Existence and regularity of multiple solutions for infinitely degenerate nonlinear elliptic equations with singular potential

In this paper, we study the Dirichlet problem for a class of infinitely degenerate nonlinear elliptic equations with singular potential term. By using the logarithmic Sobolev inequality and Hardy's inequality, the existence and regularity of multiple nontrivial solutions have been proved.     

Existence of three nontrivial solutions for nonlinear Neumann hemivariational inequalities

In this paper we consider a nonlinear Neumann problem driven by the p$p$-Laplacian with a nonsmooth potential (hemivariational inequality). Using minimax methods based on the nonsmooth critical point theory together with suitable truncation techniques, we show that the problem has at least three nontrivial smooth solutions. Two of these solutions have constant sign (one is positive, the other negative).     

Multiple solutions for a semilinear equation involving singular potential and critical exponent

Variational methods are used to prove the existence of positive and sign-changing solutions for a semilinear equation involving singular potential and critical exponent in any bounded domain.*supported in part by Tian Yuan Foundation of NNSF (A0324612)**Supported by 973 Chinese NSF and Foundation of Chinese Academy of Sciences.***Supported in part by NNSF of China.Received: September 23, 2002; revised: November 30, 2003     

Solutions to nonlinear Neumann problems with an inverse square potential

Let Ω be an open bounded domain in with smooth boundary . We are concerned with the critical Neumann problem

Multiple positive solutions for nonlinear eigenvalue problems with the p-Laplacian

In this paper we consider a nonlinear eigenvalue problem driven by the p$p$-Laplacian differential operator and with a nonsmooth potential. Using degree theoretic arguments based on the degree map for certain operators of monotone type, we show that the problem has at least two nontrivial positive solutions as the parameter λ>0$\lambda >0$ varies in a half-line.     

Existence and multiplicity of solutions for semilinear hemivariational inequalities at resonance

In this paper we study semilinear hemivariational inequalities at resonance. Assuming a partially anticoercive nonsmooth potential we prove an existence and a multiplicity theorem. The method of the proof is variational and in the multiplicity result, we employ the nonsmooth local linking theorem.     

Results for degenerate nonlinear elliptic equations involving a Hardy potential

Consider the problem

     

Multiple positive and sign-changing solutions for a singular Schrödinger equation with critical growth

Variational methods are used to prove the existence of multiple positive and sign-changing solutions for a Schrödinger equation with singular potential having prescribed finitely many singular points. Some exact local behavior for positive solutions obtained here are also given. The interesting aspects are two. One is that one singular point of the potential V(x)$V(x)$ and one positive solution can produce one sign-changing solution of the problem. The other is that each sign-changing solution changes its sign exactly once.     

Existence of a pair of positive solutions for a class of nonlinear eigenvalue problems

In this paper, we consider a nonlinear elliptic equation driven by the p-Laplacian and with a parameter λ > 0. Using a combination of variational and degree theoretic methods, we show that there exists λ^* > 0 such that, if λ > λ^*, then the problem has two positive smooth solutions. Our result extends earlier ones by Rabinowitz (semilinear equations) and Guo (nonlinear equations).      

Multiplicity of positive solutions for a singular and critical problem

In this paper, we study the following singular, critical elliptic problem :

     

Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian

We consider a nonlinear elliptic equation driven by the p-Laplacian with Dirichlet boundary conditions. Using variational techniques combined with the method of upper-lower solutions and suitable truncation arguments, we establish the existence of at least five nontrivial solutions. Two positive, two negative and a nodal (sign-changing) solution. Our framework of analysis incorporates both coercive and p-superlinear problems. Also the result on multiple constant sign solutions incorporates the case of concave-convex nonlinearities.     

Existence and multiplicity of solutions for Neumann problems

In this paper we examine semilinear and nonlinear Neumann problems with a nonsmooth locally Lipschitz potential function. Using variational methods based on the nonsmooth critical point theory, for the semilinear problem we prove a multiplicity result under conditions of double resonance at higher eigenvalues. Our proof involves a nonsmooth extension of the reduction method due to Castro-Lazer-Thews. The nonlinear problem is driven by the p-Laplacian. So first we make some observations about the beginning of the spectrum of (−Δp,W1,p(Z)). Then we prove an existence and multiplicity result. The existence result permits complete double resonance. The multiplicity result specialized in the semilinear case (i.e. p=2) corresponds to the super-sub quadratic situation.