By A. Bahri
(This examine notice monitors new phenomena in selected variational difficulties of nonlinear research and geometry. the writer develops extra his principles which first seemed in a prior quantity within the sequence Pseudo-Orbits of touch shape. The linked variational challenge, that's identified to endure positive factors of noncompactness is studied from a dynamical platforms standpoint. Professor Bahri describes flow-lines of the gradient move and illustrates that the Palais-Smale may be happy alongside flow-lines whereas it fails for the variational difficulties. a brand new idea is usually brought, the idea that of severe issues at infinity. He then proceeds to turn out lifestyles theorems for the equations which were thought of. this article will be of curiosity to specialist mathematicians and learn scholars in utilized and natural mathematics).
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Additional info for Critical Points at Infinity in Some Variational Problems
Example text
12 The function f is continuously Frechet differentiable and lim fS(x) = f(x) for all x E H. 52) Then ~ E af(x). ious substitution we may write fS as = S -n f n R f(Ane)pn((A~1 Pnx - e)s-1)de from which we see that fS is Fr~chet differentiable and Vf s is continuous on H. 51) follows. ) - f(Pnx s - SAnT n ,).. )a n ,).. n i i and an,).. >- 0, i=Oa n ,).. n '" 1.. } we may assume that Tn,A 1 + T and a n n,A1 + a n for).. ~ fO(P x -EA Tni,PnzJani, z 1=0 E n E n E H. 11 part (iii) we infer that (~,z) < fO(x,z) for all z in H.
E. x E n if u1 = 0 (~ is a positive constant). 1). e. x En}. e. in r. Let f be a fixed element of V'. 37) has a unique solution y E K. 38) AoY> f, Y > y = ~ 1. n ~ ~n+ H"-l, , in n oy = av o~. 41 ) where AO is the differential operator AOY =- N 2: i,j=1 ( a .. 42) oy/ov = 2: a .. ) is the normal derivative and on+ is the boundary 1 J Xi 46 J of (l+. 37) (for instance y E C(n». ) there exists p > 0 such that y ± pa > ~ on~. 37) we see that L:f (l (a .. y a lJ xi. aoya)dx=(f,a)forallaEC~(~-t). 38) in the sense of distributions.
A 2 > O. Let V. = H (Q) and A:V + VI be defined by a (y , z ) = (y,Az) + if a > 1 If a a and 1 > a IQ > I a . ( x )y z. 33) o. 2 a and a 2 (y ,Az) ~ i,j=1 = 1 11 0, we take V = HO(Q) and A:HO(Q) + H (Q) defined by = a (y ,z) = ~ i,j=1 I a .. (x) y z dx lJ Xi Xj 1 for all y, z E HO(Q). 34 ) Here a 0' a.. e. 35) where w > 0 and /I • II N is the Eucl idean norm in RN. e. x E n if u1 = 0 (~ is a positive constant). 1). e. x En}. e. in r. Let f be a fixed element of V'. 37) has a unique solution y E K.