The concentration-compactness principle in the calculus of variations. The locally compact case. II. (English) Zbl 0704.49004

Lions, P.-L.

Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 223-283 (1984).

[For part I see the author, ibid. 109-145 (1984; Zbl 0541.49009) which is also covered by the following review.]
Let H be a function space on \({\mathbb{R}}^ N\), and let J,\({\mathcal E}:\) \(H\to {\mathbb{R}}\), \[ {\mathcal E}(u)=\int e(x,Au(x))dx,\quad J(u)=\int j(x,Bu(x))dx, \] where e: \({\mathbb{R}}^ N\times {\mathbb{R}}^ m\to {\mathbb{R}}\), j: \({\mathbb{R}}^ N\times {\mathbb{R}}^ n\to [0,\infty [\), and A: \(H\to E\), B: \(H\to F\) (E,F are function spaces defined on \({\mathbb{R}}^ N\) with values in \({\mathbb{R}}^ m\), \({\mathbb{R}}^ n\), respectively) commute with a translation of \({\mathbb{R}}^ N\); we consider the minimization problem inf\(\{\) \({\mathcal E}(u):\) \(u\in H\), \(J(u)=1\}\). Because of the loss of boundedness of domains, the classical convexity-compactness methods fail to treat the problem and thus the author presents a new method to solve it. He derives a general principle in a heuristic form and it is rigorously justified on all problems studied in the paper. He first imbeds the problem into a one-parameter family of problems \[ I_{\lambda}=\inf \{{\mathcal E}(u):\;u\in H,\quad J(u)=\lambda \},\quad \lambda >0; \] he supposes \(j(x,q)\to j^{\infty}(q)\), \(e(x,p)\to e^{\infty}(p)\) as \(x\to \infty\) for all \(p\in {\mathbb{R}}^ m\), \(q\in {\mathbb{R}}^ n\); and he considers \[ I^{\infty}_{\lambda}=\inf \{{\mathcal E}^{\infty}(u):\;u\in H,\quad J^{\infty}(u)=\lambda \},\text{ where } {\mathcal E}^{\infty}(u)=\int e^{\infty}(Au(x))dx,\quad J^{\infty}(u)=\int j^{\infty}(B(u(x))dx; \] he assumes \(\{\) \(u\in H:\) \(J(u)=\lambda \}\neq \emptyset\), \(I_{\lambda}>-\infty\) for \(\lambda\in]0,1]\) and that minimizing sequences for \(I_{\lambda}\), \(I^{\infty}_{\lambda}\) are bounded in H.
The concentration-compactness principle is the following: In the case when e and j depend on the first variable, for each \(\lambda >0\) all minimizing sequences for I are relatively compact if and only if the strict subadditivity condition \(I_{\lambda}<I_{\alpha}+I^{\infty}_{\lambda -\alpha}\) holds for all \(\alpha\in [0,\lambda [\); in the case when e and j do not depend on the first variable, for each \(\lambda >0\) all minimizing sequences for I are relatively compact up to a translation if and only if the strict subadditivity condition \(I_{\lambda}<I_{\alpha}+I^{\infty}_{\lambda -\alpha}\) holds for all \(\alpha\in]0,\lambda [\) (he remarks that the weak subadditivity condition is always satisfied). The proof is based upon a compactness lemma obtained with the help of the notion of the concentration function of a measure.
The author gives a rigorous proof of the previous principle in several examples: The rotating star problem: \[ \inf \{\int [j(\rho (x))+k(x)\rho (x)]dx-(1/2)\int \rho (x)\rho (y)f(x-y)dxdy:\;\rho \geq 0,\quad \rho \in L^ 1({\mathbb{R}}^ 3),\quad \int \rho (x)dx=\lambda \} \] where K, f are given, j is a convex function, \(\lambda >0\); the Choquard-Pekar problem: \[ \inf \{\int [(1/2)| \nabla u(x)|^ 2+(1/2)V(x)u(x)^ 2]dx- (1/4)\int u(x)^ 2u(y)^ 2(1/| x-y|)dxdy\}, \]
\[ subject\quad to\quad u\in H^ 1({\mathbb{R}}^ N)\text{ and } \int u(x)^ 2dx=1; \] the standing waves in nonlinear Schrödinger equations: \[ \inf \{\int [| \nabla u(x)|^ 2-F(x,u(x))]dx:\;u\in H^ 1({\mathbb{R}}^ N),\quad | u|^ 2_{L^ 2({\mathbb{R}}^ N)}=1\} \] (e.g. \(F(x,t)=| t|^ p)\) and inf\(\{\int [| \nabla u(x)|^ 2+V(x)u(x)^ 2]dx:\) \(u\in H^ 1({\mathbb{R}}^ N)\), \(\int K(x)| u(x)|^ pdx=1\}\), \(p>1\); nonlinear field equations: \[ \inf \{\int | \nabla u(x)|^ 2dx:\;F(x,u(x))dx=\lambda \}; \] unconstrained problems (e.g., Hartree-Fock problems); Euler equations and minimization over manifolds; problems with multiple constraints; problems in unbounded domains other than \({\mathbb{R}}^ N\) (strips, half-spaces, exterior domains, etc.); problems invariant by translation only in some particular directions (e.g., vortex rings, rotating stars).

MSC:

49J27	Existence theories for problems in abstract spaces
49J10	Existence theories for free problems in two or more independent variables
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

concentration-compactness principle; rotating star problem; Choquard- Pekar problem; Schrödinger equations; nonlinear field equations; Hartree-Fock problems; minimization over manifolds

Citations:

Zbl 0541.49009

References:

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