Nonexistence of Stable Solutions to Quasilinear Elliptic Equations on Riemannian Manifolds

We prove nonexistence of nontrivial, possibly sign changing, stable solutions to a class of quasilinear elliptic equations with a potential on Riemannian manifolds, under suitable weighted growth conditions on geodesic balls.     

界域上具Hardy临界指数项的半线性椭圆方程的多解存在性

考虑了无界域上一类具Hardy临界指数项的半线性椭圆方程,通过证明局部(P．S．)条件和能量估计,运用伪指标理论得到了这类方程多解的存在性．     

Multiplicity of solutions for a class of fourth elliptic equations

We show that there exist at least three nontrivial solutions for a class of fourth elliptic equations under Navier boundary conditions by linking approaches.     

Dirichlet problem for semilinear edge-degenerate elliptic equations with singular potential term

In this paper, we introduce weighted p-Sobolev spaces on manifolds with edge singularities. We give the proof for the corresponding edge type Sobolev inequality, Poincaré inequality and Hardy inequality. As an application of these inequalities, we prove the existence of nontrivial weak solutions for the Dirichlet problem of semilinear elliptic equations with singular potentials on manifolds with edge singularities.     

Homogeneous Solutions of Fully Nonlinear Elliptic Equations in Four Dimensions

We prove that there is no nontrivial homogeneous order 2 analytic outside zero solutions of fully nonlinear, uniformly elliptic equations in ?. © 2013 Wiley Periodicals, Inc.     

Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations

In this paper we study boundary value problems for semilinear equations involving strongly degenerate elliptic differential operators. Via a Pohozaev??s type identity we show that if the nonlinear term grows faster than some power function then the boundary value problem has no nontrivial solution. Otherwise when the nonlinear term grows slower than the same power function, by establishing embedding theorems for weighted Sobolev spaces associated with the strongly degenerate elliptic equations, then applying the theory of critical values in Banach spaces, we prove that the problem has a nontrivial solution, or even infinite number of solutions provided that the nonlinear term is an odd function.     

Sobolev type embedding and quasilinear elliptic equations with radial potentials

We study the existence of nontrivial radial solutions for quasilinear elliptic equations with unbounded or decaying radial potentials. The existence results are based upon several new embedding theorems we establish in the paper for radially symmetric functions.     

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.     

Soil water redistribution and extraction flow models: Conservation laws

We determine all the nontrivial conservation laws for soil water redistribution and extraction flow equations which are modelled by a class of (2+1) nonlinear evolution partial differential equations with three arbitrary elements. It is shown that for arbitrary elements in the model equation there exist trivial conservation laws. We point out that nontrivial conservation laws exist for certain classes of equations which admit point symmetries.     

Boundary layer solutions to functional elliptic equations

The goal of this paper is to study a class of nonlinear functional elliptic equations using very simple comparison principles. We first construct a nontrivial solution and then study its asymptotic behaviour when the diffusion coefficient goes to 0.     

含距离位势的拟线性椭圆方程解的存在性

变分原理证明了一类含距离位势的拟线性椭圆方程齐次Dirichlet边界条件下第一特征值问题的可解性.进一步,利用临界点理论得到了一类含距离位势的非线性椭圆方程非平凡解的存在性.     

具Hardy-Sobolev临界指数的半线性椭圆方程的多解存在性

本文考虑一类具Hardy-Sobolve临界指数的半线性椭圆方程,通过证明局部(P．S．)条件和能量估计,运用伪指标理论得到了这类方程多解的存在性(见文[1-13])．     

Nonlinear Bound States on Weakly Homogeneous Spaces

We prove the existence of ground state solutions for a class of nonlinear elliptic equations, arising in the production of standing wave solutions to an associated family of nonlinear Schrödinger equations. We examine two constrained minimization problems, which give rise to such solutions. One yields what we call F _λ-minimizers, the other energy minimizers. We produce such ground state solutions on a class of Riemannian manifolds called weakly homogeneous spaces, and establish smoothness, positivity, and decay properties. We also identify classes of Riemannian manifolds with no such minimizers, and classes for which essential uniqueness of positive solutions to the associated elliptic PDE fails.     

Remarks on non homogeneous elliptic Kirchhoff equations

Extending the results in (Ambrosetti and Arcoya, Positive solutions of elliptic Kirchhoff equations (to appear)), the existence of solutions for some classes of nonlinear elliptic, possibly degenerate, non homogeneous Kirchhoff equations are proved. We handle smooth nonlinearities as well as the case in which the nonlinearity has an upward jump discontinuity.     

Mountain pass solutions for the coupled systems of second and fourth order elliptic equations

In this paper, by the mountain pass theorem, we give an existence theorem of nontrivial solutions for a coupled system of second and fourth order elliptic equations.     

Applications of local linking to asymptotically linear elliptic problems at resonance

We prove the existence of nontrivial solutions for asymptotically linear elliptic problems at resonance without assuming that the linearized equations at zero and infinity are different. The proof is based on a penalization technique and Morse index estimates for critical points produced by local linking. Received October 10, 1997     

Quantitative uniqueness estimates for the general second order elliptic equations

In this paper we study quantitative uniqueness estimates of solutions to general second order elliptic equations with magnetic and electric potentials. We derive lower bounds of decay rate at infinity for any nontrivial solution under some general assumptions. The lower bounds depend on asymptotic behaviors of magnetic and electric potentials. The proof is carried out by the Carleman method and bootstrapping arguments.     

A note on the Ambrosetti–Rabinowitz condition for an elliptic system

In this note, we study the existence and multiplicity of solutions for a system of coupled elliptic equations. We introduce a revised Ambrosetti–Rabinowitz condition, and show that the system has a nontrivial solution or even infinitely many solutions.     

Verification of Second-Order Sufficient Optimality Conditions for Semilinear Elliptic and Parabolic Control Problems

We study optimal control problems for semilinear parabolic equations subject to control constraints and for semilinear elliptic equations subject to control and state constraints. We quote known second-order sufficient optimality conditions (SSC) from the literature. Both problem classes, the parabolic one with boundary control and the elliptic one with boundary or distributed control, are discretized by a finite difference method. The discrete SSC are stated and numerically verified in all cases providing an indication of optimality where only necessary conditions had been studied before.     

一个带Hardy项的椭圆方程的多重解

通过对偶变分方法证明了一个带Hardy项和临界非线性的非线性椭圆方程的非平凡解的存在性和多重性.