# Existence and Multiplicity of Solutions to a Boundary Value Problem for Impulsive Differential Equations

Authors: Chunyan He; Yongzhi Liao; Yongkun Li
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We investigate the existence and multiplicity of solutions to a boundary value problem for impulsive differential equations. By using critical point theory, some criteria are obtained to guarantee that the impulsive problem has at least one solution, at least two solutions, and infinitely many solutions. Some examples are given to illustrate the effectiveness of our results.


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## Body

## 1. Introduction

In this paper, we will investigate the existence and multiplicity of solutions to the boundary value problem for impulsive differential equations:(1)-(ρ(t)|u′(t)|p-2u′(t))′+s(t)|u(t)|p-2u(t)=f(t,u(t)),t≠ti,a.e.t∈[0,T],Δ(ρ(ti)|u′(ti)|p-2u′(ti))=Ii(u(ti)),i=1,2,…,k,u(0)=u(T)=0,
where p≥2, ρ(t), s(t)∈L∞[0,T], with ess inft∈[0,T]ρ(t)>0, ess inft∈[0,T]s(t)>0, and 1≤ρ(t)<+∞; 0<s(t)<+∞; 0=t0<t1⋯<tk<tk+1=T, Δ(ρ(ti)|u′(ti)|p-2u′(ti))=ρ(ti+)|u′(ti+)|p-2u′(ti+)-ρ(ti-)|u′(ti-)|p-2u′(ti-), f:[0,T]×R→R is continuous, Ii:R→R are continuous.Recently, there have been many papers concerned with boundary value problems for impulsive differential equations. Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time. The theory of impulsive differential systems has been developed by numerous mathematicians (see [1–6]).Impulsive and periodic boundary value problems have been studied extensively in the literature. There have been many approaches to study periodic solutions of differential equations, such as the method of lower and upper solutions, fixed point theory, and coincidence degree theory (see [7–10]). However, the study of solutions for impulsive differential equations using variational method has received considerably less attention (see, [11–18]). Variational method is, to the best of our knowledge, novel and it may open a new approach to deal with nonlinear problems with some type of discontinuities such as impulses.Teng and Zhang in [15] studied the existence of solutions to the boundary value problem for impulsive differential equations
(2)-(|u′(t)|p-2u′(t))′=f(t,u(t),u′(t)),t≠ti,a.e.t∈[0,T],Δu′(ti)=Ii(u(ti)),i=1,2,…,l,u(0)=u(T)=0.
By using variational methods and iterative methods they showed that there exists a solution for problem (2).In this paper, we will need the following conditions.(A)
F(t,u) is measurable in t for every u∈ℝ and continuously differentiable in u for a.e. t∈[0,T] and there exist a∈C(ℝ+,ℝ+),b∈L1(0,T;ℝ+) such that
(3)|F(t,u)|≤a(|u|)b(t),|f(t,u)|≤a(|u|)b(t),
for all u∈ℝ and a.e. t∈[0,T], where F(t,u)=∫0u‍f(t,s)ds.(H0)
There exist constantsa,b>0 and r∈[0,p-1) such that
(4)|f(t,u)|≤a+b|u|r,for(t,u)∈[0,T]×R.(H1)
There exist constantsc>0 and τ∈[p,+∞) such that
(5)|f(t,u)|≤c(|u|τ-1+1),for(t,u)∈[0,T]×R.(H2)
There exist constantsai,bi>0 and ri∈[0,p-1)(i=1,2,…,k) such that
(6)|Ii(u)|≤ai+bi|u|ri,foru∈R.(H3)
There exist constantsμ>p and M>0 such that
(7)0<μF(t,u)≤uf(t,u),fort∈[0,T],|u|≥M.(H4)
limu→0f(t,u)/|u|p-1=0 uniformly, for t∈[0,T] and u∈R.(H5)
there existλ>p and β>λ-p such that
(8)limsup|x|→∞F(t,x)|x|λ<∞uniformly,fora.e.t∈[0,T],liminf|x|→∞xf(t,x)-2F(t,x)|x|β>0,uniformlyfora.e.t∈[0,T].(H6)
there existsζij>0 such that
(9)2∫0t‍Ii(s)ds-Ii(t)t≥0∀i∈{1,2,…,k},|t|≥ζij.We recall some facts which will be used in the proof of our main results. It has been shown, for instance, in [19] that the set of all eigenvalues of the following problem:
(10)-(|u′(t)|p-2u′(t))′+λ|u(t)|p-2u(t)=0,t≠ti,t∈[0,T],u(0)=u(T)=0
is given by the sequence of positive numbers
(11)λk=(p-1)(kπpT)p,fork=1,2,…,
where
(12)πp=2∫01‍1(1-sp)1/pds.
Each eigenvalue λk is simple with the associated eigenfunction
(13)φk(x)=sinp(kπpxT),for0≤x≤T.Xk denotes the eigenspace associated to λk, then W01,p(0,T)=⨁i∈NXi¯.An outline of this paper is given as follows. In the next section, we present some preliminaries including some basic knowledge and critical point theory. In Section3, by using the critical point theory, we will establish some sufficient conditions for the existence of solutions of system (1). In Section 4, some examples are given to verify and support the theoretical findings.


## 2. Preliminaries

In this section, we recall some basic facts which will be used in the proofs of our main results. In order to apply the critical point theory, we make a variational structure. From this variational structure, we can reduce the problem of finding solutions of (1) to the one of seeking the critical points of a corresponding functional.In [20], the Sobolev space W01,p(0,T) be the endowed with the norm
(14)∥u∥W01,p=(∫0T‍|u′(t)|pdt)1/p,u∈W01,p(0,T).Throughout the paper, we also consider the norm(15)∥u∥=(∫0T‍(ρ(t)|u′(t)|p+s(t)|u(t)|p)dt)1/p,u∈W01,p(0,T).
By Poincaré inequality:
(16)∫0T‍|u′(t)|pdt≥C∫0T‍|u(t)|pdt,
where C=λ1 is precisely the largest C>0 for which the above inequality holds true. Then ∫0T‍|u′(t)|pdt-λ1∫0T‍|u(t)|pdt≥0 while it minimizes and equals to zero exactly on the ray generated by the first eigenfunction sinp(πpt/T).Let us recall that(17)|u|p=(∫0T‍|u(t)|pdt)1/p,∥u∥∞=maxt∈[0,T]|u(t)|.
We denote by |·|p the usual Lp-norm. The n-dimensional Lebesgue measure of a set E⊆Rn is denoted by |E|. By the Sobolev embedding theorem, W01,p(0,T)↪Lr[0,T] continuously for r∈[1,+∞), and there exists γr>0 such that
(18)|u|r≤γr∥u∥,∀u∈W01,p(0,T).Lemma 1.
There existC1, C2>0 such that
(19)C1∥u∥W01,p≤∥u∥≤C2∥u∥W01,p,∀u∈W01,p(0,T).Proof.
Since essinft∈[0,T]ρ(t)>0, ess inft∈[0,T]s(t)>0, 1≤ρ(t)<+∞, and 0<s(t)<+∞, we have that ess inft∈[0,T]ρ(t):=m1≥1, ess inft∈[0,T]s(t):=m2>-λ1, where λ1 is a positive number and that there exists n1∈(0,1) such that -m2≤λ1(1-n1). Thus, by Poincaré inequality, we have
(20)(1-n1)∫0T‍|u′(t)|pdt≥(1-n1)λ1∫0T‍|u(t)|pdt≥-m2∫0T‍|u(t)|pdt,
for all u∈W01,p(0,T). Thereby, for every u∈W01,p(0,T),
(21)∥u∥p=∫0T‍ρ(t)|u′(t)|pdt+∫0T‍s(t)|u(t)|pdt≥m1∫0T‍|u′(t)|pdt=m1∥u∥W01,pp=C1p∥u∥W01,pp.
On the other hand, by Poincaré inequality, one has(22)∥u∥p=∫0T‍ρ(t)|u′(t)|pdt+∫0T‍s(t)|u(t)|pdt≤∥ρ∥∞∫0T‍|u′(t)|pdt+∥s∥∞∫0T‍|u(t)|pdt≤(∥ρ∥∞+∥s∥∞λ1)∫0T‍|u′(t)|pdt=(∥ρ∥∞+∥s∥∞λ1)∥u∥W01,pp=C2p∥u∥W01,pp.
TakeC1=(m1+n1-1)1/p, C2=(∥ρ∥∞+∥s∥∞/λ1)1/p, then
(23)C1∥u∥W01,p≤∥u∥≤C2∥u∥W01,p.
The proof is complete.Lemma 2.
There existsC3>0 such that if u∈W01,p(0,T), then
(24)∥u∥∞≤C3∥u∥.Proof.
Ifu∈W01,p(0,T), it follows from the mean value theorem that
(25)1T∫0T‍u(s)ds=u(ξ),
for some ξ∈(0,T). Hence, for t∈[0,T], by Hölder inequality and Poincaré inequality
(26)|u(t)|=|u(ξ)+∫ξt‍u′(s)ds|≤|u(ξ)|+∫0T‍|u′(s)|ds≤1T|∫0T‍u(s)ds|+T1/q(∫0T‍|u′(t)|pds)1/p≤((CT)1/p+T1/q)(∫0T‍|u′(t)|pds)1/p≤((CT)1/p+T1/q)1C1∥u∥.
Hence, ∥u∥∞≤((C/T)1/p+T1/q)(1/C1)∥u∥=C3∥u∥. The proof is complete.Takev∈W01,p(0,T) and multiply the two sides of the equality
(27)-(ρ(t)|u′(t)|p-2u′(t))′+s(t)|u(t)|p-2u(t)=f(t,u(t))
by v and integrate it from 0 to T, we have
(28)-∫0T‍(ρ(t)|u′(t)|p-2u′(t))′v(t)dt+∫0T‍s(t)|u(t)|p-2u(t)v(t)dt=∫0T‍f(t,u(t))v(t)dt.
Moreover,
(29)-∫0T(ρ(t)|u′(t)|p-2u′(t))′v(t)dt=-∑i=0k∫titi+1(ρ(t)|u′(t)|p-2u′(t))′v(t)dt=∑i=0k(∫titi+1‍ρ(ti+1-)|u′(ti+1-)|p-2u′(ti+1-)v(ti+1-)∑i=0k-ρ(ti+)|u′(ti+)|p-2u′(ti+)v(ti+)∑i=0k-∫titi+1ρ(t)|u′(t)|p-2u′(t)v′(t)dt)=∑i=1k(ρ(ti+)|u′(ti+)|p-2u′(ti+)-ρ(ti-)∑i=0k×|u′(ti-)|p-2u′(ti-))-ρ(T)|u′(T)|p-2u′(T)v(T)+ρ(0)|u′(0)|p-2u′(0)v(0)+∫0Tρ(t)|u′(t)|p-2u′(t)v′(t)dt=∑i=1kIi(u(ti))v(ti)+∫0Tρ(t)|u′(t)|p-2u′(t)v′(t)dt.
Combining (28), we have
(30)∫0Tρ(t)|u′(t)|p-2u′(t)v′(t)dt+∫0Ts(t)|u(t)|p-2u(t)v(t)dt+∑i=1kIi(u(ti))v(ti)=∫0Tf(t,u(t))v(t)dt.
Considering the above, we introduce the following concept solution for problem (1).Definition 3.
We say that a functionu∈W01,p(0,T) is a solution of problem (1) if the identity
(31)∫0T‍ρ(t)|u′(t)|p-2u′(t)v′(t)dt+∫0T‍s(t)|u(t)|p-2u(t)v(t)dt+∑i=1k‍Ii(u(ti))v(ti)=∫0T‍f(t,u(t))v(t)dt
holds for any v∈W01,p(0,T).
Consider the functionalφ:W01,p(0,T)→R defined by
(32)φ(u)=1p∥u∥p+∑i=1k‍∫0u(ti)‍Ii(t)dt-∫0T‍F(t,u(t))dt.
Using the continuity of f and Ii, i=1,2,…,k, one has that φ∈C1(W01,p(0,T),R). For any v∈W01,p(0,T), we have
(33)φ′(u)v=∫0T‍ρ(t)|u′(t)|p-2u′(t)v′(t)dt+∫0T‍s(t)|u(t)|p-2u(t)v(t)dt+∑i=1k‍Ii(u(ti))v(ti)-∫0T‍f(t,u(t))v(t)dt.
Thus, the solutions of problem (1) are the corresponding critical points of φ.Definition 4.
LetX be a normed space. A minimizing sequence for a function φ:X→(-∞,+∞) is a sequence uk such that φ(uk)→infφ whenever k→+∞.Definition 5.
LetX be a Banach space and let φ:X→(-∞,+∞). φ is said to be sequentially weakly lower semicontinuous if liminfk→+∞φ(xk)≥φ(x) as xk⇀x in X.Definition 6.
LetE be a Banach space and let c∈R. For any sequence {uk} in E, if φ(uk) is bounded and φ′(uk)→0 as k→+∞ possesses a convergent subsequence, then we say that φ satisfies the Palais-Smale condition (denoted by PS condition for short). We say that φ satisfies the Palais-Smale condition at level c (denoted by (PS)c condition for short) if there exists a sequence {uk} in E such that φ(uk)→c and φ′(uk)→0 as k→+∞ implies that c is a critical value of φ.Definition 7.
LetE be a Banach space and let φ:E→(-∞,+∞). φ is said to be coercive if φ(u)→+∞ as ∥u∥→+∞.Lemma 8 (see [12]).
Ifφ is sequentially weakly lower semicontinuous on a reflexive Banach space X and has a bounded minimizing sequence, then φ has a minimum on X.Definition 9.
LetX be a real Banach space with a direct sum decomposition X=X1⊕X2. The functional φ∈C1(X,R) is said to have a local linking at 0, with respect to (X1,X2), if, for some r>0,(i)
φ(u)≥0,u∈X1,∥u∥≤r,(ii)
φ(u)≤0,u∈X2,∥u∥≤r.
Ifφ has a local linking at 0, then 0 is critical point (the trivial one). Suppose, furthermore, that there are two sequences of finite dimensional subspaces X11⊂X21⊂⋯⊂X1 and X12⊂X22⊂⋯⊂X2 such that
(34)X1=⋃n‍Xn1¯,X2=⋃n‍Xn2¯.Definition 10 (see [13, Definition 2.2]).
LetI∈C1(X,ℝ). The functional I satisfies the (C)* condition if every sequence (uαn) such that αn is admissible and
(35)uαn∈Xαn,sup|I(uαn)|<∞,(1+∥uαn∥)I′(uαn)→0
contains a subsequence which converges to a critical point of I.Lemma 11 (see [13]).
Suppose thatφ∈C1(X,R) satisfies the following assumptions: (1)
φ satisfies the (C)* condition,(2)
φ has a local linking at 0,(3)
φ maps bounded sets into bounded sets,(4)
for everym∈N,  φ(u)→-∞ as ∥u∥→+∞, u∈Xm1⊕X2.Then φ has at least two critical points.Lemma 12 (see [14]).
Let E be a Banach space. Letφ∈C1(X,R) be an even functional which satisfies the PS condition and φ(0)=0. If E=V⊕Y, where V is finite dimensional, and φ satisfies the following conditions:(1)
there exist constantsρ,α such that φ|∂Bρ∩Y≥α, where Bρ={x∈E:∥x∥<ρ},(2)
for each finite-dimensional subspaceW⊂E there is R=R(W) such that φ(u)≤0, for all u∈W with ∥u∥≥R.Then φ has an unbounded sequence of critical values.


## 3. Existence of Periodic Solutions

Theorem 13.
Assume that(A),  (H0), and (H2) are satisfied, then problem (1) has at least one solution.Proof.
LetM1=max{a1,a2,…,ak}, M2=max{b1,b2,…,bk}. By Lemma 2, we have
(36)φ(u)=1p∥u∥p+∑i=1k‍∫0u(ti)‍Ii(t)dt-∫0T‍F(t,u(t))dt≥1p∥u∥p-∑i=1k‍∫0u(ti)‍(ai+bi|u|ri)dt-∫0T‍(a|u|+b|u|r+1)dt≥1p∥u∥p-kM1∥u∥∞-M2∑i=1k‍∥u∥∞ri+1-aT∥u∥∞-bT∥u∥∞r+1≥1p∥u∥p-kM1C3∥u∥-M2∑i=1k‍C3ri+1∥u∥ri+1-aTC3∥u∥-bTC3r+1∥u∥r+1,
for all u∈W01,p(0,T). This implies that lim∥u∥→∞φ(u)=∞, and φ is coercive.
On the other hand, we show thatφ is weakly lower semicontinuous. If {uk}k∈N⊂W01,p(0,T), uk⇀u, then we have that {uk}k∈N converges uniformly to u on [0,T] and liminfk→∞∥uk∥≥∥u∥. Thus
(37)liminfk→∞φ(uk)=liminfk→∞(1p∥uk∥p+∑i=1k‍∫0uk(ti)‍Ii(t)dt-∫0T‍F(t,uk(t))dt∑i=1k‍)≥1p∥u∥p+∑i=1k‍∫0u(ti)‍Ii(t)dt-∫0T‍F(t,u(t))dt=φ(u).
By Lemma8, φ has a minimum point on W01,p(0,T), which is a critical point of φ. Hence, problem (1) has at least one solution. The proof is complete.We readily have the following corollary.Corollary 14.
Assume that(A),  (H0), and (H2) are satisfied and f and the impulsive functions Ii(i=1,2,…,k) are bounded. Then problem (1) has at least one solution.Lemma 15.
Assume that(H1), (H2), and (H3) are satisfied, then φ(u) satisfies the PS condition.Proof.
Assume that{un}⊂W01,p(0,T) satisfies that φ(un) is bounded and φ′(un)→0 as n→+∞. We will prove that the sequence {un} is bounded.
It follows from(H1), (H2), (H3), and Lemma 2, we have
(38)μφ(un)-φ′(un)un=(μp-1)∥un∥p+∑i=1k‍[μ∫0un(ti)‍Ii(t)dt-Ii(un(ti))un(ti)]-∫0T‍(μF(t,un(t))-f(t,un(t))un(t))dt≥(μp-1)∥un∥p-(μ+1)×(kM1C3∥un∥+M2∑i=1k‍C3ri+1∥un∥ri+1)-∫{|un|≤u0}‍(μF(t,un(t))-f(t,un(t))un(t))dt≥(μp-1)∥un∥p-(μ+1)×(kM1C3∥un∥+M2∑i=1k‍C3ri+1∥un∥ri+1)-C4.
Hence, {un} is bounded in W01,p(0,T).
SinceW01,p(0,T) is a reflexive Banach space, passing to a subsequence if necessary, we may assume that there is a u∈W01,p(0,T) such that
(39)un⇀uinW01,p(0,T),un→uinLp[0,T],{un}convergesuniformlytouon[0,T].
Notice that
(40)(φ′(un)-φ′(u),un-u)=∫0T‍ρ(t)(|un′(t)|p-2un′(t)-|u′(t)|p-2u′(t))×(un′(t)-u′(t))dt+∫0T‍s(t)(|un(t)|p-2un(t)-|u(t)|p-2u(t))×(un(t)-u(t))dt+∑i=1k‍(Ii(un(ti))-Ii(u(ti)))(un(ti)-u(ti))-∫0T‍(f(t,un(t))-f(t,u(t)))(un(t)-u(t))dt.
Recalling the following well-known inequality: for any x,y∈RN,
(41)(|x|p-2x-|y|p-2y)(x-y)≥cp|x-y|p,p≥2,
for some constant cp (Lemma  4.2 in [21]) and using Schwarz inequality, we have
(42)cp∫0T‍ρ(t)|un′(t)-u′(t)|pdt+cp∫0T‍s(t)|un(t)-u(t)|pdt≤∥φ′(un)-φ′(u)∥∥un-u∥-∑i=1k‍(Ii(un(ti))-Ii(u(ti)))(un(ti)-u(ti))+∫0T‍(f(t,un(t))-f(t,u(t)))(un(t)-u(t))dt.
Since
(43)(Ii(un(ti))-Ii(u(ti)))(un(ti)-u(ti))≤(|Ii(un(ti))|+|Ii(u(ti))|)∥un-u∥∞.
By the assumption (H1), we have
(44)∫0T‍(f(t,un(t))-f(t,u(t)))(un(t)-u(t))dt≤∫0T‍(|f(t,un(t))|+|f(t,u(t))|)|un(t)-u(t)|dt≤c∫0T‍(2+|un|τ-1+|u|τ-1)|un(t)-u(t)|dt≤C5∥un-u∥∞(1+|un|τ-1τ-1+|u|τ-1τ-1).
From (39), it follows that un→u in W01,p(0,T). Thus, φ(u) satisfies the PS condition. The proof is complete.Lemma 16.
Assume that(A), (H2), (H5), and (H6) are satisfied, then φ satisfies the (C)* condition.Proof.
Let{uαn} be a sequence in W01,p(0,T) such that αn is admissible and
(45)uαn∈Xαn,sup|φ(uαn)|<+∞,(1+∥uαn∥)φ′(uαn)→0,
then there exist a constant C4>0 such that
(46)|φ(uαn)|≤C4,(1+∥uαn∥)φ′(uαn)≤C4,
for all large n. On the other hand, by (H5), there are constants C5>0 and ρ1>0 such that
(47)F(t,x)≤C5|x|λ,
for all |x|≥ρ1 and a.e. t∈[0,T]. By (A) one has
(48)|F(t,x)|≤maxs∈[0,ρ1]a(s)b(t),
for all |x|≤ρ1 and a.e. t∈[0,T]. It follows from (47) and (48) that
(49)|F(t,x)|≤maxs∈[0,ρ1]a(s)b(t)+C5|x|λ.
From(H2) and Lemma 2, we have that
(50)|∑i=1k‍∫0|u(ti)|‍Ii(t)dt|≤∑i=1k‍∫0|u(ti)|‍(ai+bi|t|ri)dt≤M1k∥u∥∞+M2∑i=1k‍∥u∥∞ri+1≤M1kC3∥u∥+M2C3∑i=1k‍∥u∥ri+1
for all u∈W01,p(0,T), where M1={a1,a2,…,ak}, M2={b1,b2,…,bk}. Combining (49), (50), and Hölder’s inequality, we have
(51)1p∥uαn∥p=φ(uαn)-∑i=1k‍∫0uαn(ti)‍Ii(t)dt+∫0T‍F(t,uαn(t))dt≤C4+M1kC3∥u∥+M2C3∑i=1k‍∥u∥ri+1+C5∫0T‍|uαn(t)|λdt+maxs∈[0,ρ1]a(s)∫0T‍b(t)dt≤C4+M1kC3∥u∥+M2C3∑i=1k‍∥u∥ri+1+C5∫0T‍|uαn(t)|λdt+C6,
for all large n, where C6=maxs∈[0,ρ1]a(s)∫0T‍b(t)dt. On the other hand, by (H5), there exist C7>0 and ρ2>0 such that
(52)xf(t,x)-2F(t,x)≥C7|x|β,
for all |x|≥ρ2 and a.e. t∈[0,T]. By (A),
(53)|xf(t,x)-2F(t,x)|≤C8b(t),
for all |x|≤ρ2 and a.e. t∈[0,T], where C8=(2+ρ2)maxs∈[0,ρ2]a(s). Combining (52) and (53), one has
(54)(f(t,x),x)-2F(t,x)≥C7|x|β-C7ρ2β-C8b(t),
for all x∈ℝN and a.e. t∈[0,T]. According to (H6), there exists C9>0 such that
(55)2∫0t‍Iij(s)ds-Iij(t)t≥-C9∀i=1,2,…,k,t∈ℝ.
Thus by (46), (54), and (55), we obtain
(56)(p+1)C4≥pφ(uαn)-〈φ′(uαn),uαn〉=p∑i=1k‍∫0uαn(ti)‍Ii(t)dt-uαnIi(uαn)+∫0T‍[uαn(t)f(t,uαn(t))-pF(t,uαn(t))]dt=∑i=1k‍(p∫0uαn(ti)‍Ii(t)dt-Ii(uαn(ti))uαn(ti))+∫0T‍[(∇F(t,uαn(t)),uαn(t))-pF(t,uαn(t))]dt≥-kC9+C7∫0T‍|uαn|βdt-C7ρ2βT-C8×∫0T‍b(t)dt,
for all large n. From (56), ∫0T‍|uαn|βdt is bounded. If β>λ, by Hölder’s inequality, we have
(57)∫0T‍|uαn|λdt≤T(β-λ)/β(∫0T‍|uαn|βdt)λ/β.
Since ξij∈[0,1) for all i=1,2,…,k, by (51) and (57), {uαn} is bounded in W01,p(0,T). If β≤λ, by Lemma 2, we obtain
(58)∫0T‍|uαn(t)|λdt=∫0T‍|uαn(t)|β|uαn(t)|λ-βdt≤∥uαn∥∞λ-β∫0T‍|uαn(t)|βdt≤C1λ-β∥uαn∥λ-β∫0T‍|uαn(t)|βdt.
Since ξij∈[0,1), λ-β<2, by (51) and (58), {uαn} is also bounded in W01,p(0,T). Hence, {uαn} is also bounded in W01,p(0,T). Going if necessary to a subsequence, we can assume that uααn⇀u in W01,p(0,T). As the same the proof of Lemma 15, Therefore, uαn→u in W01,p(0,T). Hence φ satisfies the (C)* condition.Theorem 17.
Assume that(A), (H1), (H2), (H3), (H5), and (H6) are satisfied and the following conditions hold.(H7)
There exist constantsδ>0, K>0, γ>0 such that
(59)γp|u|p+2kM1T|u|≤F(t,u)≤12pγpp|u|p+K|u|p+1,fort∈[0,T],|u|≤δ.(H8)
Ii(u)(i=1,2,…,k) is nondecreasing.Then the problem (1) has at least two critical points.Proof.
LetXn1=span{λk+1,λk+2,…,λk+n}, Xn2=X2=(X1)⊥ for n∈N. Then Xj=⋃n∈N‍Xnj¯,j=1,2. If u∈X1 one has |u|p≤γp∥u∥, and if u∈X2, we have ∥u∥p≤γ|u|pp.
Step 1.  φ has a local linking at 0 with respect to (X1,X2).
Foru∈X1, using (H7) we have
(60)∑i=1k‍∫0u(ti)‍Ii(t)dt≥0.
It follows from (H8) that
(61)φ(u)=1p∥u∥p+∑i=1k‍∫0u(ti)‍Ii(t)dt-∫0T‍F(t,u(t))dt≥1p∥u∥p-12pγpp|u|pp-K|u|p+1p+1≥1p∥u∥p-12pγppγpp∥u∥p-Kγp+1p+1∥u∥p+1=12p∥u∥p-Kγp+1p+1∥u∥p+1.
Thus, φ(u)≥0 for u∈X1 with ∥u∥≤r1, where r1>0 is small enough.
Foru∈X2, with ∥u∥≤r2:=δ/C3, we have |u|≤|u∥∞≤C3∥u∥≤δ since dimX2=k<+∞. Thus, From (H2) and (H8) it follows that
(62)φ(u)=1p∥u∥p+∑i=1k‍∫0u(ti)‍Ii(t)dt-∫0T‍F(t,u(t))dt≤1p∥u∥p+∑i=1k‍∫0u(ti)‍(ai+bi|u|ri)-∫0T‍(γp|u|p+2kM1T|u|)dt≤γp|u|pp+kM1∥u∥∞+M2×∑i=1k‍∥u∥∞ri+1-γp|u|pp-2kM1∥u∥∞≤-kM1∥u∥∞+M2∑i=1k‍∥u∥∞ri+1≤0.
Take r=min{r1,r2}. We know that φ has a local linking at 0 with respect to (X1,X2).
Step  2.  φ maps bounded sets into bounded sets.
Assume∥u∥≤R0, where R0 is a constant. By (H8), one has
(63)φ(u)≤1p∥u∥p+∑i=1k‍∫0u(ti)‍Ii(t)dt+∫0T‍|F(t,u(t))|dt≤1p∥u∥p+kM1∥u∥∞+M2∑i=1k‍∥u∥∞ri+1+12p∥u∥p+Kγp+1p+1∥u∥p+1≤32p∥u∥p+kM1C3∥u∥+M2×∑i=1k‍C3ri+1∥u∥ri+1+Kγp+1p+1∥u∥p+1≤32p∥R0∥p+kM1C3∥R0∥+M2×∑i=1k‍C3ri+1∥R0∥ri+1+Kγp+1p+1∥R0∥p+1<∞,
which implies that φ maps bounded sets into bounded sets.
Step 3.   For every m∈N, φ(u)→-∞ as ∥u∥→+∞,u∈Xm1⊕X2.
By(H5), for any M0>0, there exists a constant h(M0) such that F(t,u(t))≥M|u|p-h(M0) for all (t,u)∈[0,T]×R. Since dim(Xm1⊕X2) is of finite dimension, there exists γ>0 such that ∥u∥p≤γ|u|pp for all u∈Xm1⊕X2, which implies
(64)φ(u)≤1p∥u∥p+∑i=1k‍∫0u(ti)‍Ii(t)dt-M|u|pp+h(M)T≤1p∥u∥p+kM1∥u∥∞+M2×∑i=1k‍∥u∥∞ri+1-Mγ∥u∥p+h(M)T≤(1p-Mγ)∥u∥p+kM1C3∥u∥+M2∑i=1k‍C3ri+1∥u∥ri+1+h(M)T.
Choosing M>γ/p, we have φ(u)→-∞ as ∥u∥→+∞,u∈Xm1⊕X2.
Summing up the above, and by Lemma16, φ satisfies all the assumptions of Lemma 11. Hence by Lemma 11, problem (1) has at least one nontrivial solution. The proof of Theorem 17 is completed.Theorem 18.
Assume that(A), (H1), (H2), (H3), and (H4) are satisfied and the following conditions hold.(H9)
f(t,u)=-f(t,-u)for(t,u)∈[0,T]×R.(H10)
Ii(u)=-Ii(-u)(i=1,2,…,k) and nondecreasing.Then the problem (1) has an infinite number of nontrivial solutions.Proof.
φ
∈
C
1
(
W
0
1
,
p
(
0
,
T
)
,
R
), by (H9) and (H10), φ is an even functional and φ(0)=0.
First, we verify the condition (2) of Lemma12.
(65)μu≤f(t,u)F(t,u),u≥M0,μu≥f(t,u)F(t,u),u≤-M0.
Integrating (65) for u from [M0,u] and [u,-M0], respectively, we have
(66)μlnuM0≤lnF(t,u)F(t,u0),u≥M0,μlnM0-u≥lnF(t,u)F(t,-u0),u≤-M0.
That is,
(67)F(t,u)≥F(t,M0)(uM0)μ,u≥M0,(68)F(t,u)≥F(t,-M0)(-uM0)μ,u≤-M0.
Combining (67) and (68), we have
(69)F(t,u)≥α1|u|μ,|u|≥M0,
where
(70)α1=u0-μmin{mint∈[0,T]F(t,M0),mint∈[0,T]F(t,-M0)}>0.
On the other hand, by the continuity of F(t,u), F(t,u) is bounded on [0,T]×[-M0,M0], there exists K1>0 such that
(71)F(t,u)≥-K1≥α1|u|μ-α1|M0|μ-K1,|u|≤M0.
Combining (69) and (71), we have
(72)F(t,u)≥α1|u|μ-α2,∀(t,u)∈[0,T]×R,
where α2=α1+K1.
For arbitrary finite-dimensional subspaceW⊂W01,p(0,T), and any u∈W, there exists C10=C10(W)>0 such that
(73)|u|μ≥C10∥u∥.
By (H2), (72), (73), and Lemma 2, we have
(74)φ(u)=1p∥u∥p+∑i=1k‍∫0u(ti)‍Ii(t)dt-∫0T‍F(t,u(t))dt≤1p∥u∥p+∑i=1k‍∫0u(ti)‍(ai+bi|u|ri)dt-∫0T‍(α1uμ-α2)dt≤1p∥u∥p+kM1∥u∥∞+M2∑i=1k‍∥u∥∞ri+1-α1|u|μμ+α2T≤1p∥u∥p+kM1C3∥u∥+M2∑i=1k‍C3ri+1∥u∥ri+1-α1C10μ∥u∥μ+α2T,
for every u∈W. This implies that φ(u)→-∞ as u∈W and ∥u∥→∞. So there exists R(W)>0 such that φ≤0 on for all u∈W with ∥u∥≥R.
In the following, we verify the condition (1) of Lemma12.
LetV=X1⨁X2, Y=⨁i=3∞Xi¯, then W01,p(0,T)=V⨁Y and V is finite dimensional. Using (H10) we have
(75)∑i=1k‍∫0u(ti)‍Ii(t)dt≥0.
By (H3) and (H4), we have
(76)limu→0F(t,u)up=0.
Hence, for ϵ=1/2pγpp, there exists δ>0 such that for every u with |u|≤δ,
(77)|F(t,u)|≤12pγpp|u|p.
Hence, for any u∈Y with ∥u∥≤δ/C3, ∥u∥∞≤δ, by (18), (54) and (55), we have
(78)φ(u)=1p∥u∥p+∑i=1k‍∫0u(ti)‍Ii(t)dt-∫0T‍F(t,u(t))dt≥1p∥u∥p-∫0T‍F(t,u(t))dt≥1p∥u∥p-12pγpp|u|pp≥1p∥u∥p-12pγppγpp∥u∥p=12p∥u∥p.
Take α=(1/2p)(δp/C3p),ρ=δ/C3, then
(79)φ(u)≥α,∀u∈∂Bρ∩Y.
Hence, by Lemma 12 and Lemma 15, φ possesses infinite critical points, that is, problem (1) has infinite nontrivial solutions. The proof is complete.


## 4. Example

Example 19.
Letp=2, ρ(t)=1, s(t)=t, t1=1/2. Consider the boundary value problem
(80)-u′′(t)+tu(t)=t+u(t),a.e.t∈[0,1],u(0)=u(1)=0,Δu′(t1)=u′(t1+)-u′(t1-)=1+u(t).
It is easy to see that conditions (H0) and (H2) of Theorem 13 hold. According to Theorem 13, problem (80) has at least one solution.Example 20.
Letp=2, ρ(t)=1, s(t)=2-t, t1=1/3. Consider the boundary value problem
(81)-u′′(t)+(2-t)u(t)=(u(t))2,a.e.t∈[0,1],u(0)=u(1)=0,Δu′(t1)=u′(t1+)-u′(t1-)=1+u(t).
It is easy to check that all the conditions of Theorem 17 are satisfied. Thus, according to Theorem 17, problem (81) has at least two critical points.Example 21.
Letp=3, ρ(t)=1+2t, s(t)=2+t, t1=1/2. Consider the boundary value problem
(82)-((1+2t)|u′(t)|u′(t))′+(2+t)u(t)=(u(t))3,a.e.t∈[0,1],u(0)=u(1)=0,Δu′(t1)=u′(t1+)-u′(t1-)=u(t)3.
It is easy to check that all the conditions of Theorem 18 are satisfied. Therefore, according to Theorem 18, problem (82) has infinite nontrivial solutions.



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*Source: 401740-2013-02-12.xml*