The Harnack inequality in ℝ2 for quasilinear elliptic equations

We use maximum principle techniques to obtain a Harnack inequality for two-dimensional elliptic operators     

Runge property and multiplicity of solutions for elliptic equations in ℝN with a monotone nonlinearity

In this paper, we describe a method for extending (in some approximated sense) solutions of a nonlinear P.D.E. on a domain , to solutions in a domain containing . Such an extension property, the Runge property, is well known for a large class of linear problems including elliptic equations. We prove here the Runge property for semilinear problems of the kind -u+g(u)=f, with f L _loc ¹ (^N). (As a consequence, we get infinitely many solutions for these problems). The proof is based on a homotopy method, and requires a refinement of the linear results: We prove that the Runge extension v on of a solution u in for a linear elliptic equation Lu=f can be choosen in order to depend continuously on u and the coefficients of L.     

Multiple solutions for a class of fractional quasi-linear equations with critical exponential growth in ℝN

This paper deals with a class of non-local equations involving the fractional p-Laplacian, where the non-linear term is assumed to have critical exponential growth. More specifically, by applying variational methods together with a suitable Trudinger-Moser inequality for fractional Sobolev space, we obtain the existence of at least two positive weak solutions.     

Existence of solutions of certain quasilinear elliptic equations in ℝN without conditions at infinity

This paper deals with conditions for the existence of solutions of the equations

Infinitely many solutions for a class of sublinear Schrödinger–Maxwell equations

In this paper we study the existence of infinitely many solutions for a class of sublinear Schrödinger–Maxwell equations. The proof is based on the variant fountain theorem established by Zou. Recent results from the literature are extended.     

Long-term analysis of degenerate parabolic equations in ℝ N

Longtime behavior of degenerate equations with the nonlinearity of polynomial growth of arbitrary order on the whole space R^N is considered. By using -trajectories methods, we proved that weak solutions generated by degenerate equations possess an (L_U² (R^N), L_loc² (R^N))-global attractor. Moreover, the upper bounds of the Kolmogorov ε-entropy for such global attractor are also obtained.     

Existence and symmetry for elliptic equations in ℝn with arbitrary growth in the gradient

We define the Polish space R of non-degenerate rank-1 systems. Each non-degenerate rank-1 system can be viewed as a measure-preserving transformation of an atomless, σ-finite measure space and as a homeomorphism of a Cantor space. We completely characterize when two non-degenerate rank-1 systems are topologically isomorphic. We also analyze the complexity of the topological isomorphism relation on R, showing that it is \({F_\sigma }\) as a subset of R× R and bi-reducible to E₀. We also explicitly describe when a non-degenerate rank-1 system is topologically isomorphic to its inverse.     

Multiple solutions for a class of nonlinear elliptic equations on the Sierpinski gasket

This paper investigates a class of nonlinear elliptic equations on a fractal domain. We establish a strong Sobolev-type inequality which leads to the existence of multiple non-trivial solutions of △u+ c(x)u = f(x, u), with zero Dirichlet boundary conditions on the Sierpihski gasket. Our existence results do not require any growth conditions of f(x,t) in t, in contrast to the classical theory of elliptic equations on smooth domains.     

Asymptotic behavior of positive solutions of semilinear elliptic equations in ℝ n , II

In this paper, we will analyze further to obtain a finer asymptotic behavior of positive solutions of semilinear elliptic equations in R^n by employing the Li＇s method of energy function.     

Sign-changing solutions for p-biharmonic equations with Hardy potential in ℝN

In this article, by using the method of invariant sets of descending flow, we obtain the existence of sign-changing solutions of p-biharmonic equations with Hardy potential in ?^N.     

Variational inequalities for a system of elliptic–parabolic equations

We create a general framework for mathematical study of variational inequalities for a system of elliptic–parabolic equations. In this paper, we establish a solvability theorem concerning the existence of solutions for the vector-valued elliptic–parabolic variational inequality with time-dependent constraint. Moreover, we give some applications of the system, for example, time-dependent boundary obstacle problem and time-dependent interior obstacle problem.     

The venttse’ problem for nonlinear elliptic equations

The solvability of the fully nonlinear stationary Venttsel' problem is established. The equation and the boundary condition are assumed to be uniformly elliptic. Bibliography: 12 titles. Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 3–26.     

On the Agmon-Miranda maximum principle for solutions of strongly elliptic equations in domains of ℝn with conical points

In this paper the Agmon-Miranda maximum principle for solutions of strongly elliptic differential equations Lu = 0 in a bounded domain G with a conical point is considered. Necessary and sufficient conditions for the validity of this principle are given both for smooth solutions of the equation Lu = 0 in G and for the generalized solution of the problem Lu = 0 in G, D _k ^v u = gk on G (k = 0,...,m-1). It will be shown that for every elliptic operator L of order 2m > 2 there exists such a cone in ⁿ (n4) that the Agmon-Miranda maximum principle fails in this cone.     

Decay rates of Volterra equations on ℝ N

This note is concerned with the linear Volterra equation of hyperbolic type

A note on existence of antisymmetric solutions for a class of nonlinear Schr?dinger equations

We consider the nonlinear Schr?dinger equation