Three solutions for a p-Laplacian boundary value problem with impulsive effects

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Abstract

In this paper, a p-Laplacian boundary value problem with impulsive effects is considered. Multiplicity of solutions is obtained by three critical points theorem. An example is presented to illustrate main result.

Introduction

In this paper, we are interested in ensuring the existence of at least three solutions for the following p-Laplacian boundary value problem(ρ(t)Φp(u(t)))-s(t)Φp(u(t))+λf(t,u(t))=0,a.e.t(a,b),α1u(a+)-α2u(a)=0,β1u(b-)+β2u(b)=0,with the impulsive conditionsΔ(ρ(tj)Φp(u(tj)))=Ij(u(tj)),j=1,2,,l,where Φp(x)≔∣xp−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 andΔ(ρ(tj)Φp(u(tj)))=ρ(tj+)Φp(u(tj+))-ρ(tj-)Φp(u(tj-)),where 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, +∞)) andabr(s)ds+η>0,such thatf(t,x)=r(t)Φμ(x)+h(t,x),Ii(x)=ηΦμ(x)+g(x).

  • (C2)

    There exist c  L1([a, b], [0, +∞)), d  C([a, b], [0, +∞)), ξ  0, such thath(t,x)c(t)+d(t)Φp(x),g(x)ξΦp(x).

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)) whereG(x)=112+524|x|32.In fact, limx0+G(x)/Φ3(x)=+. However (C1) and (C2) imply thatlimx0+Ii(x)Φp(x)limx0+η|x|μ-p+ξ=ξ.

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|Ij(x)|aj+bj|x|γjfor everyxR,j=1,2,,l.

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 normu=abρ(t)|u(t)|p+s(t)|u(t)|pdt1/p,which 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 byJ(u,λ)=φ1(u)+λφ2(u),whereφ1(u)=upp+j=1l0u(tj)Ij(x)dx+ρ(a)α2p-1pα1p-1|u(a)|p+ρ(b)β2p-1pβ1p-1|u(b)|p,φ2(u)=-abF(t,u(t))dtandF(t,x)=0xf(t,y)dyfor all(t,x)[a,b]×R.Using 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.K1(b-a)δ1η1+δ2η2+α1α2δ1+β1β2δ2(b-a)1η1+1η2-1,K2|δ1|paa+(b-a)η1ρ(t)dt+|K1|pa+(b-a)η1b-b-aη2ρ(t)dt+|δ2|pb-(b-a)η2bρ(t)dt,K3maxα1α2|δ1|,b-aη1+α1α2|δ1|,b-aη2+β1β2|δ2|,β1β2|δ2|,g1(t)δ1t+α1α2-a,g3(t)δ2t-β1β2-b,g2(t)K1t-a-b-aη1+δ1b-aη1+α1α2,EC2cpMp-C4cM(C1+C3)K2+K3pabs(t)dt+C4K2+K3pabs(t)dt1/p,Ωaa+(b-a)η1F(t,g1(t))dt+a+(b-a)η1b-b-aη2F(t,g2(t))dt+b-(b-a)η2bF(t,g3(t))dt

Acknowledgement

The authors would like to thank the referee for valuable suggestions.

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Project supported by Hunan Provincial Innovation Foundation For Postgraduate (No. CX2010B116) and the National Natural Science Foundation of China (No. 10971229).

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