Existence And Nonexistence Of Positive Solutions For The Kirchhoff Type Equation

In this paper,we study the existence and nonexistence of positive solutions for the Kirchhoff type equation with nonlinearity having prescribed asymptotic behavior.Our results also include degenerate cases in which the nonlinear term is resonant at infinity.To our best knowledge,our resonance conditions for positive solutions are also different from the existing results.We study the following Kirchhoff type equation(?)(1-1)Where a?0,b?0 and ? is a smooth bounded domain of RN(N?3).We assume that the following conditions are satisfied uniformly for x??:(f1)f?C(?×R,R)with f(x,0)=0 and(?)(f2)there exists ??R,such that#12(f3)f3/f(x,t)is nondecreasing with respect to t>0;(f4)there exists ?>0 such that f(x,t)?b?1t3,for 0?t??;(f5)one of the following is satisfied:1)? is an open ball in RN for N?3;2)?(?)R2 is symmetric in x and y,and convex in x and y;3)?(?)R2 is convex.According to the above assumptions,we can get the following results:Theorem1.Assume that(f1),(f2)and(f5)hold.(?)If a>0,b>0.?>?1,and(f4)holds,then(1-1)has at least one positive solutions.(?)If a=0,b>0,?>?1 and(f3)holds,then(1-1)has a positive solution u?H01(?)if and only if there is some c>0 such that#12Theorem2.If a?0,b>0,?+?,(f1)-(f4)hold and there exists c>0 such that|f(x,t)|?c(1+|t|?-1),for some(?)Then(1-1)has at least one positive solution.Theorem3.Assume that(f1),(f2)and(f3)hold,then(1-1)has no positive solution in each of the following cases:(a)a=0,b>0,?<?1;(b)a>0,b>0,???1.