Analysis and Computation of Solutions for a Class of Nonlinear SBVPs Arising in Epitaxial Growth

In this work, the existence and nonexistence of stationary radial solutions to the elliptic partial differential equation arising in the molecular beam epitaxy are studied. Since we are interested in radial solutions, we focus on the fourth-order singular ordinary differential equation. It is non-self adjoint, it does not have exact solutions, and it admits multiple solutions. Here, λ∈R measures the intensity of the flux and G is stationary flux. The solution depends on the size of the parameter λ. We use a monotone iterative technique and integral equations along with upper and lower solutions to prove that solutions exist. We establish the qualitative properties of the solutions and provide bounds for the values of the parameter λ, which help us to separate existence from nonexistence. These results complement some existing results in the literature. To verify the analytical results, we also propose a new computational iterative technique and use it to verify the bounds on λ and the dependence of solutions for these computed bounds on λ.

Infinitely Many Rotationally Symmetric Solutions to a Class of Semilinear Laplace–Beltrami Equations on Spheres

We show that a class of semilinear Laplace–Beltrami equations on the unit sphere in ℝn has inûnitely many rotationally symmetric solutions. The solutions to these equations are the solutions to a two point boundary value problem for a singular ordinary differential equation. We prove the existence of such solutions using energy and phase plane analysis. We derive a Pohozaev-type identity in order to prove that the energy to an associated initial value problem tends to infinity as the energy at the singularity tends to infinity. The nonlinearity is allowed to grow as fast as |s|p-1s for |s| large with 1 < p < (n + 5)/(n − 3).

The Reduction of a Type of Singular Partial Differential Equations

Using the theory with respect to the existence of holomorphic solutions to a singular ordinary differential equation in the complex domain, this paper shows how to constructs biholomorphic trans-formations in different cases to reduce partial differential equations with singularity at the origin to more simple forms.

Multiple positive solutions to mixed boundary value problems for singular ordinary differential equations on the whole line

This paper is concerned with the mixed boundary value problem of the second order singular ordinary differential equation[Φ(ρ(t)x'(t))]' + f(t, x(t), x'(t)) = 0, t ∈ R,limt→−∞ x(t) = ∫−∞+∞ g(s, x(s), x'(s)) ds,limt→+∞ ρ(t)x'(t) = ∫−∞+∞h(s, x(s), x' (s)) ds.Sufficient conditions to guarantee the existence of at least one positive solution are established. The emphasis is put on the one-dimensional p-Laplacian term [Φ(ρ(t)x'(t))]' involved with the nonnegative function ρ satisfying ∫−∞+∞1/ρ(s) ds = +∞.

The radial-hedgehog solution in Landau–de Gennes' theory for nematic liquid crystals

We study the radial-hedgehog solution in a three-dimensional spherical droplet, with homeotropic boundary conditions, within the Landau–de Gennes theory for nematic liquid crystals. The radial-hedgehog solution is a candidate for a global Landau–de Gennes minimiser in this model framework and is also a prototype configuration for studying isolated point defects in condensed matter physics. The static properties of the radial-hedgehog solution are governed by a non-linear singular ordinary differential equation. We study the analogies between Ginzburg–Landau vortices and the radial-hedgehog solution and demonstrate a Ginzburg–Landau limit for the Landau–de Gennes theory. We prove that the radial-hedgehog solution is not the global Landau–de Gennes minimiser for droplets of finite radius and sufficiently low temperatures and prove the stability of the radial-hedgehog solution in other parameter regimes. These results contain quantitative information about the effect of geometry and temperature on the properties of the radial-hedgehog solution and the associated biaxial instabilities.