Solutions of semilinear elliptic equations with asymptotic linear nonlinearity

The existence, uniqueness and asymptotic behavior of positive solutions of the semilinear elliptic equations with asymptotic linear nonlinearity was studied. It was proved that if ε > 0, then there existed exactly one solution uλ(.,ε) for all λ ∈ (0,λ1) where λ1 represents first eignvalue of -δ in ω with Dirichlet boundary condition. The results showed that if s < t and small enough, δū+λfε(ū) < -s + ||h(ū)||∞ ≤ 0 was found by choosing δ = s.