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Large versus bounded solutions to sublinear elliptic problems

Damek Ewa, Ghardallou Zeineb

Bulletin of the Polish Academy of Sciences. Mathematics

2019

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.

Initial data generating bounded solutions of linear discrete equations

Bastinec J., Diblik J., Ruzickova M.

Opuscula Mathematica

2006

Vol. 26, no. 3

395-406

A lot of papers are devoted to the investigation of the problem of prescribed behavior of solutions of discrete equations and in numerous results sufficient conditions for existence of at least one solution of discrete equations having prescribed asymptotic behavior are indicated. Not so much attention has been paid to the problem of determining corresponding initial data generating such solutions. We fill this gap for the case of linear equations in this paper. The initial data mentioned are constructed with use of two convergent monotone sequences. An illustrative example is considered, too.

On bounded solutions of second order systems

Herzog G.

Demonstratio Mathematica

2006

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.

Bounded solutions for nonlinear elliptic equations in unbounded domains

Stańczy R.

Journal of Applied Analysis

2000

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.