Existence and multiplicity results for some quasilinear elliptic equation with weights

Using variational methods, we show the existence and multiplicity of solutions of singular boundary value problems of the type

     

Positive solutions for Neumann problems with indefinite and unbounded potential

We consider semilinear Neumann equations with an indefinite and unbounded potential. We establish the existence and uniqueness of positive solutions. We show that our setting incorporates as special cases several parametric equations of interest (such as the equidiffusive logistic equation).     

Positive solutions for critical quasilinear elliptic equations with mixed dirichlet-neumann boundary conditions

The existence and multiplicity of positive solutions are studied for a class of quasilinear elliptic equations involving Sobolev critical exponents with mixed Dirichlet-Neumann boundary conditions by the variational methods and some analytical techniques.     

Hardy-Sobolev critical singular elliptic equations with mixed Dirichlet-Neumann boundary conditions

Three important inequalities (the Poincaré, Hardy and generalized Poincaré inequalities) on the mixed boundary conditions are firstly established by some analytical techniques. Then the existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with mixed Dirichlet-Neumann boundary conditions involving Hardy terms and Hardy-Sobolev critical exponents by using the variational methods.     

On a nonlinear elliptic problem with critical potential in R~2

Consider the existence of nontrivial solutions of homogeneous Dirichlet problem for a nonlinear elliptic equation with the critical potential in R2. By establishing a weighted inequality with the best constant, determine the critical potential in R2, and study the eigenvalues of Laplace equation with the critical potential. By the Pohozaev identity of a solution with a singular point and the Cauchy-Kovalevskaya theorem, obtain the nonexis tence result of solutions with singular points to the nonlinear elliptic equation. Moreover, for the same problem, the existence results of multiple solutions are proved by the mountain pass theorem.     

ON CRITICAL SINGULAR QUASILINEAR ELLIPTIC PROBLEM WHEN n = p

This article deals with the problem where n = p. The authors prove that a Hardy inequality and the constant (p/p-1)p is optimal. They also prove the existence of a nontrivial solution of the above mentioned problem by using the Mountain Pass Lemma.     

Existence of positive solutions to fractional elliptic problems with Hardy potential and critical growth

In this work, we study the following critical problem involving the fractional Laplacian: where s ∈ (0,1), N > 2s, , and is the fractional critical exponent, 0 < μ < Λ_N,s, the sharp constant of the Hardy‐Sobolev inequality. For suitable assumptions on g(x) and K(x), we consider the existence and multiplicity of positive solutions depending on the value of p. Moreover, we obtain an existence result for the problem when λ = 0.     

Multiple positive solutions for a semilinear elliptic equation with critical Sobolev exponent

In this paper, we study a class of semilinear elliptic equations with Hardy potential and critical Sobolev exponent. By means of the Ekeland variational principle and Mountain Pass theorem, multiple positive solutions are obtained.     

On a class of critical singular quasilinear elliptic problem with indefinite weights

In this paper, the eigenvalue problem for a class of quasilinear elliptic equations involving critical potential and indefinite weights is investigated. We obtain the simplicity, strict monotonicity and isolation of the first eigenvalue λ₁. Furthermore, because of the isolation of λ₁, we prove the existence of the second eigenvalue λ₂. Then, using the Trudinger-Moser inequality, we obtain the existence of a nontrivial weak solution for a class of quasilinear elliptic equations involving critical singularity and indefinite weights in the case of 0<λ<λ₁ by the Mountain Pass Lemma, and in the case of λ₁≤λ<λ₂ by the Linking Argument Theorem.     

Existence of solutions for singular critical semilinear elliptic equation

This paper is devoted to the existence of solutions for a singular critical semilinear elliptic equation. Some existence and multiplicity results are obtained by using mountain pass arguments and analysis techniques. The results of Ding and Tang (2007) and Kang (2007) and related are improved.     

含有奇异位势和临界指数的拟线性椭圆方程的G-对称解

本文讨论一类奇异拟线性椭圆型方程
-div(|x|^-ap|▽u|^p-2▽u)=μ+h(x)/|x|^(a+1)p|u|^p-2u+k(x)|u|^p-2u/|x|^bq,x∈R^N,
其中1 < p < N, 0 ≤ a < N-p/p, a ≤ b < a + 1, 0 ≤ μ < μ = (N-p/p-a)^p, q=p^*(a, b) = Np/N-(1+a-b)p,h 和k 是R^N上的连续有界函数, 且关于O(N) 的闭子群G满足某些对称性条件. 应用变分方法和Caffarelli-Kohn-Nirenberg 不等式, 在h与k满足适当条件下, 证得了一些G-对称解的存在性和多重性结果.     

Infinitely many solutions to elliptic systems with critical exponents and Hardy potentials

In this paper,a system of elliptic equations is investigated,which involves Hardy potential and multiple critical Sobolev exponents.By a global compactness argument of variational method and a fine analysis on the Palais-Smale sequences created from related approximation problems,the existence of infinitely many solutions to the system is established.     

Gradient estimates for solutions to quasilinear elliptic equations with critical sobolev growth and hardy potential

This note is a continuation of the work[17].We study the following quasilinear elliptic equations(■)where 1 p N,0 ≤μ ((N-p)/p)~p and Q ∈ L~∞(R~N).Optimal asymptotic estimates on the gradient of solutions are obtained both at the origin and at the infinity.     

Infinitely many radial solutions to elliptic problems with critical Sobolev and Hardy terms

Let Ω RN be a ball centered at the origin with radius R > 0 and N 7, 2* = 2N/N-2. We obtain the existence of infinitely many radial solutions for the Dirichlet problem -△u = μ |x|2 u |u|2*-2u λu in Ω, u = 0 on аΩ for suitable positive numbers μ and λ. Such solutions are characterized by the number of their nodes.     

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.     

Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities

We study a time-independent nonlinear Schrödinger equation with an attractive inverse square potential and a nonautonomous nonlinearity whose power is the critical Sobolev exponent. The problem shares a strong resemblance with the prescribed scalar curvature problem on the standard sphere. Particular attention is paid to the blow-up possibilities, i.e. the critical points at infinity of the corresponding variational problem. Due to the strong singularity in the potential, some new phenomenon appear. A complete existence result is obtained in dimension 4 using a detailed analysis of the gradient flow lines.

     

A global compactness result for singular elliptic problems involving critical Sobolev exponent

Let be a bounded domain such that . Let be a (P.S.) sequence of the functional . We study the limit behaviour of and obtain a global compactness result.