Asymptotic Behavior of the Bifurcation Diagrams for Semilinear Problems with Application to Inverse Bifurcation Problems

Abstract EN

We consider the nonlinear eigenvalue problem u ″ ( t ) + λ f ( u ( t ) ) = 0 , u ( t ) > 0 , t ∈ I = : ( - 1,1 ) , u ( 1 ) = u ( - 1 ) = 0 , where f ( u ) is a cubic-like nonlinear term and λ > 0 is a parameter.

It is known by Korman et al.

(2005) that, under the suitable conditions on f ( u ) , there exist exactly three bifurcation branches λ = λ j ( ξ ) ( j = 1,2 , 3 ), and these curves are parameterized by the maximum norm ξ of the solution u λ corresponding to λ .

In this paper, we establish the precise global structures for λ j ( ξ ) ( j = 1,2 , 3 ), which can be applied to the inverse bifurcation problems.

The precise local structures for λ j ( ξ ) ( j = 1,2 , 3 ) are also discussed.

Furthermore, we establish the asymptotic shape of the spike layer solution u 2 ( λ , t ) , which corresponds to λ = λ 2 ( ξ ) , as λ → ∞ .