数学物理学报(英文版)
數學物理學報(英文版)
수학물이학보(영문판)
Acta Mathematica Scientia
2003年
1期
1-15
,共15页
Multiple solutions%variational method%elliptic equations
In this paper, the authors study the existence and nonexistence of multiplepositive solutions for problem-△ u + u = f(x, u) + μh(x), x ∈RN,u ∈ H1(RN), (*)μwhere h ∈ H-1(RN), N ≥ 3, |f(x,u)| ≤ C1up-1 +C2u with C1 ＞ 0, C2 ∈ [0,1) beingsome constants and 2 ＜ p ＜ +∞. Under some assumptions on f and h, they prove thatthere exists a positive constantμ* ＜ +∞ such that problem (*)μ has at least one positivesolution uμ ifμ∈ (0,μ*), there are no solutions for (*)μ ifμ＞μ* and uμ is increasingwith respect toμ∈ (0,μ*); furthermore, problem (*)μ has at least two positive solutionforμ∈ (0,μ*) and a unique positive solution forμ =μ* ifp ≤ 2N/N-2.
