Three solutions to a perturbed nonlinear discrete Dirichlet problem

Three Positive Solutions to a Discrete Focal Boundary Value Problem

We are concerned with the discrete focal boundary value problem ∆3x(t−k) = f(x(t)), x(a) = ∆x(t2) = ∆2x(b+ 1) = 0. Under various assumptions on f and the integers a, t2, and b we prove the existence of three positive solutions of this boundary value problem. To prove our results we use fixed point theorems concerning cones in a Banach space.

Dirichlet Duality and the Nonlinear Dirichlet Problem

We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form F.Hessu/ D 0 on a smoothly bounded domain  b Rn. In our approach the equation is replaced by a subset F Sym.Rn/ of the symmetric n nmatrices with @F fF D 0g. We establish the existence and uniqueness of continuous solutions under an explicit geometric “F -convexity” assumption on the boundary @. We a...

A Computer Algebra Approach to the Discrete Dirichlet Problem

Radial Solutions to a Dirichlet Problem Involving Critical Exponents

In this paper we show that, for each λ > 0, the set of radially symmetric solutions to the boundary value problem −∆u(x) = λu(x) + u(x)|u(x)|, x ∈ B := {x ∈ R : ‖x‖ < 1}, u(x) = 0, x ∈ ∂B, is bounded. Moreover, we establish geometric properties of the branches of solutions bifurcating from zero and from infinity.

Positive solutions of a nonlinear three-point boundary value problem

We study the existence of positive solutions to the boundary-value problem u + a(t)f(u) = 0, t ∈ (0, 1) u(0) = 0, αu(η) = u(1) , where 0 < η < 1 and 0 < α < 1/η. We show the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed point theorem in cones.