Abstract
Multiplicity of solutions for Dirichlet boundary conditions of second-order quasilinear equations with impulsive effects
Highlights
Consider the following problem with impulses−u (t) + a(t)u(t) − (|u(t)|2) u(t) =f (t, u(t)), t ∈ J,∆(u) = Ij(u(tj)), j = 1, 2, . . . , m, (1.1)u(0) = u(T) = 0, where t0 = 0 < t1 < t2 < · · · < tm < tm+1 = T, J = [0, T] \ {t1, t2, . . . , tm}, f ∈ C([0, T] × R; R), Ij ∈ C(R; R), a(t) ∈ L∞[0, T], ∆(u) = u (t+j ) − u (t−j ) and u (t±j ) = limt→t±j u (t), j = 1, 2, . . . , m.This problem is derived from a class of quasilinear Schrödinger equation
U(0) = u(T) = 0, where t0 = 0 < t1 < t2 < · · · < tm < tm+1 = T, J = [0, T] \ {t1, t2, . . . , tm}, f ∈ C([0, T] × R; R), Ij ∈ C(R; R), a(t) ∈ L∞[0, T], ∆(u) = u (t+j ) − u (t−j ) and u (t±j ) = limt→t±j u (t), j = 1, 2, . . . , m
When we look for the standing wave solution whose form is Ψ(t, x) = e−iwtu(x), w ∈ R of the following quasilinear Schrödinger equation i∂tΨ = −Ψ + W(x)Ψ − (|Ψ|2) Ψ − μ|Ψ|q−1Ψ, x ∈ R, (1.2)
Summary
It is generally known that critical point theory is a classical method to deal with the existence and multiplicity of solutions for differential equations (see [3, 7, 12, 16, 21, 26, 30]). Critical point theory has been proved to be an effective tool to investigate boundary value problems for impulsive differential equations. There are few articles which considered the multiplicity of standing wave solutions for the impulsive Dirichlet boundary value problem involving the quasilinear term (|u|2) u. Motivated by the works mentioned above, in this paper, our purpose is to investigate the multiplicity of solutions for Dirichlet boundary conditions of second-order quasilinear equations with impulsive effects (1.1). By making use of the variant fountain theory (see [30]), the multiplicity of solutions for the problem (1.1) are obtained
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