## 2,068 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…

### On Some Discrete Variational Problems

- Mathematics
- 2001

Making use of a discrete version of the P.-L. Lions concentration-compactness principle, we establish some results on the existence of nontrivial solutions for nonlinear stationary discrete equations…

### 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…

## References

SHOWING 1-10 OF 44 REFERENCES

### The Concentration-Compactness Principle in the Calculus of Variations. (The limit case, Part I.)

- Mathematics
- 1985

After the study made in the locally compact case for variational problems with some translation invariance, we investigate here the variational problems (with constraints) for example in RN where the…

### 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…

### 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…

### Orbital stability of standing waves for some nonlinear Schrödinger equations

- Mathematics, Physics
- 1982

We present a general method which enables us to prove the orbital stability of some standing waves in nonlinear Schrödinger equations. For example, we treat the cases of nonlinear Schrödinger…