By Hartwig Hirschfeld
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Extra resources for Arabic Chrestomathy in Hebrew Characters
Since S n Ni. is a compact set, we can choose p(K) sufficiently large so that 10p,K,wl 2: 101/2 for all p 2: p(K) and w E S n N k . 6) implies that (gt) is satisfied. 4 Exercises 1. As can be easily checked by elementary methods, the linear resonant ODE problem (P2) in the introduction has no solution. 1 be used? 2. In view of the Fredholm alternative, if p : - - t IR is a continuous function on a bounded, smooth domain f2 C IR N , a necessary and sufficient condition for existence of a solution to the linear problem -~u == AIU + p(x), U == 0 on ao, is that n k p(x)(h(x) dx = 0, where cPl (x) is an eigenfunction corresponding to AI.
2) . 4) TJ(l, ~C+E) C ~C-E Now, pick h E f such that uED and define h(u) == TJ(l, h(u)). 2) and the fact that 2£ < b - a we have TJ(l, h(u)) == u if u E aD [since ~laD :::; a < b - 2£], hence h E f. 4) imply that m~~(h(u)) :::; c - £ , uED which contradicts the definition of c. Therefore, c is a critical value of~. 1. 1) is clearly satisfied when ~ is such that ~(v) ~ -00 as IIvll ~ 00, v E V, and ~(w) ~ +00 as Ilwll ~ 00, w E W. This situation is typical in many applications. 1) x; is the kt h eigenvalue 3 Application to a Resonant Problem g: n X 1R ---+ 43 1R is continuous and uniformly bounded, say Ig(x,s)1 ~ M "Ix E 0, "Is E 1R.
1. 1. 1, the gradient mapping \7
Arabic Chrestomathy in Hebrew Characters by Hartwig Hirschfeld