Potential bounds in problems on quasilinear elliptic equations

We study quasilinear elliptic equations with strong nonlinear terms and systems of such equations. The methods developed by the authors in [1], [2] are used to prove the existence of solutions for boundary—value problems using some information on behavior of potential bounds for nonlinearities; the L–characteristics of elliptic operators and their fractional powers play an important role. New conditions are suggested for the existence of classical solutions of quasilinear second order elliptic equations.     

Elliptic equations for measures on infinite dimensional spaces and applications

We introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces. This concept allows one to consider equations whose coefficients are not globally integrable. By using a suitably extended Lyapunov function technique, we derive a priori estimates for the solutions of such equations and prove new existence results. As an application, we consider stochastic Burgers, reaction-diffusion, and Navier-Stokes equations and investigate the elliptic equations for the corresponding invariant measures. Our general theorems yield a priori estimates and existence results for such elliptic equations. We also obtain moment estimates for Gibbs distributions and prove an existence result applicable to a wide class of models. Received: 23 January 2000 / Revised version: 4 October 2000 / Published online: 5 June 2001     

非线性参数椭圆系统正解的存在性与多解性

本文讨论了一类非线性含参数椭圆系统正解的存在性与多解性,通过线性算子的谱半径,给出其正径向解存在与多解的条件,本质上改进和推广了文[1－3]的结果．     

Study of weak solutions for a fourth-order parabolic equation with variable exponent of nonlinearity

The aim of this paper is to study the existence and uniqueness of weak solutions for an initial boundary problem of a fourth-order parabolic equation with variable exponent of nonlinearity. First, the authors of this paper apply Leray-Schauder’s fixed point theorem to prove the existence of solutions of the corresponding nonlinear elliptic problem and then obtain the existence of weak solutions of nonlinear parabolic problem by combining the results of the elliptic problem with Rothe’s method. In addition, the authors also discuss the regularity of weak solutions in the case of space dimension one.     

A variational theory of the Hessian equation

By studying a negative gradient flow of certain Hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of nonhomogenous Hessian equations. To achieve this we derive uniform, first‐ and second‐order a priori estimates for the elliptic and parabolic Hessian equations. Our results generalize well‐known results for semilinear elliptic equations and the Monge‐Ampère equation. © 2001 John Wiley & Sons, Inc.     

Elliptic Quadratic Operator Equations

The paper is devoted to the study of elliptic quadratic operator equations over the finite dimensional Euclidean space. We provide necessary and sufficient conditions for the existence of solutions of elliptic quadratic operator equations. The iterative Newton-Kantorovich method for stable solutions is also presented.

     

On a class of nonlinear variational–hemivariational inequalities

A three critical points theorem for nondifferentiable functions is pointed out and an existence result of multiple solutions for a Neumann elliptic variational–hemivariational inequality involving the p-laplacian is established. As an application, a Neumann problem for elliptic equations with discontinuous nonlinearities is studied.     

On solvability of variational inequalities with discontinuous semimonotone operators

By using the method of monotone operators, a theorem on the existence of the solution with a special property is obtained for an elliptic variational inequality with discontinuous semimonotone operator; this theorem is then used to prove the existence of a semicorrect solution of a variational inequality with a differential semilinear high-order operator of elliptic type with a nonsymmetric linear part and discontinuous nonlinearity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 443–447, March, 1993.     

Sub–supersolution method and extremal solutions for higher order quasi-linear elliptic hemi-variational inequalities

In the present paper, we generalize the sub–supersolution method for a class of higher order quasi-linear elliptic hemi-variational inequalities. Using the notion of sub and supersolution, we prove the existence, comparison, compactness and extremality results for the higher order quasi-linear elliptic hemi-variational inequality under considerations.     

Existence of solutions for some non‐Fredholm integro‐differential equations with the bi‐Laplacian

We prove the existence of solutions for some semilinear elliptic equations in the appropriate H⁴ spaces using the fixed‐point technique where the elliptic equation contains fourth‐order differential operators with and without Fredholm property, generalizing the previous results.     

Uniform convergence of finite volume element method with Crouzeix–Raviart element for non-self-adjoint and indefinite elliptic problems

In this paper, we consider the finite volume element method based on the Crouzeix–Raviart element and prove the existence, uniqueness and uniform convergence of the finite volume element approximations for the non-self-adjoint and indefinite elliptic problems under minimal elliptic regularity assumption.     

Principal eigenvalues with indefinite weight functions

Both existence and non-existence results for principal eigenvalues of an elliptic operator with indefinite weight function have been proved. The existence of a continuous family of principal eigenvalues is demonstrated.

     

p-Laplacian Inclusions via Fixed Points for Multifunctions in Posets

We consider the Dirichlet problem for elliptic differential inclusions involving the p-Laplacian and governed by multivalued nonlinearities bounded above and below by single-valued monotone functions not necessarily continuous. The aim of this note is to provide existence and comparison results for the problem under consideration without imposing further regularity assumptions. The main tools used in the proof of our results are existence and comparison results for extremal solutions of (single-valued) nonlinear elliptic equations, and an abstract comparison principle for fixed points of multivalued operators in partially ordered sets (posets).     

一类四阶非线性椭圆问题变号解的存在性

通过建立一个新空间,在新空间中讨论了一个含Hardy位势的四阶非线性椭圆问题变号解的存在性.在一个环绕定理下,得到了问题变号解的存在性.     

Existence results for a nonlinear elliptic system modelling a temperature dependent electrical resistor

Summary The electrical resistance in many substances varies greatly with the temperature which in turns depends on the Joule effect. This nonlinear effect is described under steady conditions by a system of nonlinear elliptic equations with a quadratic growth in the gradient. This work gives a theorem of existence for the related elliptic boundary value problem.     

Existence and uniqueness of solutions for quasilinear elliptic systems

We obtain necessary and sufficient conditions for the existence of positive solutions for a class of sublinear Dirichlet quasilinear elliptic systems.

     

Multiple positive solutions for Robin problem involving critical weighted Hardy–Sobolev exponents with boundary singularities

This paper deals with the existence of positive solutions for Robin elliptic problems involving critical weighted Hardy–Sobolev exponents with boundary singularities. Using the Caffarelli–Kohn–Nirenberg inequalities and variational methods, we prove the existence and multiplicity of positive solutions.     

Degree Theory for Second Order Nonlinear Elliptic Operators and its Applications

We introduce, along the lines of [4], an integer valued degree for second order fully nonlinear elliptic operators which is invariant under homotopy within elliptic operators. We also give some applications to the bifurcation problem for nonlinear elliptic equations. Applications to the existence of solutions of certain fully nonlinear elliptic equations on compact manifolds can be found in [7].     

R2 中带不定权与临界位势的非线性椭圆问题

该文讨论了R²中一类带不定权且含临界位势的二阶椭圆方程的特征值问题,然后,借此特征值问题的第一特征值性质,利用临界点理论及Trudinger-Moser不等式,证明了R² 中一类带不定权且含临界位势的二阶非线性椭圆方程非平凡解的存在性.     

On a nonlinear elliptic problem with critical potential in ?2

Consider the existence of nontrivial solutions of homogeneous Dirichlet problem for a nonlinear elliptic equation with the critical potential in ℝ². By establishing a weighted inequality with the best constant, determine the critical potential in ℝ², and study the eigenvalues of Laplace equation with the critical potential. By the Pohozaev identity of a solution with a singular point and the Cauchy-Kovalevskaya theorem, obtain the nonexistence result of solutions with singular points to the nonlinear elliptic equation. Moreover, for the same problem, the existence results of multiple solutions are proved by the mountain pass theorem.