Minimax systems

The variational approach to solving nonlinear problems eventually leads to the search for critical points of related functionals. In case of semibounded functionals, one can look for extrema. Otherwise, one is forced to use minimax methods. There are several approaches to such methods. In this paper we unify these approaches providing one theory that works for all of them. The usual approach has used Palais-Smale sequences. We show that all of them lead to Cerami sequences as well. Applications are given.     

Semilinear elliptic problems near resonance with a nonprincipal eigenvalue

We consider the Dirichlet problem for the equation −Δu=λu±f(x,u)+h(x) in a bounded domain, where f has a sublinear growth and h∈L². We find suitable conditions on f and h in order to have at least two solutions for λ near to an eigenvalue of −Δ. A typical example to which our results apply is when f(x,u) behaves at infinity like a(x)|u|^q−2u, with M>a(x)>δ>0, and 1<q<2.     

Multiplicity of positive solutions for semilinear elliptic problems in unbounded domains

In this paper, we study the effect of domain shape on the multiplicity of positive solutions for the semilinear elliptic equations. We prove a Palais-Smale condition in unbounded domains and assert that the semilinear elliptic equation in unbounded domains has multiple positive solutions.     

NontriviaL solutions of some nonlinear elliptic problems

This paper is concerned with a class of semilinear elliptic Dirichlet problems approximating degenerate equations. The aim is to prove the existence of at least 4^k?1 nontrivial solutions when the degeneration set consists of k distinct connected components     

Two solutions for a singular elliptic equation with critical growth at infinity

We look for positive solutions for the singular equation $-\Delta u-\frac{1}{2}\left(x\cdot \nabla u\right)=\mu h\left(x\right)u^{q-1}+\lambda u+u^{(N+2)/(N-2)},\text{in }{\mathbb{R}}^N,$ where $N\ge 3$, $\lambda >0$, $\mu >0$ is a parameter, $0<q<1$ and $h$ has some summability properties. By using a perturbation method and critical point theory, we obtain two solutions when $max\{1,N/4\}<\lambda <N/2$ and the parameter $\mu >0$ is small.     

临界情形的非齐次半线性椭圆型方程的多解性

考虑有界区域Ω RN 上非齐次半线性椭圆型方程 -Δu(x) =up(x) λf(x)在齐次混合边值条件 (即第三边值问题 ) u n au Ω =0下正解的存在性 ,其中α ,λ≥ 0 ,p=N 2N- 2 ,N>2 ,f(x) ∈L∞(Ω) .证明了存在常数λ >0 ,当λ∈ (0 ,λ )时 ,上述问题至少存在两个正解     

Semilinear elliptic equations and fixed points

In this paper, we deal with a class of semilinear elliptic equations in a bounded domain , , with boundary. Using a new fixed point result of the Krasnoselskii type for the sum of two operators, an existence principle of strong solutions is proved. We give two examples where the nonlinearity can be critical.

     

Sandwich pairs in critical point theory

Since the development of the calculus of variations there has been interest in finding critical points of functionals. This was intensified by the fact that for many equations arising in practice the solutions are critical points of functionals. If a functional is semibounded, one can find a Palais-Smale (PS) sequence

These sequences produce critical points if they have convergent subsequences (i.e., if satisfies the PS condition). However, there is no clear method of finding critical points of functionals which are not semibounded. The concept of linking was developed to produce Palais-Smale (PS) sequences for functionals that separate linking sets. In the present paper we discuss the situation in which one cannot find linking sets that separate the functional. We introduce a new class of subsets that accomplishes the same results under weaker conditions. We then provide criteria for determining such subsets. Examples and applications are given.

     

Multiplicity results of periodic solutions for systems of first order delay differential equation

This paper is concerned with the periodic solutions of the following delay non-autonomous systems

(1)     

Non-convex self-dual Lagrangians: New variational principles of symmetric boundary value problems

We study the concept and the calculus of Non-convex self-dual (Nc-SD) Lagrangians and their derived vector fields which are associated to many partial differential equations and evolution systems. They indeed provide new representations and formulations for the superposition of convex functions and symmetric operators. They yield new variational resolutions for large class of Hamiltonian partial differential equations with variety of linear and nonlinear boundary conditions including many of the standard ones. This approach seems to offer several useful advantages: It associates to a boundary value problem several potential functions which can often be used with relative ease compared to other methods such as the use of Euler-Lagrange functions. These potential functions are quite flexible, and can be adapted to easily deal with both nonlinear and homogeneous boundary value problems. Additionally, in most cases the solutions generated using this new method have greater regularity than the solutions obtained using the standard Euler-Lagrange function. Perhaps most remarkable, however, are the permanence properties of Nc-SD Lagrangians; their calculus is relatively manageable, and their applications are quite broad.     

半线性椭圆方程的正解

本文用特征理论及上下解方法,证明了一类半线性椭圆方程边值问题的正解的存在性,同时给出了解的估计.     

Second-order semilinear elliptic inequalities in exterior domains

We study the problem of existence and nonexistence of positive solutions of the semilinear elliptic inequalities in divergence form with measurable coefficients in exterior domains in . For W(x)?|x|^−σ at infinity we compute the critical line on the plane (p,σ), which separates the domains of existence and nonexistence, and reveal the class of potentials V that preserves the critical line. Example are provided showing that the class of potentials is maximal possible, in certain sense. The case of (p,σ) on the critical line has also been studied.     

关于半线性椭圆共振问题的注记

本文应用Ｍｏｒｓｅ理论和惩罚性技巧研究了一类半线性椭圆方程在无穷远处和在原点处都共振情形下非平凡解的存在性．     

A semilinear elliptic PDE not in divergence form via variational methods

In this paper we consider a semilinear equation driven by an operator not in divergence form. Precisely, the principal part of the operator is in divergence form, but it has also a lower order term depending on Du. While the right-hand side of the equation satisfies superlinear and subcritical growth conditions at zero and at infinity. The problem has not a variational structure, but, despite that, we use variational techniques in order to prove an existence and regularity result for the equation.     

Stable solutions of semilinear elliptic problems in convex domains

In this note, we consider semilinear equations , with zero Dirichlet boundary condition, for smooth and nonnegative f, in smooth, bounded, strictly convex domains of . We study positive classical solutions that are semi-stable. A solution u is said to be semi-stable if the linearized operator at u is nonnegative definite. We show that in dimension two, any positive semi-stable solution has a unique, nondegenerate, critical point. This point is necessarily the maximum of u. As a consequence, all level curves of u are simple, smooth and closed. Moreover, the nondegeneracy of the critical point implies that the level curves are strictly convex in a neighborhood of the maximum of u. Some extensions of this result to higher dimensions are also discussed.