A REMARK ON THE RESONANCE SET FOR A SEMILINEAR ELLIPTIC EQUATION

We study the resonance set SIGMA of pairs (alpha, beta) is-an-element- of R2 for which the problem DELTAu + alphau+ - betau- = 0 has a nontri vial solution u is-an-element-of H0(1)(OMEGA). We show that if lambda0 is an eigenvalue of multiplicity two of - DELTA, then SIGMA and]lambd a-, lambda[2 has measure zero, where lambda, lambdaBAR are the neighbo uring eigenvalues of lambda0. Moreover, we have that, if the operator DELTA + alphaI(u>0) + betaI(u<0) has a kernel of dimension one for (al pha, beta) is-an-element-of SIGMA and u not-equal 0 such that DELTAu alphau+ - betau- = 0, then (alpha, beta) is an isolated point on SIGM A and L, where L is the straight line parallel to the diagonal of R+ x R+ through (alpha, beta).