## 2,072 Citations

### 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…

### An abstract version of the concentration compactness principle

- Mathematics
- 2001

We prove an abstract version of concentration compactness principle in Hilbert space and show its applications to a range of elliptic problems on unbounded domains.

### The concentration-compactness principle for the nonlocal anisotropic $p$-Laplacian of mixed order

- Mathematics
- 2021

In this paper, we study the existence of minimizers of the Sobolev quotient for a class of nonlocal operators with an orthotropic structure having different exponents of integrability and different…

### The concentration-compactness principles for Ws,p(·,·)(ℝN) and application

- Mathematics
- 2020

Abstract We obtain a critical imbedding and then, concentration-compactness principles for fractional Sobolev spaces with variable exponents. As an application of these results, we obtain the…

### Existence and multiplicity of solutions for discontinuous elliptic problems in ℝN

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2021

Abstract This paper concerns with the existence of multiple solutions for a class of elliptic problems with discontinuous nonlinearity. By using dual variational methods, properties of the Nehari…

### Solutions of p-Kirchhoff type problems with critical nonlinearity in R N

- Mathematics
- 2018

In this paper, we are interested in the existence of weak solutions for the fractional p-Laplacian equation with critical nonlinearity in RN. By using fractional version of concentration compactness…

### Abstract Concentration Compactness and Elliptic Equations on Unbounded Domains

- Mathematics
- 2001

The notion of concentration compactness, used in numerous applications (cf. P.-L.Lions [4],[5]), was formulated originally in terms of specific functional spaces. In fact, much of the method can be…

### Concentration and compactness in nonlinear Schrödinger–Poisson system with a general nonlinearity☆

- Mathematics
- 2009

### Existence of positive solutions of some nonlinear elliptic problems in unbounded domains

- Mathematics
- 2006

### On a variational problem with lack of compactness related to the Strichartz inequality

- Mathematics
- 2004

Abstract.We study a variational problem from nonlinear fiber optics which strongly lacks compactness, due to the absence of a priori bounds in spaces different from $L^2({\mathbb R})$. A method is…

## References

SHOWING 1-10 OF 44 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 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…

### 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…

### 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 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…

### The existence of a non-minimal solution to the SU (2) Yang-Mills-Higgs equations on ℝ3. Part I

- Mathematics
- 1982

This paper (Part I) and the sequel (Part II) prove the existence of a smooth, non-trivial, finite action solution to the SU (2) Yang-Mills-Higgs equations on ℝ3 in the Bogomol'nyi-Prasad-Sommerfield…