + \(\frac{2}{N}\) < p₊.     

Singular solutions of some nonlinear parabolic equations

We consider the existence and uniqueness of singular solutions for equations of the formu ₁=div(|Du|^p−2 Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2. Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the existence of very singular solutions, i.e. singular solutions such that ∫_|x|≤r u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result. In the case ϕ(u)=u ^q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal. Dedicated to Professor Shmuel Agmon     

Optimal estimates for blowup rate and behavior for nonlinear heat equations

We first describe all positive bounded solutions of where \input amstex \loadmsbm $(y,s)\in \Bbb R^N\times \Bbb R$ , 1 < p, and (N − 2)p ≤ N + 2. We then obtain for blowup solutions u(t) of uniform estimates at the blowup time and uniform space-time comparison with solutions of u′ = u^p. © 1998 John Wiley & Sons, Inc.     

Oscillatory and asymptotic behaviour of fourth order nonlinear difference equations

This paper considers a class of fourth order nonlinear difference equations Δ²(r _n Δ²(y _n ) + Δ²(r _n ,f(n,n _n )=0,n ∈N(n ₀) wheref(n, y) may be classified as superlinear, sublinear, strongly superlinear and strongly sublinear. In superlinear and sublinear cases, necessary and sufficient conditions are obtained for the difference equation to admit the existence of nonoscillatory solutions with special asymptotic properties. In strongly superlinear and strongly sublinear cases, sufficient conditions are given for all solutions to be oscillatory. Partially Supported by the National Science Foundation of China     

Asymptotic behavior of solutions to viscous conservation laws with slowly varying external forces

This paper considers the existence and large time behavior of solutions to the convection-diffusion equation u _t −Δu+b(x)·∇(u|u| ^{q −1})=f(x, t) in ℝ ⁿ ×[0,∞), where f(x, t) is slowly decaying and q≥1+1/n (or in some particular cases q≥1). The initial condition u ₀ is supposed to be in an appropriate L ^p space. Uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term.This work is partially supported by an AMO Grant     

Unboundedness of the large solutions of some asymmetric oscillators at resonance

In this paper, we consider the unboundedness of solutions of the following differential equation (φ_p(x′))′ + (p ? 1)[αφ_p(x⁺) ? βφ_p(x^?)] = f(x)x′ + g(x) + h(x) + e(t) where φ_p(u) = |u|^{p? 2} u, p > 1, x^± = max {±x, 0}, α and β are positive constants satisfying with m, n ∈ N and (m, n) = 1, f and g are continuous and bounded functions such that lim_x→±∞g(x) ? g(±∞) exists and h has a sublinear primitive, e(t) is 2π_p‐periodic and continuous. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Entire solutions of quasilinear elliptic equations

We study entire solutions of non-homogeneous quasilinear elliptic equations, with Eqs. (1) and (2) below being typical. A particular special case of interest is the following: Let u be an entire distribution solution of the equation Δpu=|u|^q−1u, where p>1. If q>p−1 then u≡0. On the other hand, if 0<q<p−1 and u(x)=o(|x|^{p/(p−q−1)}) as |x|→∞, then again u≡0. If q=p−1 then u≡0 for all solutions with at most algebraic growth at infinity.     

Global solutions of the fast diffusion equation with gradient absorption terms

We investigate the existence of nonnegative weak solutions to the problem ut=Δ(um)−p|∇u| in Rn×(0,∞) with ⁺(1−2/n)<m<1. It will be proved that: (i) When 1<p<2, if the initial datum u₀∈D(Rn) then there exists a solution; (ii) When 1<p<(2+mn)/(n+1), if the initial datum u₀(x) is a bounded and nonnegative measure then the solution exists; (iii) When (2+mn)/(n+1)?p<2, if the initial datum is a Dirac mass then the solution does not exist. We also study the large time behavior of the L¹-norm of solutions for 1<p?(2+mn)/(n+1), and the large time behavior of t1/β‖u(⋅,t)−Ec(⋅,t)L^∞_‖ for (2+mn)/(n+1)<p<2.     

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)     

Existence of multiple solutions for semilinear elliptic equations in the annulus

The existence of radial solutions of Δu ＋ λg（｜x｜）f（u） = 0 in annuli with Dirichlet（Dirichlet/Neumann） boundary conditions is investigated.It is proved that the problems have at least two positive radial solutions on any annulus if f is superlinear at 0 and sublinear at ∞.     

Extinction and non-extinction for a polytropic filtration equation with a nonlocal source

In this article, the authors establish the conditions for the extinction of solutions, in finite time, of the fast diffusive polytropic filtration equation u _t ?=?div(|?u ^m | ^p?2?u ^m )?+?a∫_Ω u ^q (y,?t)dy with a, q, m?>?0, p?>?1, m(p???1)?R ^N (N?>?2). More precisely speaking, it is shown that if q?>?m(p???1), any non-negative solution with small initial data vanishes in finite time, and if 0?q?m(p???1), there exists a solution which is positive in Ω for all t?>?0. For the critical case q?=?m(p???1), whether the solutions vanish in finite time or not depends on the comparison between a and μ, where μ?=?∫?_Ωφ ^p?1(x)dx and φ is the unique positive solution of the elliptic problem ?div(|?φ| ^p?2?φ)?=?1, x?∈?Ω; φ(x)?=?0, x?∈??Ω.     

A note on positive evanescent solutions for a certain class of elliptic problems

This paper is devoted to the existence and properties of solutions of the following class of nonlinear elliptic differential equations Δu(x)+f(x,u(x))+g(‖x‖)x⋅∇u(x)=0, x∈Rn, ‖x‖>R. We prove existence of positive solutions vanishing at positive infinity. Our approach is based on the subsolution and supersolution method. The nonlinearity f covers both sublinear and superlinear cases and does not necessarily satisfy f(x,0)≡0. The asymptotic behavior of solutions is also described.     

Oscillatory of unstable type second-order neutral difference equations

We consider the problem of oscillation and nonoscillation solutions for unstable type second-order neutral difference equation: Δ²(x(n) − p(n)x(n −τ)) =q(n)x(g(n)). (1) In this paper, we obtain some conditions for the bounded solutions of Eq(1) to be oscillatory and for the existence of the nonoscillatory solutions.     

A priori bounds for semilinear equations and a new class of critical exponents for Lipschitz domains

A priori bounds for positive, very weak solutions of semilinear elliptic boundary value problems −Δu=f(x,u) on a bounded domain Ω⊂Rn with u=0 on ∂Ω are studied, where the nonlinearity 0?f(x,s) grows at most like sp. If Ω is a Lipschitz domain we exhibit two exponents p_* and p^*, which depend on the boundary behavior of the Green function and on the smallest interior opening angle of ∂Ω. We prove that for 1<p<p_* all positive very weak solutions are a priori bounded in L^∞. For p>p^* we construct a nonlinearity f(x,s)=a(x)sp together with a positive very weak solution which does not belong to L^∞. Finally we exhibit a class of domains for which p_*=p^*. For such domains we have found a true critical exponent for very weak solutions. In the case of smooth domains is an exponent which is well known from classical work of Brezis, Turner [H. Brezis, R.E.L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977) 601-614] and from recent work of Quittner, Souplet [P. Quittner, Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Ration. Mech. Anal. 174 (2004) 49-81].