05-5.dvi

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1 (468) , R n ; C,<<1, ([1], c. 3), Lu(x) P ; n (a ij (x)u xi ) xj + c(x)u(x) i1 7j= a ij C 1 (), a ij (x) =a ji (x), c C () Lu(x)+g(x1 7 u(x)) = p(x)1 7 x 1 7 (.1) Buj ; =1 7 (.) g(x1 7 u) (): g :R! R (mod ) ([], c. 157), x g(x1 7 ) R g(x1 7 u) [g ; (x1 7 u)1 7g + (x1 7 u)], g ; (x1 7 u) = inf g(x1 7 s), s!u g + (x1 7 u) = sup g(x1 7 s)1 7 s!u p(x) (.) L ; nx i1 7j=1 uj ; =1 7 a ij (x)u xi cos(n1 7 x j )j ; =1 7 cos(n1 7 x j ) n L (x)+(x)u(x)j ; =1 7 (.3) C 1 (;) ([1], c. 3) ; (.1){(.) u W q (), q 1, (.1) x Bu(x) ; , Lu(x)=1 7 x 1 7 (.4) Buj ; = (.5) , x a L q (), q> n n+, p(x) L q(). jg(x1 7 u)j a(x) 8u R1 7 (.6) 43

2 [3], , g(x1 7 u) g(u) R, g(u) = g 1 7 u!1 g ; <g(u) <g u R, N(L) (.4){ (.5) , (.1){(.) , p g + (x)dx + g ; (x)dx < p(x) (x)dx < g + (x)dx + g ; (x)dx1 7 < > N(L) (.1){(.) , g(x1 7 u) , [4], { ,1 71 7[5], [6] [7], [8] g(x1 7 u) g(u) , g(u) (.1){(.) g(u) u, g(u;) > g(u+). 1 7 [9] K.-C. Chang, Palais{Smale ((P. S.)) , , u W m () T W m (), > ; u(x) [g ; (x1 7 u(x))1 7g + (x1 7 u(x))] (.7) x, , , m , g(x1 7 u) R, un( )1 7 kuk!+1 dx u(x) g(x1 7 s)ds = ;11 7 N() u(x) =, [1] K.-C. Chang g(x1 7 u), (1 7g) , u(x) (.7) ;u(x) = g(x1 7 u(x)), x , g(x1 7 u) u , u = '(x) , '(x) +g(x1 7 '(x)) =, { '(x) = [g ; (x1 7 '(x))1 7g + (x1 7 '(x))] , (.1){(.) ( (.1){(.) , x u(x) g(x1 7 )) u [11] (.1), , [1], , g(x1 7 u) R u <

3 , g(x1 7 u;) g(x1 7 u+) u R x , [1] (.1){(.) g(x1 7 u) u , [9], [1] g(x1 7 u) u, g(x1 7 u;) > g(x1 7 u+) ( "), [1] , " (g(x1 7 u;) <g(x1 7 u+)) p ( [3]) X , Y P X 1 7 Y, P P , { , X X X (1 7 ), y Y x Y hy1 7 xi Qx Ax + P TPx; p =1 7 (1.1) A X N(A), T : Y! Y ([13], c. 53) Y ( , ), p X , T : Y! Y , f : Y! R, f(x + h) ; f(x) = f T. ht (x + th)1 7hidt 8x1 7 h1 7 Y: x X [14] ( ) Q : X! X, h X , suphq(x + th)1 7hi < ( suphq(x + v)1 7hi < ): t!+ v! ([9]) R ' : X! R, x X , '(x )= 1 ) { ' x ([15], c. 34), x ' ( ) x X Q : X! X, x Q ([16], c. 79): hq(x + th)1 7hi = hqx 1 7hi 8h X: (1.) t! ) X , Y, 1 7 P X 1 7 Y 1 7 ) A : X! X , , , (Ax1 7 x) 8x X1 7 45

4 133) T : Y! Y Y, M >, ktxkm 8x Y 1 7 4) p X (f(x) ; (p1 7 x)) = (1.3) xn(a)1 7 kxk! f T x X, '(x ) = inf '(x), '(x) =(Ax1 7 x)=+f(x);(p1 7 x), X x ;Ax +p P (ST)(Px )1 7 (1.4) ST T [17] (1:1) Q, x (1:1) P TP ) Y f T , u1 7 v Y jf(u) ; f(v)j = 1 ht (v + t(u ; v))1 7u; vidt M ) jht (v + t(u ; v))1 7u; vijdt Mku ; vk Y (1.1) x P TP ht (x + t!+ th)1 7hihTx1 7hi 8h X, x (1:1) , hq(x+th)1 7hihQx1 7 hi 8h X 1 7, t! , x (1.) , h X, hq(x + th)1 7hi < hqx1 7 hi hqx1 7 hi x Q t! hqx1 7 hi > hq(x+t(;h))1 7 (;h)i hqx1 7 ;hi = ;hqx1 7 hi <, t! , x Q : ( ) Q : X! X x X, Q ([13], c. 3) , A X, X A : N(A)1 7 X + = fx X j (Ax1 7 x) > g[fg1 7 X ; = fx X j (Ax1 7 x) < g[fg: X ;, X A , ) ), 3) :1 1 7 X Y 1 7 ) A : X! X , , , N(A) X ; A ) p X f T. (f(x) ; (p1 7 x)) = ;11 7 (1.5) xn(a)1 7 kxk!+1 46

5 '(x) =(Ax1 7 x)=+f(x) ; (p1 7 x) x (1:4) (1:1) , ' (1:1) Q (.1){(.) , (.1) A Ay [18] ( Ay), fs i 1 7 i Ig, S i = f(x1 7 u) R n+1 ju = ' i (x)1 7 x g, ' i Wloc1 71() , x g(x1 7 u;) <g(x1 7 u+) (g(x1 7 u;) 6= g(x1 7 u+)) i I, u = ' i (x) 1 7 (L' i (x)+g(x1 7 ' i (x)+) ; p(x))(l' i (x)+g(x1 7 ' i (x);) ; p(x)) > : (1.6) , (.1) A A1y ( A1y), , (1.6), L' i (x) + g(x1 7 ' i (x)) = p(x) (.1){(.) p L q (), q>n=(n + ), J p : X! R, X = W 1 () X = W 1 () , : J (u) = 1 G p (u) = nx i1 7j=1 J p (u) =J (u)+g p (u)1 7 dx (.3) a ij (x)u xi u xj dx + 1 u(x) g(x1 7 s)ds ; c(x)u (x)dx1 7 p(x)u(x)dx1 7 (1.7) J p (u) =J (u)+ 1 (s)u (s)ds + G p (u): (1.8) ; ) (:4){(:5) N(L) ) Bu u, J (u) 8u W 1 (), L, J (u) 8u W 1 (), L + (x)u, J (u)+ 1 (s)u (s)ds 8u W 1 ()1 7 ; 3) () 1 7 (:6)1 7 4) p L q () , u X, G p(u) =+1: (1.9) un(l)1 7kuk!+1 J p (u ) = inf X J p(u)1 7 (1.1) u W q (), ;Lu (x)+p(x) [g ; (x1 7 u(x))1 7 g + (x1 7 u(x))] (1.11) 47

6 x (:) (:1) Ay (A1y), u, (1:1), ( ) (:1){(:) ) 1 7 3) :3, p L q () (1:9), G p(u) =;1: (1.1) un(l)1 7 kuk! J p (u) 1 7 X , u W q (), (1:11) (:) (:1) Ay ( A1y), J p (u) ( ) (:1){(:) { [3] (1.9), (1.1) , N(L) x g(x1 7 u) =g (x) ( u! N(L)) , g(x1 7 u) (.6) { : p L q () g + (x)dx + g ; (x)dx < p(x) (x)dx < g + (x)dx + g ; (x)dx1 7 (1.13) < > > < g ; (x)dx + < g + (x)dx < p(x) (x)dx < > g ; (x)dx + > (1:13) ((1:14)) (1:9) ((1:1)). 1 7 [19] , g(x1 7 u) x f : R! R f !1 f(s)ds= = f , () = f(s)ds= ; f = (f(s) ; f )ds=: g + (x)dx: (1.14) < f(s)= s!1 +() = "> f(s)=f +, >!+1 s! , jf(s) ; f + j <"= s> >, jf(s) ; f + jds= 1 <"= > j + ()j jf(s) ; f + jds= " , ;() =.!;1 jf(s) ; f + jds= < " + ;!; R " < " + " = ": = ,

7 (1.13) 1 7 F () = R dx F ()= = R (x) (g(x1 7 s) ; p(x))ds = dx (x) (x)6= (x) (x) + g(x1 7 s)ds;p(x) = (x)< 1 (x) (x) > (x) x (x) > , g (x), 1 7 (x) < R(x) 1 (x) R (x) 1!1 (x) 1 (x) (x) g(x1 7 s)ds (x) dx ; (x) 1!1 (x) g(x1 7 s)ds dx+ p(x) (x)dx: R g(x1 7 s)ds = g(x1 7 s)ds = g (x) , g(x1 7 s)ds a(x) x, (x)6=, F ()= = (x)g (x)dx + (x)g (x)dx ; p(x) (x)dx!1 > (a(x) (.6)) (1.13) F () = ! , (1.14): g(x1 7 u) (:1) , (:6) x g(x1 7 u) =g (x), u! N(L) , N(L), x < g ; (x) <g(x1 7 s) <g + (x)(g + (x) <g(x1 7 s) <g ; (x)) 8s R: (1.15) (1:13) ((1:14)) (:1){(:) W q () , u Wq () (:1){(:) (.1) , R Lu(x) (x)dx = R u(x)l (x)dx =, R g(x1 7 u(x)) (x)dx = R p(x) (x)dx (1.15) (1.13) ((1.14)): ([6], ) , g(x1 7 u) , 1 7 (.) (1.9) ((1.1)) u W q () (.1){(.) A, X N(A) 1 7 X + (X A) > , (Ax1 7 x) kxk 8x X +, kk X A , (Ax1 7 x)= T (1.1) : '(x) =(Ax1 7 x)=+f(x) ; (p1 7 x) Q. 49

8 ) A ([13], c. ) , '(x) =+1: (.1) kxk! x X, x = x 1 + x 1 7x 1 N(A), x X +, '(x) = (Ax 1 7x )= +(f(x 1 + x ) ; f(x 1 )) + (f(x 1 ) ; (p1 7 x)) kx k ; (M 1 + kpk)kx k +(f(x 1 ) ; (p1 7 x 1 )), M 1 = MkP k, M ( jf(x);f(y)j Mkx;yk Y M 1 kx;yk 8x1 7 y X) "> (1.3) d > , f(x);(p1 7 x) x N(A) 1 7 kxk d, d 1 >, kxk d 1, x N(A) f(x) ; (p1 7 x) >"+ (M1+kpk) d >d t>d t ; (M 1 + kpk)t >"; min 1 7 inf xn(a)1 7 kxkd (f(x) ; (p1 7 x)) : , jf(x)j jf(x) ; f()j + jf()j M 1 kxk + jf()j 8x N(A) kxk = (kx 1 k + kx k ) 1=, x = x 1 + x 1 7x 1 N(A), x X +, kxk > p maxfd 1 1 7d g , kx 1 k >d 1, kx k >d '(x) > ; (M1+kpk) + " + (M1+kpk) = ", t ; (M 1 + kpk)t (M1+kpk) 8t R '(x) >"; min 1 7 inf (f(x) ; (p1 7 x)) + f(x 1 ) ; (p1 7 x 1 ) " , xn(a)1 7 kxkd x X 1 7 kxk > p maxfd 1 1 7d g '(x) > " "> (.1) ([], ) , x X, '(x ) = inf X '(x)1 7 (.) x (1.4) , Q , x X, (.), (1.1) Q ([], c. 13): '(x) =(Ax1 7 x)=+f(x) ; (p1 7 x) ([9]) E , ' : E! R (P:S:) , E = E 1 E, E , b 1 < b N E , 'j E b 1 b N) ' , ' : E! R (P.S.) [9], (x n ) E, ('(x n )) (x n )= kwk E!, min n) '(x) =(Ax1 7 x)=+f(x);(p1 7 x) X, A , 1 7 f Y ( ) X Y, A , , N(A) A p (1.5) , ' (P.S.) ([9], ) , ' , , ', ) , X A N(A), X ; 1 7 X , (Ax1 7 x) kxk 8x X (Ax1 7 x) ;kxk 8x X ; : 5

9 (f(x) ; (p1 7 x)) = X 1 = X ;, X = N(A) xn(a)1 7 kxk!+1 X +, X = X 1 X 1 7 X x X, x = x + x 1, x N(A), x 1 X '(x) =(Ax1 7 x)=+(f(x + x 1 ) ; f(x )) + (f(x ) ; (p1 7 x )) ; (p1 7 x 1 ) kx 1k ; (M 1 + kpk)kx 1 k +(f(x ) ; (p1 7 x )) ; (M 1 + kpk) M 1 = MkP k, M ) , x X 1 + inf (f(v) ; (p1 7 v)) = b1 7 vn(a) '(x) =(Ax1 7 x)=+f(x) ; f() + f() ; (p1 7 x) ; kxk +(M 1 + kpk)kxk + jf()j: r >, b 1 = , sup '(x) < b, 1 7, , xx1 7 kxk=r (f(x) ; (p1 7 x)) = ;11 7 (.3) xn(a)1 7 kxk! X 1 = X ; N(A), X = X X = X 1 X 1 7 X x X '(x) =(Ax1 7 x)=+(f(x) ; f()) + (f() ; (p1 7 x)) kxk ; ; (M 1 + kpk)kxk;jf()j ; (M 1 + kpk) ;jf()j = b : ' X x X 1, x = x + x, x N(A), x X ;, '(x) =(Ax1 7 x)=+(f(x + x ) ; f(x )) + (f(x ) ; (p1 7 x )) ; (p1 7 x ) ; kx k ; (M 1 + kpk)kx k +(f(x ) ; (p1 7 x )): (.3) d 1 > , x N(A) ckx kd 1 f(x ) ; (p1 7 X ) <b ; (M1+kpk) d >, kx kd, x X ; ; kx k +(M 1 + kpk)kx k <b ; sup (f() ; (p1 7 )): N(A) , x X 1, x = x + x, x N(A)1 7x X ;,1 7kxk = p kx k + kx k p max d 1, d = r, kx kd 1, kx kd '(x) <b ; sup N(A) '(x) < (M 1 + kpk) + b ; (M 1 + kpk) (f() ; (p1 7 )) + f(x ) ; (p1 7 x ) b , b 1 = sup xx 11 7 kxk=r '(x) < b, kxk = r X , 'j X , , x X , ) ' ) , X ;(Ax 1 7)+(p1 7 ) sup hx1 7 h! t!+ 51 = b 1 7 f(x + h + t) ; f(x + h) : (.4) t

10 fj X P TP, (.4) f(x + h + t) ; f(x + h) t = 1 hp TP(x + h + t)1 7id: sup hx1 7 h! t! , (.5) hp TP(x + h + t)1 7id +(Ax 1 7) ; (p1 7 ) 8 X: (.5) y = ;Ax +p S(P TP)(x )1 7 (.6) S(P TP) F = P TP [17] SF(x ) (F (x n )) 1 7 X, (x n ) x , (.6) , X 1 7 " > , , hz1 7 i;hy 1 7 i < ;" 8z SF(x ): sup hf (x + h + s )1 7 i;hy 1 7 i < ;": (.7) hx1 7 h! t! (.7) >, (F (x + h + s )1 7 ) ; (y 1 7 ) < ;", khk <1 7<s< sup hx1 7 h! t!+ 1 hf (x + h + t )1 7 id ;hy 1 7 i;" (.5) = , '(x) , x (.6) [1], S(P TP)x P (ST)(Px ) ([], c. 16), (.6) (1.4) , (1.1) , x ' P TP (1.1) , , x ' P TP, Q, X, suphq(x + h)1 7 i < hx h! s 1 7 " , h X, khk <" hq(x + h)1 7 i < ;s , h X, t>1 7khk + tk k <",1 71 7hQ(x + h + t )1 7 i < ;s 8 [1 7 1] (.5) = ;s , x ', (.5) X , x Q (.5), , Q(x )=: hq(x )1 7i 8 X: 5

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