# A nonlinear elliptic equation with singular potential and by Badiale M., Benci V., Rolando S.

By Badiale M., Benci V., Rolando S.

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Now we consider general n. 22) Let r be as above. -1 , Crj,i(z)a-l11 IWn,j,R(a+z)I 5 ... an-1 -211 IWn,j,c(a, z)I <_ cr,j,I(z)al where Re(z) E D(n, r), and cr,j,i(z) is a function of polynomial growth. Proof. We use induction on n. We first consider the real case. Consider the function , rn_1). Let z = Wn+1,j,n(a, z). Let r = (rl, , rn) E Rn and r' = (rl, (zl)... , zn+l) E Cn+1, z1 + ... zn+1 = 0, and Re(z) = x = (xl, ... , xn+l) E D(n + 1, r). Let ll be a positive integer, and 0 < m1, m2, m3 ll.

N-l,v(t(u), ) on D1,... j _1. Suppose that aic(ui1)c(ui+11)-1 E 7r;, where 1 < 0 for some i. Then Wn-1,v(a2C(u21)C(u31)-1, ... , an-1C(un-11)C(un1)-1, ) = 0. Since c(ull) is always integral, hl + ji+i < 1. If l < -1, Ic(u21)Ivn-l-zn+1 ... IC(unl)Iv1-zn+1'n_1,v(t(u), Z) X < a1u21 + a2u31 + ... + an-lunl >v du21 ... 3) to ui+11. If 1 = -1, and ui1 ov, hl + ji+l < -1, so again, the above integral is zero. 7). D. 8). By definition, (Ixv-7-1tj+lc(uj+11)Ivn Cn-1(t(u),z) = 2t2c(u21)Ivn 17r ... i)i,j, ItnC(unl)Ivn Cn_1(t(u), z) _ Irv-2t2C(2121)Ivl ...

3). ,k mg-YT), 2 Eisenstein series on GL(n) 33 where r runs through all the permutations satisfying r-1(i) < T-1(j) if i < j, j I, provided the right hand side converges absolutely and locally uniformly. j. 9), and l(r) _ l(Th) + ... + l(Tl). 9) can be written uniquely in the above form. Proof. We use induction on #I. Let I' = {il, ,ih_l}, and r' = Th_1 T1. Suppose that (i, j) E ITh. Then j = Th 1(ih). SO T(i,i) = T'(i - 1,ih). Since i - 1 < 2h, T'(i - 1) < T'(ih) = ih. Therefore, ITh CIT. Suppose that (i, j) E Th 1IT,.