On asymptotics of solutions to semilinear elliptic equations near the first eigenvalue of the nonperturbed problem

We writef=(g) iff(x)cg(x) with some positive constantc for allx from the domain of functionsf andg. We show that at least (n²/r) entries must be changed in an arbitrary (generalized) Hadamard matrix in order to reduce its rank belowr. This improves the previously known bound (n²/r²). If we additionally know that the changes are bounded above in absolute value by some numbern/r, then the number of these entries is bounded below by (n³/(r²)), which improves upon the previously known bound (n²/²).Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 535–540, April, 1998.The research of the first author was supported by the Russian Foundation for Basic Research under grants No. 96-01-00094 and No. 96-15-96102 and by the INTAS Foundation under grant No. 93-1376. The research of the second author was supported by the Russian Foundation for Basic Research under grants No. 96-01-01222 and No. 96-15-96090.