Entire Solutions Of A Class Of Semilinear Elliptic Equations

In this paper, we consider the semilinear elliptic equation:The existence, uniqueness and asymptotic behavior are considered. Our framework includes two cases: the solution is a entire bounded solution; the solution is a entire large solution. This equation has always been the highlight in Partial Differential Equation.We notice that when the solution blow-up on∞, it is very important to study the asymptotic behavior of entire large solution. We compare the condition ofρand f and summarize the method and tools which are frequently used. We also expound some important proofs in detail.In the argument of entire large solution, we discussed that when f doesn't have the property of monotonous, the equation also has solution.Supposeρis a positive continuous function satisfies -â–³U =ρ(x), x∈N has ground state solution; f(u), f₁(u), f₂(u), u∈[0, +∞) satisfies thatHere f₁(u) > 0, u∈(0, +∞), f(0) = 0, and f is a locally Lipschitz function, f₁,f₂ are nondecreasing, locally Lipschitz functions, such thatThen equationhas positive solutions.