EXISTENCE AND BIFURCATION OF THE POSITIVE SOLUTIONS FOR A SEMILINEAR EQUATION WITH CRITICAL EXPONENT

In this paper, we consider the semilinear elliptic equation ()(mu) -D elta u + u = u(p-1) + uf(x), u > 0, u is an element of H-1 (R(N)), N > 2. For p = 2N/(N - 2), we show that there exists a positive constant mu > 0 such that (*)(mu) possesses at least one solution if mu is an element of (0, mu) and no solutions if mu > mu*. Furthermore, (*)(mu) possesses a unique solution when mu = mu, and at least two solutions when mu is an element of (0, mu) and 2 < N < 6. For N greater than o r equal to 6, under some monotonicity conditions on f ((1.6)) we show that there exist two constants 0 < mu* less than or equal to mu** < m u such that problem (*)(mu) possesses a unique solution for mu is an element of (0, mu*), and at least two solutions if mu is an element o f (mu*, mu*). (C) 1996 Academic Press, Inc.