Multiple solutions for an inhomogeneous semilinear elliptic equation in RN

In this paper, the authors study the existence and nonexistence of multiple positive solutions for problem { -Deltau+u = f(x,u) muh(x), x is an element of R-N, (8)(mu) mu is an element of H-1 (R-N), where h is an element of H-1(R-N), N greater than or equal to 3, \f (x, u)\ less than or equal to C(1)u(p-1) + C(2)u with C-1 > 0, C-2 is an element of [0, 1) being some constants and 2 < p < +infinity. Under some assumptions on f and h, they prove that there exists a positive constant mu* < +infinity such that problem (*)(mu) has at least one positive solution mu(mu) if mu is an element of (0, mu*), there are no solutions for (*)(mu) if mu > mu* and mu(mu) is increasing with respect to mu is an element of (0, mu*); furthermore, problem (*)(mu) has at least two positive solution for mu is an element of (0, mu*) and a unique positive solution for mu = mu* if p less than or equal to (2N)/(N-2).