## 2,854 Citations

The concentration-compactness principle in the calculus of variations. The locally compact case

- Mathematics
- 1984

Compactness and quasilinear problems with critical exponents

- Mathematics
- 2005

Abstract A compactness result is revised in order to prove the pointwise convergence of the gradients of a sequence of solutions to a general quasilinear inequality (anisotropic or not, degenerate or…

Lower Semicontinuity of Functionals via the Concentration-Compactness Principle

- Mathematics
- 2001

In this paper we show the weak lower semicontinuity of some classes of functionals, using the concentration-compactness principle of P. L. Lions. These functionals involve an integral term, and we do…

Concentration-compactness principles for Moser–Trudinger inequalities: new results and proofs

- Mathematics
- 2013

We are concerned with the best exponent in Concentration-Compactness principles for the borderline case of the Sobolev inequality. We present a new approach, which both yields a rigorous proof of the…

-L On the existence of a positive solution of semilinear elliptic equations in unbounded domains

- 2018

. - We prove here the existence of a positive solution, under general conditions, for semilinear elliptic equations in unbounded domains with a variational structure. The solutions we build cannot be…

A Remark on the Concentration Compactness Principle in Critical Dimension

- Mathematics
- 2020

We prove some refinements of concentration compactness principle for Sobolev space $W^{1,n}$ on a smooth compact Riemannian manifold of dimension $n$. As an application, we extend Aubin's theorem for…

An improvement on the concentration-compactness principle

- Mathematics
- 2001

In this paper we first improve the concentration-compactness lemma by proving that the vanishing case is a special case of dichotomy, then we apply this improved concentration-compactness lemma to a…

A concentration-compactness principle at infinity and positive solutions of some quasilinear elliptic equations in unbounded domains

- Mathematics
- 2005

The concentration-compactness principle for fractional order Sobolev spaces in unbounded domains and applications to the generalized fractional Brezis–Nirenberg problem

- MathematicsNonlinear Differential Equations and Applications NoDEA
- 2018

In this paper we extend the well-known concentration-compactness principle for the Fractional Laplacian operator in unbounded domains. As an application we show sufficient conditions for the…

## References

SHOWING 1-10 OF 41 REFERENCES

Compactness and Topological Methods for some Nonlinear Variational Problems of Mathematical Physics

- Mathematics
- 1982

Existence and non-existence results for semilinear elliptic problems in unbounded domains

- Mathematics, PhilosophyProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 1982

Synopsis In this paper, we prove various existence and non-existence results for semilinear elliptic problems in unbounded domains. In particular we prove for general classes of unbounded domains…

Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation

- Mathematics
- 1977

The equation dealt with in this paper is in three dimensions. It comes from minimizing the functional which, in turn, comes from an approximation to the Hartree-Fock theory of a plasma. It describes…

Existence of Stationary States in Nonlinear Scalar Field Equations

- Mathematics
- 1980

We report on some recent results concerning existence of solutions for nonlinear scalar field equations that lead to semilinear elliptic boundary value problems in ℝN. Such problems arise in a wide…

Action minima among solutions to a class of Euclidean scalar field equations

- Mathematics
- 1978

We show that for a wide class of Euclidean scalar field equations, there exist non-trivial solutions, and the non-trivial solution of lowest action is spherically symmetric. This fills a gap in a…

Existence of solitary waves in higher dimensions

- Mathematics
- 1977

The elliptic equation Δu=F(u) possesses non-trivial solutions inRn which are exponentially small at infinity, for a large class of functionsF. Each of them provides a solitary wave of the nonlinear…

Symmetry of positive solutions of nonlinear elliptic equations in R

- Mathematics
- 1981

(1993). Radial symmetry of positive solutions of nonlinear elliptic equations in Rn. Communications in Partial Differential Equations: Vol. 18, No. 5-6, pp. 1043-1054.