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Tj /F1 1 Tf 1.0074 0 Td (2)Tj 0.6132 0.68985 TD (;)Tj /F6 1 Tf -24.4842 -2.7156 TD (u)Tj /F1 1 Tf 0.9417 0 Td [(is)-383(the)-351(unkno)32(wn)-361(and)]TJ /F6 1 Tf 9.1104 0 Td (f)Tj /F1 1 Tf 0.60225 0 Td (\()Tj /F6 1 Tf 0.3942 0 Td (x)Tj /F1 1 Tf 0.5694 0 Td [(\))-351(is)-383(a)-361(giv)34(en)-373("forcing")-349(term.)-559(The)-361(solutions)-362(of)-361((1))-361(corresp)-33(ond)-340(to)-362(critical)]TJ -11.61795 -1.2045 TD [(p)-33(oin)33(ts)-318(of)-328(the)-329(functional)]TJ 0 -2.74845 TD ((3))Tj /F6 1 Tf 11.13615 0 Td (F)Tj /F1 1 Tf 0.7884 0 Td (\()Tj /F6 1 Tf 0.3942 0 Td (u)Tj /F1 1 Tf 0.5694 0 Td 0.263 Tc (\)=)Tj 1.82865 0.6789 TD 0 Tc (1)Tj ET q 5.52 0 0 -0.48 233.268 268.188 cm /Im1 Do Q BT /F1 1 Tf 10.95891 0 0 10.95891 233.28 257.64 Tm (2)Tj /F4 1 Tf 0.7884 2.04765 TD (Z)Tj /F9 1 Tf 7.67123 0 0 7.67123 248.04 255.24 Tm (\n)Tj /F5 1 Tf 10.95891 0 0 10.95891 256.68 265.2 Tm (jr)Tj /F6 1 Tf 1.10595 0 Td (u)Tj /F5 1 Tf 0.5694 0 Td (j)Tj /F9 1 Tf 7.67123 0 0 7.67123 278.04 269.76 Tm (2)Tj /F5 1 Tf 10.95891 0 0 10.95891 285.48 265.2 Tm <00>Tj /F1 1 Tf 2.37615 0.6789 TD (1)Tj ET q 33 0 0 -0.48 297.708 268.188 cm /Im1 Do Q BT /F1 1 Tf 10.95891 0 0 10.95891 297.72 257.64 Tm (\()Tj /F6 1 Tf 0.3942 0 Td (p)Tj /F1 1 Tf 0.7227 0 Td 0.219 Tc [(+1)219(\))]TJ /F4 1 Tf 2.17905 2.04765 TD 0 Tc (Z)Tj /F9 1 Tf 7.67123 0 0 7.67123 339.96 255.24 Tm (\n)Tj /F5 1 Tf 10.95891 0 0 10.95891 348.48 265.2 Tm (j)Tj /F6 1 Tf 0.27375 0 Td (u)Tj /F5 1 Tf 0.5694 0 Td (j)Tj /F7 1 Tf 7.67123 0 0 7.67123 360.72 269.76 Tm (p)Tj /F9 1 Tf 0.59443 0 Td (+1)Tj /F5 1 Tf 10.95891 0 0 10.95891 379.44 265.2 Tm <00>Tj /F4 1 Tf 1.0074 1.3578 TD (Z)Tj /F9 1 Tf 7.67123 0 0 7.67123 396.6 255.24 Tm (\n)Tj /F6 1 Tf 10.95891 0 0 10.95891 405.12 265.2 Tm 0.109 Tc (fu)Tj /F1 1 Tf -30.3972 -2.94555 TD 0 Tc [(w)33(ell)-339(de\fned)-319(on)-329(the)-318(Sob)-33(olev)-317(space)]TJ /F6 1 Tf 14.9796 0 Td (H)Tj /F9 1 Tf 7.67123 0 0 7.67123 246.12 236.88 Tm (1)Tj -0.1095 -0.876 TD (0)Tj /F1 1 Tf 10.95891 0 0 10.95891 251.04 232.92 Tm [((\n))-329(\(b)-33(ecause)-296(of)-328(the)-318(restriction)-318((2))-318(on)]TJ /F6 1 Tf 16.5783 0 Td (p)Tj /F1 1 Tf 0.5037 0 Td (\).)Tj -32.2149 -1.7739 TD [(The)-361(Ljusternik-Sc)32(hnirelman)-351(critical)-372(p)-33(oin)33(t)-362(theory)-350(\(Mosco)33(w,)-372(1930\))-360(pro)32(vides)-361(the)-362(existence)-361(of)-361(in-)]TJ -1.2045 -1.2045 TD [(\fnitely)-372(man)32(y)-372(solutions)-362(for)-360((1))-362(when)]TJ /F6 1 Tf 16.2498 0 Td (f)Tj /F5 1 Tf 0.9417 0 Td <11>Tj /F1 1 Tf 1.12785 0 Td [(0.)-558(The)-361(in)32(v)55(ariance)-372(of)]TJ /F6 1 Tf 9.3732 0 Td (F)Tj /F1 1 Tf 1.1607 0 Td [(under)-352(symmetry)-372(is)-373(essen)33(tial)-372(for)]TJ -28.85325 -1.19355 TD [(this)-362(t)33(yp)-33(e)-361(of)-361(theory)-351(to)-350(apply)88(.)-548(Nothing)-361(w)33(as)-372(kno)33(wn)-361(\(except)-340(for)-361(dimension)]TJ /F6 1 Tf 31.96305 0 Td (N)Tj /F1 1 Tf 1.23735 0 Td [(=)-340(1)-361(b)32(y)-371(classical)-361(ODE)]TJ -33.2004 -1.2045 TD [(tec)33(hniques\))-374(when)-372(departing)-374(from)-383(symmetry)88(.)-624(It)-384(w)33(as)-394(an)-383(op)-33(en)-373(problem)-373(whether)-373(ev)33(en)-383(one)-384(solution)]TJ T* [(p)-33(ersisted)-319(when)-340(adding)-329(a)-339(forcing)-328(term)]TJ /F6 1 Tf 16.6878 0 Td (f)Tj /F1 1 Tf 0.9417 0 Td [(in)-351((1).)-459(This)-351(w)33(as)-339(the)-329(problem)-340(I)-340(ga)33(v)34(e)-351(Abbas)-340(as)-339(a)-339(\frst)-330(step)]TJ -17.6295 -1.19355 TD [(to)33(w)33(ards)-329(his)-329(PhD.)]TJ 1.2045 -1.78485 TD [(Within)-340(a)-340(short)-340(time)-339(Abbas)-351(came)-339(up)-341(with)-339(a)-350(remark)55(able)-329(partial)-339(result)-340(and)-340(inno)32(v)55(ativ)34(e)-339(strategies)]TJ -1.2045 -1.1935 TD [(in)33(v)33(olving)-415(a)-405(com)33(bination)-384(of)-404(Algebraic)-393(T)87(op)-33(ology)-392(and)-395(Analysis.)-667(He)-405(pro)33(v)33(ed)-394([CRAS,)-415(1980)-393(and)-395(JF)110(A)]TJ 0 -1.2045 TD [(1981])-272(that)-264(problem)-263((1))-274(has)-274(in\fnitely)-274(man)32(y)-284(solutions)-263(for)-273(a)-285(dense)-264(set)-274(of)]TJ /F6 1 Tf 30.2658 0 Td (f)Tj /F8 1 Tf 7.67123 0 0 7.67123 410.28 105.84 Tm (0)Tj /F6 1 Tf 10.95891 0 0 10.95891 413.28 101.88 Tm (s)Tj /F1 1 Tf 0.47085 0 Td [(.)-427(Subsequen)31(tly)-262(A.)-295(Bahri)-274(-)]TJ ET endstream endobj 5 0 obj << /Length 14625 >> stream /GS4 gs BT /F2 1 Tf 8.76713 0 0 8.76713 535.32 735.36 Tm 0 g (3)Tj /F1 1 Tf 10.95891 0 0 10.95891 72 709.08 Tm (H.)Tj 1.32495 0.00001 TD [(Berest)33(yc)33(ki)-284([T)88(rans.)-438(AMS,)-296(1981])-283(pro)33(v)33(ed)-285(that)-274(there)-275(exists)-296(some)-284(1)]TJ /F6 1 Tf 27.4407 0 Td 0.274 Tc (
Tj
/F1 1 Tf
0.9198 0 Td
[(2\),)-295(suc)32(h)]TJ
-39.3105 -1.20451 TD
[(that)-384(for)-394(ev)34(ery)]TJ
/F6 1 Tf
6.5481 0 Td
(p)Tj
/F5 1 Tf
0.8979 0 Td
(2)Tj
/F1 1 Tf
1.0512 0 Td
(\(1)Tj
/F6 1 Tf
0.8979 0 Td
0.164 Tc
(;p)Tj
/F7 1 Tf
7.67123 0 0 7.67123 185.28 694.2 Tm
0 Tc
(N)Tj
/F1 1 Tf
10.95891 0 0 10.95891 193.56 695.88 Tm
[(\))-405(and)]TJ
/F11 1 Tf
2.8032 0 Td
(every)Tj
/F6 1 Tf
2.6937 0 Td
(f)Tj
/F1 1 Tf
0.60225 0 Td
[(,)-416(equation)-383((1))-394(admits)-395(in\fnitely)-394(man)32(y)-405(solutions.)-646(Later)-394(on)]TJ
-17.1915 -1.20451 TD
[(A.)-339(Bahri-)-317(P)88(.L.)-339(Lions)-329([CP)88(AM,)-338(1988])-316(impro)32(v)34(ed)-329(this)-319(result)-318(and)-318(established)-308(that)-307(the)-307(same)-329(conclusion)]TJ
0 -1.19355 TD
[(holds)-406(when)]TJ
/F6 1 Tf
5.3874 0 Td
0.416 Tc
[(p<)11(N)393(=)]TJ
/F1 1 Tf
3.42735 0 Td
0 Tc
(\()Tj
/F6 1 Tf
0.3942 0 Td
(N)Tj
/F5 1 Tf
1.19355 0 Td
<00>Tj
/F1 1 Tf
1.0512 0 Td
(2\))Tj
/F6 1 Tf
0.8979 0 Td
(:)Tj
/F1 1 Tf
0.6789 0 Td
[(T)88(o)-416(this)-406(da)33(y)-416(it)-416(is)-416(not)-405(kno)33(wn)-405(whether)-396(problem)-405((1))-395(admits)]TJ
/F11 1 Tf
25.76535 0 Td
[(even)-438(one)]TJ
/F1 1 Tf
-38.79585 -1.2045 TD
[(solution)-340(for)-350(ev)34(ery)]TJ
/F6 1 Tf
8.0154 0 Td
(f)Tj
/F1 1 Tf
0.95265 0 Td
[(under)-330(the)-351(natural)-329(assumption)-341((2).)-492(A.)-361(Bahri)-350(and)-340(H.)-361(Berest)33(yc)33(ki)-350([CP)88(AM,)-360(1984)]TJ
-8.96805 -1.19355 TD
[(and)-428(Acta)-437(Math,)-460(1984])-425(also)-438(successfully)-427(applied)-438(their)-428(metho)-33(d)-417(to)-438(study)-428(second)-428(order)-427(systems)-428(of)]TJ
0 -1.2045 TD
[(ODEs)-329(and)-329(p)-33(erio)-32(dic)-307(orbits)-329(of)-328(non)-318(autonomous)-307(Hamiltonian)-328(systems.)]TJ
/F0 1 Tf
0 -1.79581 TD
[(I)-33(I.)-437(Equations)-450(in)34(v)32(olving)-460(the)-438(limiting)-436(exp)-32(onen)32(t)]TJ
/F6 1 Tf
23.50965 0 Td
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/F1 1 Tf
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/F6 1 Tf
1.52205 0 Td
0 Tc
(N)Tj
/F1 1 Tf
1.17165 0 Td
0.252 Tc
[(+2)252(\))]TJ
/F6 1 Tf
1.9272 0 Td
0 Tc
(=)Tj
/F1 1 Tf
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/F6 1 Tf
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(N)Tj
/F5 1 Tf
1.14975 0 Td
<00>Tj
/F1 1 Tf
1.0293 0 Td
(2\))Tj
/F0 1 Tf
0.8979 0 Td
[(.)-678(Critical)-437(p)-33(oin)33(ts)-449(at)]TJ
-32.97045 -1.2045 TD
[(in\fnit)33(y)99(.)]TJ
/F1 1 Tf
1.2045 -1.7958 TD
[(This)-361(section)-340(includes)-351(some)-351(of)-361(the)-351(most)-350(fundamen)31(tal)-339(and)-351(in\ruen)32(tial)-350(w)33(orks)-361(of)-361(Abbas.)-525(They)-351(are)]TJ
-1.2045 -1.20451 TD
[(concerned)-264(with)-285(nonlinearities)-285(in)33(v)33(olving)-305(the)-286(limiting)-295(exp)-33(onen)33(t)]TJ
/F6 1 Tf
26.88225 0 Td
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/F1 1 Tf
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0.285 Tc
(=\()Tj
/F6 1 Tf
1.45635 0 Td
0 Tc
(N)Tj
/F1 1 Tf
1.0512 0 Td
0.143 Tc
[(+2)143(\))]TJ
/F6 1 Tf
1.8177 0 Td
0 Tc
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(N)Tj
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<00>Tj
/F1 1 Tf
0.9198 0 Td
[(2\).)-427(Here)-295(is)-296(a)-295(basic)]TJ
-34.8429 -1.19355 TD
[(example.)-437(Consider)-318(the)-329(equation)]TJ
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/F4 1 Tf
14.33355 2.3433 TD
(8)Tj
0 -0.8979 TD
(>)Tj
0 -0.29565 TD
(<)Tj
0 -1.80675 TD
(>)Tj
0 -0.29565 TD
(:)Tj
/F5 1 Tf
1.21545 2.4528 TD
<00>Tj
/F1 1 Tf
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<01>Tj
/F6 1 Tf
0.8322 0 Td
(u)Tj
/F1 1 Tf
0.8541 0 Td
(=)Tj
/F6 1 Tf
1.06215 0 Td
(u)Tj
/F7 1 Tf
7.67123 0 0 7.67123 287.28 518.52 Tm
(p)Tj
/F1 1 Tf
10.95891 0 0 10.95891 294.72 514.56 Tm
(+)Tj
/F6 1 Tf
1.0074 0 Td
(a)Tj
/F1 1 Tf
0.5256 0 Td
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/F6 1 Tf
0.3942 0 Td
(x)Tj
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(\))Tj
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(u)Tj
/F1 1 Tf
1.89435 0 Td
[(in)-340(\n)]TJ
/F6 1 Tf
1.89435 0 Td
(;)Tj
-9.44985 -1.50015 TD
0.285 Tc
(u>)Tj
/F1 1 Tf
1.9053 0 Td
5.146 Tc
[(0i)5146(n)4806(\n)]TJ
/F6 1 Tf
7.54455 0 Td
0 Tc
(;)Tj
-9.44985 -1.50015 TD
(u)Tj
/F1 1 Tf
0.8541 0 Td
0.274 Tc
[(=0)-4872(o)274(n)]TJ
/F6 1 Tf
8.09205 0 Td
0 Tc
(@)Tj
/F1 1 Tf
0.58035 0 Td
(\n)Tj
/F6 1 Tf
0.7227 0 Td
(;)Tj
/F1 1 Tf
-27.80205 -2.52945 TD
[(where)-416(\n)-427(is)-427(a)-427(smo)-33(oth)-416(b)-34(ounded)-395(domain)-427(of)]TJ
/F10 1 Tf
18.8559 0 Td
(R)Tj
/F7 1 Tf
7.67123 0 0 7.67123 286.56 457.92 Tm
(N)Tj
/F6 1 Tf
10.95891 0 0 10.95891 294.96 453.96 Tm
0.164 Tc
(;N)Tj
/F5 1 Tf
1.78485 0 Td
0 Tc
<15>Tj
/F1 1 Tf
1.21545 0 Td
(3)Tj
/F6 1 Tf
0.5037 0 Td
(;)Tj
/F1 1 Tf
0.7008 0 Td
(and)Tj
/F6 1 Tf
2.04765 0 Td
(a)Tj
/F1 1 Tf
0.5256 0 Td
(\()Tj
/F6 1 Tf
0.3942 0 Td
(x)Tj
/F1 1 Tf
0.5694 0 Td
(\))Tj
/F5 1 Tf
0.82125 0 Td
(2)Tj
/F6 1 Tf
1.10595 0 Td
(C)Tj
/F1 1 Tf
0.7884 0 Td
(\()Tj
ET
q
7.92 0 0 -0.48 413.868 463.188 cm
/Im1 Do
Q
BT
/F1 1 Tf
10.95891 0 0 10.95891 413.88 453.96 Tm
[(\n)11(\))-427(is)-427(suc)32(h)-427(that)-417(the)-417(linear)]TJ
-31.19655 -1.2045 TD
[(op)-33(erator)]TJ
/F5 1 Tf
4.0296 0 Td
<00>Tj
/F1 1 Tf
0.77745 0 Td
<01>Tj
/F5 1 Tf
1.04025 0 Td
<00>Tj
/F6 1 Tf
0.9855 0 Td
(a)Tj
/F1 1 Tf
0.8541 0 Td
[(is)-328(co)-33(erciv)33(e.)-426(A)-328(sup)-34(erb)-308(result)-318(of)-328(A.)-328(Bahri)-317(-)-328(J.-M.)-338(Coron)-317([CRAS,)-338(1985)-306(and)-318(CP)88(AM,)]TJ
-7.6869 -1.2045 TD
[(1988])-360(asserts)-362(that)-351(if)-371(\n)-372(has)-362(\\non)32(trivial)-371(top)-33(ology")-349(\(when)]TJ
/F6 1 Tf
24.7908 0 Td
(N)Tj
/F1 1 Tf
1.25925 0 Td
0.34 Tc
(=3)Tj
/F6 1 Tf
1.6206 0 Td
0 Tc
(;)Tj
/F1 1 Tf
0.60225 0 Td
[(\n)-372(not)-362(con)33(tractible)-351(su\016ces,)-372(when)]TJ
/F6 1 Tf
-28.2729 -1.19355 TD
(N)Tj
/F5 1 Tf
1.19355 0 Td
<15>Tj
/F1 1 Tf
1.0512 0 Td
[(4)-328(the)-318(assumption)-319(is)-340(more)-328(tec)33(hnical\),)-318(and)-318(if)]TJ
/F6 1 Tf
19.0749 0 Td
(a)Tj
/F1 1 Tf
0.5256 0 Td
(\()Tj
/F6 1 Tf
0.3942 0 Td
(x)Tj
/F1 1 Tf
0.5694 0 Td
(\))Tj
/F5 1 Tf
0.66795 0 Td
<11>Tj
/F1 1 Tf
1.0512 0 Td
[(0,)-339(then)-319((4))-317(admits)-329(a)-328(solution.)]TJ
-23.3235 -1.80675 TD
[(In)-483(order)-471(to)-471(appreciate)-460(this)-482(result,)-515(its)-482(b)-34(eautiful)-470(pro)-34(of,)-502(and)-472(its)-493(impact)-471(\(almost)-471(300)-480(citations)]TJ
-1.2045 -1.19355 TD
[(according)-328(to)-351(MathSciNet\),)-328(it)-350(is)-362(imp)-33(ortan)33(t)-329(to)-350(put)-341(things)-351(in)-351(p)-33(ersp)-34(ectiv)34(e.)-482(In)-351(the)-340(mid)-362(60s)-339(P)33(ohozaev)]TJ
0 -1.2045 TD
[(established)-373(that)-373(if)-383(\n)-394(is)-383(starshap)-34(ed)-362(and)]TJ
/F6 1 Tf
17.6295 0 Td
(a)Tj
/F1 1 Tf
0.5256 0 Td
(\()Tj
/F6 1 Tf
0.3942 0 Td
(x)Tj
/F1 1 Tf
0.5694 0 Td
(\))Tj
/F5 1 Tf
0.75555 0 Td
<11>Tj
/F1 1 Tf
1.14975 0 Td
[(0,)-405(problem)-373((4))-372(admits)-384(no)-383(solution.)-591(On)-396(the)-372(other)]TJ
-21.024 -1.19355 TD
[(hand)-439(Y)88(amab)-33(e)-438(claimed)-438(the)-438(existence)-427(of)-438(a)-437(solution)-439(for)-437(an)-449(equation)-427(resem)32(bling)-438((4))-438(with)-438(ro)-33(ots)-427(in)]TJ
0 -1.2045 TD
[(Geometry)87(.)-623(T)87(rudinger)-384((1967))-372(disco)33(v)34(ered)-395(a)-394(serious)-384(\ra)33(w)-394(in)-405(Y)88(amab)-33(e's)-394(pro)-34(of;)-415(the)-384(heart)-372(of)-394(the)-384(error)]TJ
T*
[(came)-274(from)-273(the)-263(non)-275(compactness)-253(of)-273(the)-274(Sob)-33(olev)-262(em)32(b)-33(edding)]TJ
/F6 1 Tf
25.54635 0 Td
(H)Tj
/F9 1 Tf
7.67123 0 0 7.67123 361.92 333 Tm
(1)Tj
-0.1095 -0.876 TD
(0)Tj
/F5 1 Tf
10.95891 0 0 10.95891 369.96 328.92 Tm
<1A>Tj
/F6 1 Tf
1.0512 0 Td
(L)Tj
/F7 1 Tf
7.67123 0 0 7.67123 388.92 333 Tm
(p)Tj
/F9 1 Tf
0.59443 0 Td
(+1)Tj
/F1 1 Tf
10.95891 0 0 10.95891 405.12 328.92 Tm
[(.)-427(These)-274(t)33(w)33(o)-284(\\negativ)33(e")-251(facts)]TJ
-30.3972 -1.19355 TD
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[(A)33(t)-263(Rutgers,)-263(Abbas)-252(receiv)34(ed)-253(in)-252(1990)-250(the)-242(Board)-241(of)-262(T)88(rustees)-242(Aw)34(ard)-263(for)-252(Excellence,)-262(the)-241(Univ)33(ersit)33(y's)]TJ
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[(talen)33(t)-340(to)-350(\\create")-317(mathematics)-329(in)-362(fron)33(t)-339(of)-350(their)-351(ey)33(es.)-503(He)-361(had)-340(b)-34(een)-339(v)33(ery)-350(encouraging,)-339(caring,)-350(and)]TJ
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[(face.)-437(It)-329(is)-328(the)-330(last)-328(image)-328(I)-329(k)34(eep)-329(of)-328(him.)]TJ
1.2045 -2.39805 TD
[(Ha)110<7F>]TJ
1.2483 0 Td
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[(Y)88(an)33(y)33(an)-340(Li,)-350(P)33(aul)-328(Rabino)33(witz)-328(and)-329(Claude)-318(Viterb)-33(o\).)]TJ
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‹‹ Ð Ì '¹ } KQ`1.00BCopyright (C) 1997 American Mathematical Society. All Rights ReservedCMBX10Computer Modern * 7 m $ Y W V R H I X Z G C E T &