MULTIPLE SOLUTIONS FOR AN INHOMOGENEOUS SEMILINEAR ELLIPTIC EQUATION IN R~N

In this paper, the authors study the existence and nonexistence of multiple positive solutions for problem(*)μwhere h ∈ H-1(RN), N ≥ 3, |f(x,u)| ≤ C1up-1 + C2u with C1 > 0, C2∈ [0,1) being some constants and 2 < p < ∞. Under some assumptions on f and h, they prove that there exists a positive constant μ* <∞ such that problem (*)μ has at least one positive solution uμ if μ,∈ (0,μ*), there are no solutions for (*)μ if μ, > μ* and uμ is increasing with respect to μ∈ (0,μ*); furthermore, problem (*)μ has at least two positive solution for μ ∈ (0,μ*) and a unique positive solution for μ, =μ* if p ≤2N/N-2.