p-Laplacian方程边界问题的多解性(英文)

在这一篇文章中我们讨论下面这个方程:-Δpu=λf(x,u)inΩ u=0 on Ω,其中Ω是具有光滑边界的有界开集,Ω,p>n,λ>0,且f:Ω×R→R是一个Caratheodory泛函,满足下列条件,存在t>0,使得supt∈[0,t]︱f(.,t)︱∈L∞(Ω),我们可以得出上面方程存在至少三个解。     

Nonradial Solutions of a Mixed Concave-Convex Elliptic Problem

We study the homogeneous Dirichlet problem in a ball for semi-linear elliptic problems derived from the Brezis-Nirenberg one with concave-convex nonlinearities. We are interested in determining non-radial solutions which are invariant with respect to some subgroup of the orthogonal group. We prove that unlike separated nonlinearities, there are two types of solutions, one converging to zero and one diverging. We conclude at the end on the classification of non radial solutions related to the nonlinearity used.     

Eigenvalue Problem with Nonstandard Growth Conditions and Involving Concave and Convex Nonlinearities

In this paper, we are concerned with new phenomena which arise when we study some eigenvalue problem involving variable exponents. Variational method is employed to obtain some existence and multiplicity results.     

A Multiplicity Result for the p-Laplacian Involving a Parameter

We study existence and multiplicity of positive solutions for the following problem

Multiplicity of Periodic Solutions of Duffing Equations with Jumping Nonlinearities

We provide sufficient conditions for the existence and multiplicity of subharrnonic solutions for Duffing's equations with jumping nonlinearities.     

一类带有组合非线性项Kirchhoff方程解的多重性

本文研究一类全空间上的Kirchhoff型方程.当非线性项在无穷远处是线性有界时,利用变分方法建立方程解的多解性结果,并讨论次线性项对问题解的多重性的具体影响,改进了相关文献中的结论.     

Multiplicity Results for an Eigenvalue Problem for Hemivariational Inequalities in Strip-Like Domains

In this paper we study the multiplicity of solutions for a class of eigenvalue problems for hemivariational inequalities in strip-like domains. The first result is based on a recent abstract theorem of Marano and Motreanu, obtaining at least three distinct, axially symmetric solutions for certain eigenvalues. In the second result, a version of the fountain theorem of Bartsch which involves the nonsmooth Cerami compactness condition, provides not only infinitely many axially symmetric solutions but also axially nonsymmetric solutions in certain dimensions. In both cases the principle of symmetric criticality for locally Lipschitz functions plays a crucial role.Mathematics Subject Classifications (2000) 35A15, 35P30, 35J65.Supported by the EU Research Training Network HPRN-CT-1999-00118.     

临界情形下拟线性椭圆方程Nemann问题正解的多重性(英文)

本文讨论某个非线性椭圆方程的Neumann问题在临界情形下正解的多重性.通过Nehari流形的分解,我们证明该方程至少有两个不同的正解.     

包含Sobolev-Hardy指数的p-Laplace方程的多解

在本文中，我们利用Sobolev-Hardy不等式，局部PS条件和亏格理论，证明了一类带临界Sobolev-Hardy指数的奇异p-Laplace方程存在多解．     

Bifurcation of Positive Solutions for a Nonlocal Problem

In this paper, we study bifurcation of positive solutions for a nonlocal problem in a bounded domain. Using the degree argument and variational method, we obtain two results about bifurcation of positive solutions.     

Existence and Multiplicity of Solutions for Fractional Elliptic Problems with Discontinuous Nonlinearities

We consider the following fractional elliptic problem:

$$\begin{aligned} (P)\left\{ \begin{array}{ll} (-\Delta )^s u = f(u) H(u-\mu )&{} \quad \text{ in } \ \Omega ,\\ u =0 &{}\quad \text{ on } \ \mathbb{{R}}^n {\setminus } \Omega , \end{array} \right. \end{aligned}$$

where \((-\Delta )^s, s\in (0,1)\) is the fractional Laplacian, \(\Omega \) is a bounded domain of \(\mathbb{{R}}^n,(n\ge 2s)\) with smooth boundary \(\partial \Omega ,\) H is the Heaviside step function, f is a given function and \(\mu \) is a positive real parameter. The problem (P) can be considered as simplified version of some models arising in different contexts. We employ variational techniques to study the existence and multiplicity of positive solutions of problem (P).

     

Global Solutions to a Nonlocal Fisher-KPP Type Problem

We consider a nonlocal Fisher-KPP reaction-diffusion model arising from population dynamics, consisting of a certain type reaction term \(u^{\alpha} ( 1-\int_{\varOmega}u^{\beta}dx ) \), where \(\varOmega\) is a bounded domain in \(\mathbb{R}^{n}(n \ge1)\). The energy method is applied to prove the global existence of the solutions and the results show that the long time behavior of solutions heavily depends on the choice of \(\alpha\), \(\beta\). More precisely, for \(1 \le\alpha <1+ ( 1-2/p ) \beta\), where \(p\) is the exponent from the Sobolev inequality, the problem has a unique global solution. Particularly, in the case of \(n \ge3\) and \(\beta=1\), \(\alpha<1+2/n\) is the known Fujita exponent (Fujita in J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 13:109–124, 1966). Comparing to Fujita equation (Fujita in J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 13:109–124, 1966), this paper will give an opposite result to our nonlocal problem.     

Multiplicity Results for a Class of Quasilinear Elliptic Problems

By using variational methods we prove the multiplicity of weak solutions of a class of asymptotically p-linear problems.