Singularly perturbed elliptic boundary value problems

Summary The paper treats elliptic operators of the form L(ɛ∂₁, ..., ɛ∂_n), where L is a polynomial in a variables of order 2m₁, and ɛ is a small parameter. Solutionsu _ɛ of Lu=0 in a half space satisfyng conditions B_j(ɛ∂₁, ɛ∂₂, ..., ɛ∂_n)u=ɛγ_jϕ_j(x)(j=1, ..., m₁) on the boundary are constructed and estimated using H?lder norms, Poisson kernels, and an elaborate potential theory. Properties of the interior limit u₀=u _ɛ(κ) are studied. The paper is preparatory to a detailed investigation of Schauder estimates for such problems with variables coefficients. Supported in part by N. S. F. Grant GP-11660. Entrata in Redazione il 9 gennaio 1971.     

Singularly perturbed nonlinear elliptic problems on manifolds

Let be a connected compact smooth Riemannian manifold of dimension with or without smooth boundary We consider the following singularly perturbed nonlinear elliptic problem on

Singularly perturbed elliptic systems

We study the existence and concentration behavior of positive solutions for a class of Hamiltonian systems (two coupled nonlinear stationary Schrödinger equations). Combining the Legendre–Fenchel transformation with mountain pass theorem, we prove the existence of a family of positive solutions concentrating at a point in the limit, where related functionals realize their minimum energy. In some cases, the location of the concentration point is given explicitly in terms of the potential functions of the stationary Schrödinger equations.     

Degenerate elliptic boundary value problems with asymptotically linear nonlinearity

The purpose of this paper is to study a class of semilinear degenerate elliptic boundary value problems with asymptotically linear nonlinearity which include as particular cases the Dirichlet and Robin problems. Our approach is based on the global inversion theorems between Banach spaces, and is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. By making use of the variational method, we prove existence and uniqueness theorems for our problem. The results here extend three earlier theorems due to Ambrosetti and Prodi to the degenerate case.     

Solutions of elliptic problems with nonlinearities of linear growth

In this paper, we study existence of nontrivial solutions to the elliptic equation

Biharmonic equations with asymptotically linear nonlinearities

This article considers the equation △2u = f(x, u)with boundary conditions either u|(a)Ω = (a)u/(a)n|(a)Ω = 0 or u|(a)Ω = △u|(a)Ω = 0, where f(x,t) is asymptotically linear with respect to t at infinity, and Ω is a smooth bounded domain in RN, N ＞ 4. By a variant version of Mountain Pass Theorem, it is proved that the above problems have a nontrivial solution under suitable assumptions of f(x, t).     

Dirichlet problem with discontinuous nonlinearities super-linear or asymptotically linear at infinity

In the present paper, a class of Dirichlet problem with discontinuous nonlinearities super-linear or asymptotically linear at infinity is studied, and some new existence theorems of solutions are obtained.     

Singularly perturbed problems with double singularity

A class of singularly perturbed initial and boundary value problems for systems of linear differential equations with singularities of various types is studied. The asymptotics of the solutions of these problems is constructed; in contrast to known results, it involves boundary layers of new types that are dependent not only on the spectrum of the limit operator. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 494–501, October, 1997. Translated by N. K. Kulman     

Multiplicity results for asymptotically linear elliptic problems at resonance

In this paper several new multiplicity results for asymptotically linear elliptic problem at resonance are obtained via Morse theory and minimax methods. Some new observations on the critical groups of a local linking-type critical point are used to deal with the resonance case at 0.     

Multiple positive solutions for Kirchhoff type problems with singularity and asymptotically linear nonlinearities

In this paper, we devote ourselves to investigating a nonlocal problem involving singularity and asymptotically linear nonlinearities. By using the variational and perturbation methods, we obtain the existence of two positive solutions which improve the existing result in the literature.     

具有转向点曲线的椭圆型方程奇摄动边值问题

The singularly perturbed elliptic equation boundary value problem with a curve of turning point is considered. Using the method of multiple scales and the comparison theorem,the asymptotic behavior of solution for the boundary value problem is studied.     

Multiplicity results for second-order two-point boundary value problems with superlinear or sublinear nonlinearities

We consider boundary value problems for nonlinear second order differential equations of the form

     

On semilinear elliptic equations with sublinear and superlinear nonlinearities in rN

The existence of infinitely many solutions for the equation     

Singularly perturbed boundary value problems

1.IntroductionThesingularlypedurbednonlineartwO-pointboundaryvalueproblemsinthisarticleareofthefollowingtypewhereconstantparametere>0,constantsx?ER,asERe.Thestatesofthesystem(XI(t),Xz(t),YI(t),YZ(t)),whichtakevaluesinRxRexRadxRe,containonepart(XI(t),YI(t))calledslowvariablesandonepart(XZ(t),YZ(t))calledfastvariables.Welookforthesolutions(XI,X2,YI,Y2)EL'(0,T;R m m)whichsatisfysystem(1).Suchasituationiscommonintheapplications.Itappearsforexampleineconomicmodelstotakeintoaccoun…     

Singularly perturbed linear boundary-value problems with pulse effects and the regular reduced problem

A singularly perturbed linear boundary-value problem with pulse effect is considered. By using pseudo-inverse matrices, we construct an asymptotic solution with a single boundary layer.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 463–467, April, 1995.