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

In this paper, 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.     

Sign-changing solutions for some fourth-order nonlinear elliptic problems

In this paper, we consider the existence and multiplicity of sign-changing solutions for some fourth-order nonlinear elliptic problems and some existence and multiple are obtained. The weak solutions are sought by means of sign-changing critical theorems.     

Sign-Changing and Multiple Solutions Theorems for Semilinear Elliptic Boundary Value Problems with Jumping Nonlinearities

In this paper, we use the ordinary differential equation theory of Banach spaces and minimax theory, and in particular, the relative mountain pass lemma to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity, and get new multiple solutions and sign-changing solutions theorems, at last we get up to six nontrivial solutions. Received April 21, 1998, Revised November 2, 1998, Accepted January 14, 1999     

Nonlocal second-order geometric equations arising in tomographic reconstruction

In this paper, we study new nonlocal geometric equations which are related to tomographic reconstruction when using the level-set approach. We treat two additional difficulties which make the work original. On one hand, the level lines do not evolve along normal directions and the nonlocal term is not of “convolution type”. On the other hand, the speed is not necessarily bounded compared to the nonlocal term. We prove an existence and uniqueness result for our equation.     

Boundary-value Problems for Integro-differential Equations of Elliptic Type

In this paper we study the existence of solutions to the Dirichlet problem for a class of integro-differential equations of elliptic type by using the weakly continuous method.     

Diffusion–convection reaction equations with sign-changing diffusivity and bistable reaction term

We consider a reaction–diffusion equation with a convection term in one space variable, where the diffusion changes sign from the positive to the negative and the reaction term is bistable. We study the existence of wavefront solutions, their uniqueness and regularity. The presence of convection reveals several new features of wavefronts: according to the mutual positions of the diffusivity and reaction, profiles can occur either for a single value of the speed or for a bounded interval of such values; uniqueness (up to shifts) is lost; moreover, plateaus of arbitrary length can appear; profiles can be singular where the diffusion vanishes.     

Generic multiplicity for a scalar field equation on compact surfaces

We prove generic multiplicity of solutions for a scalar field equation on compact surfaces via Morse inequalities. In particular our result improves significantly the multiplicity estimate which can be deduced from the degree-counting formula in Chen and Lin (2003) [12]. Related results are derived for the prescribed Q-curvature equation.     

高阶亚线性Duffing方程的周期解

在本文中，二阶亚线性Ｄｕｆｆｉｎｇ方程周期解存在的结果被推广到高阶Ｄｕｆｆｉｎｇ方：ｘ＾（２ｎ）＋ｇ（ｘ）＝ｐ（ｔ）＝ｐ（ｔ＋２π）（ｎ≥１）和ｘ＾（２ｎ＋１）＋ｇ（ｘ）－ｐ（ｔ）＝ｐ（ｔ＋２π）。     

带Hardy位势的双调和椭圆方程的正负解及变号解

本文讨论带Hardy位势的四阶渐近线性椭圆方程,应用变分方法,我们得到了正负解及变号解的存在性.     

KIRCHHOFF TYPE EQUATIONS DEPENDING ON A SMALL PARAMETER

ＫＩＲＣＨＨＯＦＦＴＹＰＥＥＱＵＡＴＩＯＮＳＤＥＰＥＮＤＩＮＧＯＮＡＳＭＡＬＬＰＡＲＡＭＥＴＥＲ￥Ｐ．Ｄ＇ＡＮＣＯＮＡＳ．ＳＰＡＧＮＯＬＯ（ＤｅｐａｒｔｍｅｌｌｔｏｆＭａｔｈｅｍａｔｉｃｓ,ＰｉｓａＵｎｉｖｅｒｓｉｔｙ！Ｐｉｓａ,Ｉｔａｌｙ）Ａｂｓｔ...     

A Bahri–Lions theorem revisited

In 1988, A. Bahri and P.L. Lions [A. Bahri, P.L. Lions, Morse-index of some min–max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988) 1027–1037] studied the following elliptic problem:

where Ω is a bounded smooth domain of , 2<p<(2N−2)/(N−2) and f(x,u) is not assumed to be odd in u. They proved the existence of infinitely many solutions under an appropriate growth restriction on f. In the present paper, we improve this result by showing that under the same growth assumption on f the problem admits in fact infinitely many sign-changing solutions. In addition we derive an estimate on the number of their nodal domains. We also deal with the corresponding fourth order equation Δ²u=|u|^p−2u+f(x,u) with both Dirichlet and Navier boundary conditions, as well as with strongly coupled elliptic systems.     

Nonlocal symmetries and recursion operators: Partial differential and differential-difference equations

A systematic method to derive the nonlocal symmetries for partial differential and differential-difference equations with two independent variables is presented and shown that the Korteweg-de Vries (KdV) and Burger's equations, Volterra and relativistic Toda (RT) lattice equations admit a sequence of nonlocal symmetries. An algorithm, exploiting the obtained nonlocal symmetries, is proposed to derive recursion operators involving nonlocal variables and illustrated it for the KdV and Burger's equations, Volterra and RT lattice equations and shown that the former three equations admit factorisable recursion operators while the RT lattice equation possesses (2×2) matrix factorisable recursion operator. The existence of nonlocal symmetries and the corresponding recursion operator of partial differential and differential-difference equations does not always determine their mathematical structures, for example, bi-Hamiltonian representation.     

Existence of positive solution for a system of elliptic equations via bifurcation theory

In this paper we study the existence of solution for the following class of elliptic systems

(P)$\{\begin{array}{c}?\mathrm{\Delta }u=(a?{\int }_{\mathrm{\Omega }}K(x,y)f(u,v)dy)u+bv,\text{in}\mathrm{\Omega }\hfill \\ ?\mathrm{\Delta }v=(d?{\int }_{\mathrm{\Omega }}\mathrm{\Gamma }(x,y)g(u,v)dy)v+cu,\text{in}\mathrm{\Omega }\hfill \\ u=v=0,\text{on}?\mathrm{\Omega }\hfill \end{array}$

where $\mathrm{\Omega }?{\mathbb{R}}^N$ is a smooth bounded domain, $N\ge 1$, and $K,\mathrm{\Gamma }:\mathrm{\Omega }\times \mathrm{\Omega }\to \mathbb{R}$ are nonnegative functions satisfying some hypotheses and $a,b,c,d\in \mathbb{R}$. The functions f and g satisfy some conditions which permit to use Bifurcation Theory to prove the existence of solution for (P).     

On a nonlocal Cahn-Hilliard/Navier-Stokes system with degenerate mobility and singular potential for incompressible fluids with different densities

We consider a diffuse interface model describing flow and phase separation of a binary isothermal mixture of (partially) immiscible viscous incompressible Newtonian fluids having different densities. The model is the nonlocal version of the one derived by Abels, Garcke and Grün and consists in a inhomogeneous Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard equation. This model was already analyzed in a paper by the same author, for the case of singular potential and non-degenerate mobility. Here, we address the physically more relevant situation of degenerate mobility and we prove existence of global weak solutions satisfying an energy inequality. The proof relies on a regularization technique based on a careful approximation of the singular potential. Existence and regularity of the pressure field is also discussed. Moreover, in two dimensions and for slightly more regular solutions, we establish the validity of the energy identity. We point out that in none of the existing contributions dealing with the original (local) Abels, Garcke Grün model, an energy identity in two dimensions is derived (only existence of weak solutions has been proven so far).     

On the structure of global solutions of the nonlinear heat equation in a ball

In this paper, we consider the nonlinear heat equation

(NLH)

u_t−Δu=|u|^αu,