Abstract
This paper deals with the resonance problem for the one-dimensional p-Laplacian with homogeneous Dirichlet boundary conditions and with nonlinear impulses in the derivative of the solution at prescribed points. The sufficient condition of Landesman-Lazer type is presented and the existence of at least one solution is proved. The proof is variational and relies on the linking theorem.
Highlights
We are interested in the solutions of ( ) satisfying the impulse conditions in the derivative pu := u tj+ p– u tj+ – u tj– p– u tj– = Ij u(tj), j =, . . . , r
In this paper we focus on a quasilinear equation with p = and look just for sufficient conditions
In the presence of nonlinear impulses which have certain asymptotic properties, we show that the fact f ∈ φn⊥ might still be the sufficient condition for the existence of a solution to ( ) and ( )
Summary
For p = this problem is considered in [ ] where the necessary and sufficient condition for the existence of a solution of ( ) and ( ) is given. The same argument as that used for p = in [ , Section ] for the nonresonance case yields the following existence result for the quasilinear impulsive problem ( ), ( ). Let u be a weak solution of ( ), ( ), v ∈ D(tj, tj+ ) (the space of smooth functions with a compact support in (tj, tj+ ), j = , , .
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