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.     

Asymptotic behavior for elliptic problems with singular coefficient and nearly critical Sobolev growth

Let be a ball centered at the origin with radius R. We investigate the asymptotic behavior of positive solutions for the Dirichlet problem in on ∂B_R when ɛ→⁺ for suitable positive numbers μ Mathematics Subject Classification (2000) 35J60, 35B33     

A global compactness result for an elliptic equation with double singular terms

In this paper, we establish a global compactness result for (P.S.) sequences of the variational functional of the elliptic problem $\left\{\begin{array}{cc}-\Delta u-\frac{\mu }{|x{|}^2}u=\frac{1}{|x{|}^s}|u{|}^{2_s^1-2}u+\lambda u,\hfill & x\in \Omega ,\hfill \\ u=0\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$where $\Omega \subset {\mathbb{R}}^n$, $n\ge 3$, is a bounded smooth domain with $0\in \Omega$, $\mu \in [0,{(n-2)}^2∕4)$, $s\in [0,2)$ and $\lambda \in \mathbb{R}$ are constants. This extends the global compactness result of Cao and Peng (2003) to the case of elliptic problems with double singular critical terms. Our arguments adapt some refined Sobolev inequalities systematically developed quite recently by Palatucci and Pisante (2014) and blow-up analysis. In this way, our arguments turn out to be quite transparent and easy to be applied to many other problems.     

A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms

Let be a bounded domain such that . We obtain existence of sign-changing solutions for the Dirichlet problem on Ω,u=0 on ∂Ω for suitable positive numbers μ and λ.     

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.     

On a class of singular biharmonic problems involving critical exponents

This paper deals with the following class of singular biharmonic problems

     

Combined effects of singular and critical nonlinearities in elliptic problems

In this paper we consider the existence and multiplicity of positive weak solutions for a class of elliptic problems with the nonlinearity containing both singular and critical terms. By means of the concentration compactness principle due to Lions and Ekeland’s variational principle, two positive weak solutions are obtained.     

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.