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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.

On a radial periodic solution to the diffusion equation for the exterior of a ball

Tal-Figiel B.

Fasciculi Mathematici

2002

Nr 33

111-114

The subject of the paper is the construction of a periodic radial solution to the diffusion parabolic differential equation (I) delta u(x,t)-D:u(x,t) = f(x,t), where sigma = D(2)(x)+ D(2)(x2)+ D2(x3), x = (xl,x1,x1), t belonge(-infinity, inffinity), in radial coordinates (r,t) (r,[x] r=\x\= (x(2)(1) + x(2)(2) , in the exterion of the ball D(1), = {(r,t):r>R,t (-infinity, infinity)}. Equation (I) is of the from (II) D] (W(r, t)) - D, (W(r, t) = rU (r, t) (r, t). By the suitable Green function (r, t, p, s) -> G(r, t, p, s), we construct the periodic solution with respect to the variable / of equation (II) as the potential W(r, t) of the double layer.

Green functions and periodic solution to the m-parabolic equation with boundary-value conditions of Riquier type

Barański F., Zdankiewicz Z.

Fasciculi Mathematici

2001

Nr 32

5-13

The subject of the paper is the construction of the periodic solutions to the w-caloric equation P"'u(x,t) = 0, P=D(2)*(2)-D, P(2) =P(P), Pm = P(P(m~l) in the strip (mathemical formula), satisfying the periodic boundary-value conditions (mathematical formula), where h(11)(t), h1,2(t) are the periodic functions with the period p > 0.