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.

     

Existence and asymptotic properties of solutions to elliptic systems involving multiple critical exponents

This paper is concerned with a singular elliptic system, which involves the Caffarelli-Kohn-Nirenberg inequality and multiple critical exponents. By analytic technics and variational methods, the extremals of the corresponding bet Hardy-Sobolev constant are found, the existence of positive solutions to the system is established and the asymptotic properties of solutions at the singular point are proved.     

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.     

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.     

Existence results for degenerate elliptic equations with critical cone Sobolev exponents

In this paper, we study the existence result for degenerate elliptic equations with singular potential and critical cone sobolev exponents on singular manifolds. With the help of the variational method and the theory of genus, we obtain several results under different conditions.     

Existence of solutions for the p-Laplacian with critical Sobolev exponent and convection

In this article, we study the existence of solutions for the p-Laplacian involving critical Sobolev exponent and convection based on the theory of the Leray–Schauder degree for non-compact mappings.     

Existence of nodal solution for semi-linear elliptic equations with critical sobolev exponent on singular manifold

In this article, we prove that semi-linear elliptic equations with critical cone Sobolev exponents possess a nodal solution.     

Existence and multiplicity results for critical growth polyharmonic elliptic systems

In the present paper, we deal with the existence and multiplicity of nontrivial solutions for a class of polyharmonic elliptic systems with Sobolev critical exponent in a bounded domain. Some new existence and multiplicity results are obtained. Our proofs are based on the Nehari manifold and Ljusternik–Schnirelmann theory. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Nodal solutions for fourth order elliptic equations with critical exponent on compact manifolds

Using a variational method, we prove the existence of nodal solutions to prescribed scalar Q- curvature type equations on compact Riemannian manifolds with boundary. These equations are fourth-order elliptic equations with critical Sobolev growth.     

Existence and concentration result for Kirchhoff equations with critical exponent and Hartree nonlinearity

This paper is concerned with the following Kirchhoff-type equations $$ \left\{ \begin{array}{ll} \displaystyle -\big(\varepsilon^{2}a+\varepsilon b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\mathrm{d}x\big)\Delta u + V(x)u+\mu\phi |u|^{p-2}u=f(x,u), &\quad \mbox{ in }\mathbb{R}^{3},\(-\Delta)^{\frac{\alpha}{2}} \phi=\mu|u|^{p},~u>0, &\quad \mbox{ in }\mathbb{R}^{3},\\end{array} \right. $$ where $f(x,u)=\lambda K(x)|u|^{q-2}u+Q(x)|u|^{4}u$, $a>0,~b,~\mu\geq0$ are constants, $\alpha\in(0,3)$, $p\in[2,3),~q\in[2p,6)$ and $\varepsilon,~\lambda>0$ are parameters. Under some mild conditions on $V(x),~K(x)$ and $Q(x)$, we prove that the above system possesses a ground state solution $u_{\varepsilon}$ with exponential decay at infinity for $\lambda>0$ and $\varepsilon$ small enough. Furthermore, $u_{\varepsilon}$ concentrates around a global minimum point of $V(x)$ as $\varepsilon\rightarrow0$. The methods used here are based on minimax theorems and the concentration-compactness principle of Lions. Our results generalize and improve those in Liu and Guo (Z Angew Math Phys 66: 747-769, 2015), Zhao and Zhao (Nonlinear Anal 70: 2150-2164, 2009) and some other related literature.     

Perturbation methods in semilinear elliptic problems involving critical hardy-sobolev exponent

In this article, we deal with a class of semilinear elliptic equations which are perturbations of the problems with the critical Hardy-Sobolev exponent. Some existence results are given via an abstract perturbation method in critical point theory.     

On G-symmetric solutions of a quasilinear elliptic equation involving critical Hardy-Sobolev exponent

This paper deals with the class of singular quasilinear elliptic problem

     

On an obstacle problem for degenerate elliptic operators involving the critical Sobolev exponent

We establish the existence of a solution to the variational inequality (the obstacle problem) (1.1) which involves the critical Sobolev exponent. This result is also extended to an obstacle problem with a lower order perturbation. Dedicated to Professor F. Browder on the occasion of his 80-th birthday     

Sign changing solutions to a nonlinear elliptic problem involving the critical Sobolev exponent in pierced domains

We consider the problem in Ω_ε, u=0 on ∂Ω_ε, where Ω_ε:=ΩB(0,ε) and Ω is a bounded smooth domain in , which contains the origin and is symmetric with respect to the origin, N3 and ε is a positive parameter. As ε goes to zero, we construct sign changing solutions with multiple blow up at the origin.