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The law of large numbers and a functional equation

100%

Sablik M.

1998

tom 68

nr 2

165-175

We deal with the linear functional equation (E) $g(x) = ∑^r_{i=1} p_i g(c_i x)$, where g:(0,∞) → (0,∞) is unknown, $(p₁,...,p_r)$ is a probability distribution, and $c_i$'s are positive numbers. The equation (or some equivalent forms) was considered earlier under different assumptions (cf. [1], [2], [4], [5] and [6]). Using Bernoulli's Law of Large Numbers we prove that g has to be constant provided it has a limit at one end of the domain and is bounded at the other end.

Large versus bounded solutions to sublinear elliptic problems

86%

Damek E. , Ghardallou Z.

2019

tom Vol. 67, no. 1

69--82

Let L be a second order elliptic operator with smooth coefficients defined on a domain Ω ⸦ Rd (possibly unbounded), d ≥ 3. We study nonnegative continuous solutions u to the equation Lu(x) - φ (x, u(x)) = 0 on Ω, where φ is in the Kato class with respect to the first variable and it grows sublinearly with respect to the second variable. Under fairly general assumptions we prove that if there is a bounded nonzero solution then there is no large solution.

Bounded solutions for nonlinear elliptic equations in unbounded domains

72%

Stańczy R.

Journal of Applied Analysis

2000

tom Vol. 6, nr 1

129-138

We prove here the existence of a bounded, radial solution in unbounded domain of the nonlinear elliptic problem ?u = f(||x||,u) for ||x||> 1, x ∈ Rn u(x)=0 for ||x||=1 under some asymptotic and sign condition on f. Under stronger assumptions it is proved that this solution must be of constant sign. The existence of radial solutions, vanishing at ? , of some semilinear equation is also established here.

On bounded solutions of second order systems

72%

Herzog G.

2006

tom Vol. 39, nr 4

793-801

We prove existence and uniqueness of bounded solutions of u"+f(t, u) = 0, u(0) = x on [0,infinity) under quasimonotonicity and one-sided Lipschitz conditions on f.