Abstract: In this paper,we consider the existence of multiple positive solutions of the following inhomogeneous semilinear elliptic equation
-△u+u=λ(f(u)+h(x)) in Ω u∈H₀¹(Ω),u>0 in Ω (P)_λ where λ> 0,Ω=w and ω is a bounded smooth open set in R²,h(x)∈ L²(Ω),h(x)≢0,f(t)∈ C¹([0,+∞)) satisfies f(0)=f'(0)=0,f^w(t) exists and f^w(t)> 0,0 < f(t) < Cexp(αtⁿ) for some constants C,α> 0.0 < v < 2 and t∈(0,+∞),f(t)<θtf'(t) for some θ ∈(0,1). By looking for the local minimum of the corresponding energy functional we tain the first minimum positive solution and by applying mountain pass lemma around the ndboum positive solution we prove the following result:
Theorem (P)_λ has at least two positive solutions if λ∈(0,λ^*) whcre λ^*> O is some constant and (P)_λ has no positive solution if λ > λ^*.