On the number of positive solutions of elliptic systems

The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|?|^{p –2}?) + λk_i (|x |) fⁱ (u₁, …,u_n) = 0, p > 1, R₁ < |x | < R₂, u_i (x) = 0, on |x | = R₁ and R₂, i = 1, …, n, x ∈ ?^N , where k_i and fⁱ, i = 1, …, n, are continuous and nonnegative functions. Let u = (u₁, …, u_n), φ (t) = |t |^{p –2}t, fⁱ₀ = lim_{‖ u ‖→0}((fⁱ ( u ))/(φ (‖ u ‖))), fⁱ_∞= lim_{‖ u ‖→∞}((fⁱ ( u ))/(φ (‖ u ‖))), i = 1, …, n, f = (f¹, …, fⁿ), f ₀ = ∑ⁿ _{i =1} fⁱ ₀ and f _∞ = ∑ⁿ _{i =1} fⁱ _∞. We prove that either f ₀ = 0 and f _∞ = ∞ (superlinear), or f ₀ = ∞and f _∞ = 0 (sublinear), guarantee existence for all λ > 0. In addition, if fⁱ ( u ) > 0 for ‖ u ‖ > 0, i = 1, …, n, then either f ₀ = f _∞ = 0, or f ₀ = f _∞ = ∞, guarantee multiplicity for sufficiently large, or small λ, respectively. On the other hand, either f₀ and f_∞ > 0, or f₀ and f_∞ < ∞ imply nonexistence for sufficiently large, or small λ, respectively. Furthermore, all the results are valid for Dirichlet/Neumann boundary conditions. We shall use fixed point theorems in a cone. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the positive solutions of some nonlinear diffusion problems

We consider the nonlinear diffusion equationu _t –a(x, u _{x x} )+b(x, u)=g(x, u) with initial boundary conditions andu(t, 0)=u(t, 1)=0. Here,a, b, andg denote some real functions which are monotonically increasing with respect to the second variable. Then, the corresponding stationary problem has a positive solution if and only if(0, ^*) or(0, ^*]. The endpoint ^* can be estimated by , where ₁ u denotes the first eigenvalue of the stationary problem linearized at the pointu. The minimal positive steady state solutions are stable with respect to the nonlinear parabolic equation.

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Zusammenfassung Wir betrachten die nichtlineare Diffusionsgleichungu _t –a(x, u _x ) _x +b(x, u)=g(x, u) mit Randbedingungen undu (t, 0)=u (t, 1)=0. Dabei sinda, b, undg monoton wachsende Funktionen bzgl. des zweiten Argumentes. Das zugehörige stationäre Problem hat genau dann eine positive Lösung, falls (0, ^*) oder(0, ^*]. Der Endpunkt ^* kann durch abgeschätzt werden, wobei ₁ u den ersten Eigenwert des an der Stelleu linearisierten stationären Problems bezeichnet. Die minimale positive stationäre Lösung ist stabil bzgl. der obigen nichtlinearen parabolischen Gleichung.

     

Nonexistence of positive solutions to nonlinear nonlocal elliptic systems

In this paper we consider the question of nonexistence of nontrivial solutions for nonlinear elliptic systems involving fractional diffusion operators. Using a weak formulation approach and relying on a suitable choice of test functions, we derive sufficient conditions in terms of space dimension and systems parameters. Also, we present three main results associated to three different classes of systems.     

On positive solutions of perturbed nonlinear differential equations

Some results are given concerning positive solutions of equations of the form x⁽ⁿ⁾ + P(t) G(x) = Q(t, x).Let class I (II) consist of all n-times differentiable functions x(t), such that x(t)>0 and x^{(n ? 1)}(t) ? 0 (x^{(n ? 1)}(t) ? 0) for all large t. Two theorems are given guaranteeing the nonexistence of solutions in class I and II, respectively, and three theorems ensure the convergence to zero of positive solutions. A recent result of Hammett concerning the second-order case is extended to the general case.     

On positive solutions of quasilinear elliptic systems

In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems where –_p is the p-Laplace operator, p > 1 and is a C ^1,-domain in . We prove an analogue of [7, 16] for the eigenvalue problem with and obtain a non-existence result of positive solutions for the general systems.     

Global attractivity of positive periodic solutions for nonlinear impulsive systems

In this paper, the existence, uniqueness and global attractivity of positive periodic solutions for nonlinear impulsive systems are studied. Firstly, existence conditions are established by the method of lower and upper solutions. Then uniqueness and global attractivity are obtained by developing the theories of monotone and concave operators. And lastly, the method and the results are applied to the impulsive n$n$-species cooperative Lotka–Volterra system and a model of a single-species dispersal among n$n$-patches.     

The existence of positive solutions for some nonlinear equation systems

In this paper we consider the existence of positive solutions of the following boundary value problem:

     

Existence of positive bounded solutions for some nonlinear elliptic systems

We prove some existence results of positive bounded continuous solutions to the semilinear elliptic system Δu=λp(x)g(v), Δv=μq(x)f(u) in domains D with compact boundary subject to some Dirichlet conditions, where λ and μ are nonnegative parameters. The functions f,g are nonnegative continuous monotone on (0,∞) and the potentials p, q are nonnegative and satisfy some hypotheses related to the Kato class K(D).     

On positive solutions of some nonlinear fourth-order beam equations

The existence, uniqueness and multiplicity of positive solutions of the following boundary value problem is considered:

u⁽⁴⁾(t)−λf(t,u(t))=0, for 0<t<1,u(0)=u(1)=u″(0)=u″(1)=0,

where λ>0 is a constant, f :[0,1]×[0,+∞)→[0,+∞) is continuous.     

Existence of positive solutions for systems of nonlinear third-order differential equations

This paper is concerned with boundary value problems for systems of nonlinear third-order three-point differential equations. Using fixed-point theorems, the existence of positive solutions is obtained.     

On the number of solutions to systems of Pell equations

We prove that, for positive integers a, b, c and d with c≠d, a>1, b>1, the number of simultaneous solutions in positive integers to ax²−cz²=1, by²−dz²=1 is at most two. This result is the best possible one. We prove a similar result for the system of equations x²−ay²=1, z²−bx²=1.     

非自治Kolmogorov竞争系统的严格正解

In this paper, the existence of strictly positive solutions for N species normutonomous Kolmogorov competition systems is studied. By applying the Schauder‘s flxed pointtheorem some new sufficient conditions are established. In particular, for the almost periodicsystem, tile existence of strictly positive almost periodic solutiorts is obtained, Some previousresults are improved and generalized.     

On uniform estimates for positive solutions of nonlinear differential equations

The following differential equation is considered in this paper: