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The concentration-compactness principle in the calculus of variations. The locally compact case. II. (English) Zbl 0704.49004
[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 ${\bbfR}\sp N$, and let J,${\cal E}:$ $H\to {\bbfR}$, $$ {\cal E}(u)=\int e(x,Au(x))dx,\quad J(u)=\int j(x,Bu(x))dx, $$ where e: ${\bbfR}\sp N\times {\bbfR}\sp m\to {\bbfR}$, j: ${\bbfR}\sp N\times {\bbfR}\sp n\to [0,\infty [$, and A: $H\to E$, B: $H\to F$ (E,F are function spaces defined on ${\bbfR}\sp N$ with values in ${\bbfR}\sp m$, ${\bbfR}\sp n$, respectively) commute with a translation of ${\bbfR}\sp N$; we consider the minimization problem inf$\{$ ${\cal E}(u):$ $u\in H$, $J(u)=1\}$. 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 $$ I\sb{\lambda}=\inf \{{\cal E}(u):\ u\in H,\quad J(u)=\lambda \},\quad \lambda >0; $$ he supposes $j(x,q)\to j\sp{\infty}(q)$, $e(x,p)\to e\sp{\infty}(p)$ as $x\to \infty$ for all $p\in {\bbfR}\sp m$, $q\in {\bbfR}\sp n$; and he considers $$ I\sp{\infty}\sb{\lambda}=\inf \{{\cal E}\sp{\infty}(u):\ u\in H,\quad J\sp{\infty}(u)=\lambda \},\text{ where } {\cal E}\sp{\infty}(u)=\int e\sp{\infty}(Au(x))dx,\quad J\sp{\infty}(u)=\int j\sp{\infty}(B(u(x))dx; $$ he assumes $\{$ $u\in H:$ $J(u)=\lambda \}\ne \emptyset$, $I\sb{\lambda}>-\infty$ for $\lambda\in]0,1]$ and that minimizing sequences for $I\sb{\lambda}$, $I\sp{\infty}\sb{\lambda}$ 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 $I\sb{\lambda}<I\sb{\alpha}+I\sp{\infty}\sb{\lambda -\alpha}$ 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 $I\sb{\lambda}<I\sb{\alpha}+I\sp{\infty}\sb{\lambda -\alpha}$ 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: $$ \inf \{\int [j(\rho (x))+k(x)\rho (x)]dx-(1/2)\int \rho (x)\rho (y)f(x-y)dxdy:\ \rho \ge 0,\quad \rho \in L\sp 1({\bbfR}\sp 3),\quad \int \rho (x)dx=\lambda \} $$ where K, f are given, j is a convex function, $\lambda >0$; the Choquard-Pekar problem: $$ \inf \{\int [(1/2)\vert \nabla u(x)\vert\sp 2+(1/2)V(x)u(x)\sp 2]dx- (1/4)\int u(x)\sp 2u(y)\sp 2(1/\vert x-y\vert)dxdy\}, $$ $$ subject\quad to\quad u\in H\sp 1({\bbfR}\sp N)\text{ and } \int u(x)\sp 2dx=1; $$ the standing waves in nonlinear Schrödinger equations: $$ \inf \{\int [\vert \nabla u(x)\vert\sp 2-F(x,u(x))]dx:\ u\in H\sp 1({\bbfR}\sp N),\quad \vert u\vert\sp 2\sb{L\sp 2({\bbfR}\sp N)}=1\} $$ (e.g. $F(x,t)=\vert t\vert\sp p)$ and inf$\{\int [\vert \nabla u(x)\vert\sp 2+V(x)u(x)\sp 2]dx:$ $u\in H\sp 1({\bbfR}\sp N)$, $\int K(x)\vert u(x)\vert\sp pdx=1\}$, $p>1$; nonlinear field equations: $$ \inf \{\int \vert \nabla u(x)\vert\sp 2dx:\ F(x,u(x))dx=\lambda \}; $$ unconstrained problems (e.g., Hartree-Fock problems); Euler equations and minimization over manifolds; problems with multiple constraints; problems in unbounded domains other than ${\bbfR}\sp N$ (strips, half-spaces, exterior domains, etc.); problems invariant by translation only in some particular directions (e.g., vortex rings, rotating stars).

MSC:
49J27Optimal control problems in abstract spaces (existence)
49J10Free problems in several independent variables (existence)
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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