Nontrivial solutions of semilinear equations at resonance

In this paper we prove new existence results concerning nontrivial solutions to semilinear elliptic problem at resonance. The methods used here are based on combining the minimax methods and the Morse theory.     

Nontrivial solutions of Kirchhoff type problems

In this work, we study Kirchhoff type problems on a bounded domain. We consider the cases where the nonlinearity is superlinear near zero but asymptotically 4-linear at infinity, and the nonlinearity is asymptotically linear near zero but 4-superlinear at infinity. By computing the relevant critical groups, we obtain nontrivial solutions via Morse theory.     

Nontrivial solutions for discrete boundary value problems with multiple resonance via computations of the critical groups

In this paper, we study the existence of nontrivial solutions for a class of second-order difference equations with multiple resonance at both infinity and the origin by applying the critical point theory and Morse theory.     

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．     

On some semilinear elliptic equation involving exponential growth

In this paper, we establish the existence of at least two nontrivial solutions for some semilinear elliptic equation involving a nonlinearity term having a critical exponential growth condition. Our main argument is critical point theory.     

Computations of critical groups and applications to asymptotically linear wave equation and beam equation

In this paper, by using the Morse index theory for strongly indefinite functionals developed in [Nonlinear Anal. TMA, in press], we compute precisely the critical groups at the origin and at infinity, respectively. The abstract theorems are used to study the existence (multiplicity) of nontrivial periodical solutions for asymptotically wave equation and beam equation with resonance both at infinity and at zero.     

Existence and multiplicity results for Dirichlet problems with p-Laplacian

In this paper we apply Morse theory to study the existence of nontrivial solutions of p-Laplacian type Dirichlet boundary value problems.     

An elliptic resonance problem with multiple solutions

In this paper we study the existence of multiple nontrivial solutions for a semilinear elliptic equation at resonance by the minimax methods and Morse theory.     

Existence and multiplicity results for classes of elliptic resonant problems

In this paper we prove some new existence results of nontrivial solutions for classes of elliptic resonant problems. We also establish several multiplicity results. The methods used here are based on combining the minimax methods and the Morse theory especially some new observations on the critical groups of a local linking-type degenerate critical point.     

具有无界非线性项的半线性椭圆共振问题

本文中我们研究一类在高阶特征值处共振、具有无界的非线性项的渐近线性椭圆方程的非平凡解的存在性．     

Remarks on multiple solutions for elliptic resonant problems

We obtain four nontrivial solutions for an elliptic resonant problem via Morse theory and Lyapunov-Schmidt reduction method. Our result improves some recent works.     

Nontrivial solutions of boundary value problems of second-order difference equations

We employ the critical point theory to establish the existence of nontrivial solutions for some boundary value problems of second-order difference equations.     

Dirichlet problems with double resonance and an indefinite potential

We consider semilinear Dirichlet problems with an unbounded and indefinite potential and with a Carathéodory reaction. We assume that asymptotically at infinity the problem exhibits double resonance. Using variational methods, together with Morse theory and flow invariance arguments, we prove multiplicity theorems producing three, five, six or seven nontrivial smooth solutions. In most multiplicity theorems, we provide precise sign information for all the solutions established.     

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.     

High energy solutions and nontrivial solutions for fourth-order elliptic equations

In this paper, we study the existence of nontrivial solutions and infinitely many high energy solutions for a class of nonlinear fourth-order elliptic equations in RN via variational methods. Three main theorems are obtained.     

Multiple solutions of some boundary value problems with parameters

In this paper, we study the existence and multiplicity of nontrivial solutions for the following second-order Dirichlet nonlinear boundary value problem with odd order derivative: −u^″(t)+au^′(t)+bu(t)=f(t,u(t)) for all t∈[0,1] with u(0)=u(1)=0, where a,b∈R¹, f∈C¹([0,1]×R¹,R¹). By using the Morse theory, we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution, two nontrivial solutions and infinitely many solutions, separately.     

Hammerstein非线性积分方程组的非平凡解及应用

该文利用拓扑方法和锥理论研究下列Hammerstein非线性积分方程组u(x)=∫_G k(x,y)f(y,u(y),v(y)) dy,v(x)=∫_Gk(x,y)g(y,u(y),v(y)) dy.在适当的条件下,证明了上述方程组非平解的存在性,并把所得结果应用于研究非线性二阶常微分方程组边值问题的非平凡解的存在性.     

EXISTENCE OF SOLUTIONS TO 2m-ORDER PERIODIC BOUNDARY VALUE PROBLEM

In this paper, we are concerned with the existence of nontrivial solutions to a 2m-order nonlinear periodic boundary value problem. By the infinite dimensional Morse theory, under some conditions on nonlinear term, we obtain that there exist at least two nontrivial solutions.     

Solutions of the quasilinear elliptic problem with a critical Sobolev-Hardy exponent and a Hardy-type term

In the present paper, the quasilinear elliptic problem with a critical Sobolev-Hardy exponent and a Hardy-type term is considered. By means of a variational method, the existence of nontrivial solutions to the problem is obtained.     

Multiple Nontrivial Solutions for Superlinear Double Phase Problems Via Morse Theory

The aim of this paper is the study of a double phase problems involving superlinear nonlinearities with a growth that need not satisfy the Ambrosetti-Rabinowitz condition. Using variational tools together with suitable truncation and minimax techniques with Morse theory, the authors prove the existence of one and three nontrivial weak solutions, respectively.