Three solutions for a p-Laplacian boundary value problem with impulsive effects☆
Introduction
In this paper, we are interested in ensuring the existence of at least three solutions for the following p-Laplacian boundary value problemwith the impulsive conditionswhere Φp(x)≔∣x∣p−2x, p > 1, a < b, ρ, s ∈ L∞([a, b]) with ess inf [a,b]ρ > 0 and ess inf[a, b]s > 0, ρ(a+) = ρ(a) > 0, ρ(b−) = ρ(b) > 0, α1, α2, β1, β2 are positive constants, f:[a, b] × R → R is continuous, t0 = a < t1 < t2 < ⋯ < tl < tl+1 = b, Ij: R → R, j = 1, 2, … , l are continuous, λ ∈ [0, +∞) is a parameter andwhere z(y+) and z(y−) denote the right and left limits of z(y) at y, respectively. We refer to impulsive problem (1), (2), (3) as (IP).
Differential equations with impulsive effects arising from the real world describe the dynamics of processes in which sudden, discontinuous jumps occur. For the background, theory and applications of impulsive differential equations, we refer the interest readers to [1], [2], [3], [4], [5], [6], [7]. There have been many approaches to study the existence of solutions of impulsive differential equations, such as fixed point theory, topological degree theory (including continuation method and coincidence degree theory) and comparison method (including upper and lower solutions methods and monotone iterative method) and so on (see, for example, [8], [9], [10], [11] and references therein).
Recently, [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] using variational method studied the existence and multiplicity of solutions of impulsive problem. More precisely, in [13] Tian and Ge obtained sufficient conditions that guarantee the existence of at least two positive solutions of a p-Laplacian boundary value problem with impulsive effects. Two key conditions of the main result of [13] are listed as follows.
- (C1)
There exist μ > p, h ∈ C([a, b] × [0, +∞), [0, +∞)), η > 0, r ∈ C([a, b], [0, +∞)), g ∈ C([0, +∞), [0, +∞)) andsuch that
- (C2)
There exist c ∈ L1([a, b], [0, +∞)), d ∈ C([a, b], [0, +∞)), ξ ⩾ 0, such that
However, there are many cases which can’t be dealt with by the result of [13]. For example, p = 3, there is only one impulsive point t1 ∈ (a, b) and the impulsive condition is −Δ(ρ(t1)Φ3(u′(t1))) = G(u(t1)) whereIn fact, . However (C1) and (C2) imply that
In [23], Ricceri established a three critical points theorem. After that, the theorem has been used extensively and a series of results are obtained (see, for example, [24], [25], [26], [27], [28]).
Existence and multiplicity of solutions for p-Laplacian boundary value problem have been studied extensively in the literature (see, for example, [24], [29], [30], [31], [32] and references therein). However, to the best of our knowledge, existence of at least three solutions for p-Laplacian boundary value problem with impulsive effects has attracted less attention.
Motivated by the above facts, in this paper we devote to study the multiplicity of solutions of (IP) via three critical points theorem obtained by Ricceri [23].
Throughout this paper, we assume that.
- (H1)
There exist constants aj > 0, bj > 0 and γj ∈ [0, p − 1), j = 1, 2, … , l such that
The remaining part of this paper is organized as follows. Some fundamental facts will be given in Section 2. In Section 3, main result of this paper will be presented and an example will be given to illustrate the theorem.
In the following, for convenience, when we make a statement without specifying the domain of λ, we assume that it holds for all λ ∈ [0, +∞).
Section snippets
Preliminaries
Here and in the sequel, U will denote the Sobolev space W1,p([a, b]) equipped with the normwhich is equivalent to the usual one. We define the norm in C([a, b]) as ∥u∥∞ = maxt∈[a,b]∣u(t)∣.
Consider J: U × [0, +∞) → R defined bywhereandUsing the continuity of f and Ij, j = 1, 2, … , l, one has that J(u, λ) is
Main result
In this section, main result of this paper is presented. For convenience, we introduce some notations and two conditions.
Acknowledgement
The authors would like to thank the referee for valuable suggestions.
References (35)
- et al.
Existence and global attractivity of positive periodic solutions for impulsive predator-prey model with dispersion and time delays
Nonlinear Anal. RWA
(2009) - et al.
Periodic solution of a delayed ratio-dependent predator-prey model with monotonic functional response and impulse
Nonlinear Anal. TMA
(2009) - et al.
Periodic solutions for a class of higher-dimension functional differential equations with impulses
Nonlinear Anal. TMA
(2008) - et al.
Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equation
Appl. Math. Comput.
(2004) - et al.
Maximum principle for periodic impulsive first order problems
J. Comput. Appl. Math.
(1998) - et al.
Multiple nonnegative solutions for second order impulsive differential equations
Appl. Math. Comput.
(2000) - et al.
Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations
J. Math. Anal. Appl.
(2006) - et al.
Variational approach to impulsive differential equations
Nonlinear Anal. RWA
(2009) - et al.
Variational approach to impulsive differential equations with periodic boundary conditions
Nonlinear Anal. RWA
(2010) - et al.
Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects
Nonlinear Anal. TMA
(2010)
An application of variational methods to Dirichlet boundary value problem with impulses
Nonlinear Anal. RWA
Variational methods to Sturm–Liouville boundary value problem for impulsive differential equations
Nonlinear Anal. TMA
Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects
Nonlinear Anal. TMA
The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects
Nonlinear Anal.
An application of variational method to second-order impulsive differential equation on the half-line
Appl. Math. Comput.
A multiplicity result for p-Lapacian boundary value problems via critical points theorem
Appl. Math. Comput.
On discrete fourth-order boundary value problems with three parameters
J. Comput. Appl. Math.
Cited by (50)
Infinitely many rotating periodic solutions for suplinear second-order impulsive Hamiltonian systems
2019, Applied Mathematics LettersCitation Excerpt :In the past few decades, a series of nonlinear functional methods were applied to deal with the existence of solutions to boundary value problems for impulsive differential equations such as the coincidence degree theory (see [3]), comparison principles (see [4]), fixed point theorems (see [5]). Especially, in the recent years, the variational method has been successfully used to investigate the existence and multiplicity of solutions to boundary value problems for differential equations with impulsive effects (see [6–11]). On the other hand, it is well known that the existence of periodic solutions to second-order differential equations is a classical topic.
INFINITELY MANY SOLUTIONS FOR A SECOND ORDER IMPULSIVE DIFFERENTIAL EQUATION WITH p-LAPLACIAN OPERATOR
2022, Memoirs on Differential Equations and Mathematical PhysicsEXISTENCE OF SOLUTIONS FOR A CLASS OF SECOND-ORDER BOUNDARY VALUE PROBLEMS
2022, Mathematics for ApplicationsVariational methods to the p-Laplacian type nonlinear fractional order impulsive differential equations with Sturm-Liouville boundary-value problem
2021, Fractional Calculus and Applied AnalysisOn the Existence of Three Solutions for Some Classes of Two-Point Semi-linear and Quasi-linear Differential Equations
2020, Bulletin of the Iranian Mathematical Society
- ☆
Project supported by Hunan Provincial Innovation Foundation For Postgraduate (No. CX2010B116) and the National Natural Science Foundation of China (No. 10971229).