Yb. Deng et Y. Li, EXISTENCE AND BIFURCATION OF THE POSITIVE SOLUTIONS FOR A SEMILINEAR EQUATION WITH CRITICAL EXPONENT, Journal of differential equations, 130(1), 1996, pp. 179-200
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.