Multiplicity and Uniqueness of Positive Solutions for Nonhomogeneous Semilinear Elliptic Equation with Critical Exponent

In this paper, we consider the following problem {-Delta u(x) + u(x) = lambda(u(p)(x) + h(x)), x is an element of R-N, u(x) is an element of H-1(R-N), u(x) > 0, x is an element of R-N, (*) where lambda > 0 is a parameter, p = (N+2)/(N-2). We will prove that there exists a positive constant 0 < lambda* < +infinity such that (*) has a minimal positive solution for lambda is an element of (0,lambda*), no solution for lambda > lambda*, a unique solution for lambda = lambda*. Furthermore, (*) possesses at least two positive solutions when lambda is an element of (0,lambda*) and 3 <= N <= 5. For N >= 6, under some monotonicity conditions of h we show that there exists a constant 0 < lambda** < lambda* such that problem (*) possesses a unique solution for lambda is an element of(0,lambda**).