Remarks on elliptic singular perturbation problems

We show the effectiveness of viscosity-solution methods in asymptotic problems for second-order elliptic partial differential equations (PDEs) with a small parameter. Our stress here is on the point that the methods, based on stability results [3], [16], apply without hard PDE calculations. We treat two examples from [11] and [23]. Moreover, we generalize the results to those for Hamilton—Jacobi—Bellman equations with a small parameter.H. Ishii was supported in part by the AFOSR under Grant No. AFOSR 85-0315 and the Division of Applied Mathematics, Brown University.     

Multiple solutions of semilinear elliptic problems in a ball

In this paper lower bounds for the number of solutions of semilinear elliptic problems in a ball of $\text{R}$^N are given. Its hypotheses are only related to the behavior of the nonlinearities at ±∞ and at 0. Global assumptions are never made. For example, oddness is never required for the proof of multiplicity results.     

Uniqueness and nondegeneracy for some nonlinear elliptic problems in a ball

In this paper we study the uniqueness and nondegeneracy of positive solutions of nonlinear problems of the type Δ_pu+f(r,u)=0 in the unit ball B, u=0 on ∂B. Here Δ_p denotes the p Laplace operator Δ_p=div(|∇u|^p−2∇u), p>1. The main ideas rely on the Maximum Principle and an implicit function theorem that we derive in a suitable weighted space. This space is essential to deal with the case p≠2.     

Analytical solutions to the heat conduction problems for a cylinder and a ball based on determining the temperature perturbation front

Analytical solutions to the heat conduction problems for a cylinder and a ball are obtained by the integral method of heat balance. To improve the accuracy of the solutions, the temperature function is approximated by polynomials of high degrees. Their coefficients are determined via introducing additional boundary conditions, which are found from the governing differential equation and the basic boundary conditions, including those specified at the temperature perturbation front. It is shown that the additional boundary conditions, even in the second approximation, lead to a considerable improvement in the solution accuracy.     

Singular perturbation problems for systems of partial differential equations of elliptic type

Elliptic boundary value problems for systems of nonlinear partial differential equations of the form $\text{F}{}_{\mathrm{i}}\text{(x, u}{}_1\text{, u}{}_2\text{,…, u}{}_{\mathrm{N}}\text{,}\text{?u}{}_{\mathrm{i}}\text{?x}{}_{\mathrm{j}}\text{, ?}{}^{\mathrm{p}}{}_{\mathrm{i}}\text{?}{}^2\text{u}{}_{\mathrm{i}}\text{?x}{}_{\mathrm{j}}\text{?x}{}_{\mathrm{k}}\text{) = ?}{}_{\mathrm{i}}\text{(x), x ? R}{}^{\mathrm{n}}$, i = 1(1)N, j, k = 1(1)n, p_i ? 0, ? being a small parameter, with Dirichlet boundary conditions are considered. It is supposed that a formal approximation Z is given which satisfies the boundary conditions and the differential equations upto the order χ(?) = o(1) in some norm. Then, using the theory of differential inequalities, it is shown that under certain conditions the difference between the exact solution u of the boundary value problem and the formal approximation Z, taken in the sense of a suitable norm, can be made small.     

Singular perturbation problems for linear elliptic differential operators of arbitrary order. I. Degeneration to elliptic operators

Dirichlet problems of singular perturbation type for linear elliptic differential operators of arbitrary order are studied. The asymptotic validity of approximations constructed by the boundary layer method is demonstrated in the maximum norm by means of a priori estimates.     

Layered solutions for a semilinear elliptic system in a ball

We consider the following system of Schrödinger-Poisson equations in the unit ball B₁ of R³:

     

Nonlinear elliptic problems with a singular weight on the boundary

We study existence of solutions to $$-\Delta u = \frac{u^p}{|x|^2}\quad u\, >\,0 \,{\rm in }\,\Omega$$ with u?=?0 on ???, where ?? is a smooth bounded domain in ${\mathbb {R}^N}$ , N??? 3 with ${0\,\in\,\partial \Omega}$ and ${1< p < \frac{N+2}{N-2}}$ . The existence of solutions depends on the geometry of the domain. On one hand, if the domain is starshaped with respect to the origin there are no energy solutions. On the other hand, in dumbbell domains via a perturbation argument, the equation has solutions.     

Symmetry of large solutions of nonlinear elliptic equations in a ball

Let g be a locally Lipschitz continuous real-valued function which satisfies the Keller-Osserman condition and is convex at infinity, then any large solution of −Δu+g(u)=0 in a ball is radially symmetric.     

Three positive radial solutions for elliptic equations in a ball

In this work we deal with a class of second-order elliptic problems of the form $-\Delta u=\lambda k\left(\left|x\right|\right)f\left(u\right)\text{ in }\Omega$, with non-homogeneous boundary condition $u=a\text{ on }\partial \Omega$ where $\Omega$ is the ball of radius $R_0$ centered at origin, $\lambda ,a$ are positive parameters, $f\in C([0,+\infty ),[0,+\infty ))$ is an increasing function and $k\in C([0,R_0],[0,+\infty ))$ is not identically zero on any subinterval of $[0,R_0]$. We obtain via a fixed point theorem of cone expansion/compression type the existence of at least three positive radial solutions.     

Slope-changing solutions of elliptic problems on

We consider the problem on , where F is a smooth function periodic of period 1 in all its variables. We are going to find a non-degeneracy condition on F for which the following holds. If we are given a sequence of positive integers and a sequence of real numbers (the slopes), then we shall find an increasing sequence {Q_i} of integers and a solution u which is entire, periodic in (x₂,…,x_n) and which is close to the plane α₁(x₁−Q_i)+u(Q_i,0,…,0) for .