# Infinitely Many Solutions of Superlinear Elliptic Equation

Authors: Anmin Mao; Yang Li
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Via the Fountain theorem, we obtain the existence of infinitely many solutions of the following superlinear elliptic boundary value problem:−Δu=f(x,u) in Ω,u=0 on ∂Ω, where Ω⊂ℝN(N>2) is a bounded domain with smooth boundary and f is odd in u and continuous. There is no assumption near zero on the behavior of the nonlinearity f, and f does not satisfy the Ambrosetti-Rabinowitz type technical condition near infinity.


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## Body

## 1. Introduction

Consider the following nonlinear problem:(1)-Δu=f(x,u)inΩ,u=0on∂Ω,
which has been receiving much attention during the last several decades. Here Ω⊂ℝN(N>2) is a bounded smooth domain and f is a continuous function on Ω×R and odd in u. We make the following assumptions on f:(S1)
there exist constantsa1>0 and 2N/(N-2)=2*>ν>2 such that
(2)|f(x,u)|≤a1(1+|u|ν-1),∀x∈Ω,u∈R;(S2)
F(x,u)≥0, for all (x,u)∈Ω×R, and
(3)lim|u|→∞F(x,u)u2=∞,uniformlyforx∈Ω,whereF(x,u)=∫0u‍f(x,u)dx;(S3)
there exists a constantb>0 such that
(4)limsup|u|→∞f(x,u)u-2F(x,u)u2+1<b,uniformlyforx∈Ω.Note that Costa and Magalhães in [1] introduced a condition similar to (S3), which also appeared in [2].In this paper, we will study the existence of infinitely many nontrivial solutions of (1) via a variant of Fountain theorems established by Zou in [3]. Fountain theorems and their dual form were established by Bartsch in [4] and by Bartsch and Willem in [5], respectively. They are effective tools for studying the existence of infinitely many large or small energy solutions. It should be noted that the P.S. condition and its variants play an important role for these theorems and their applications.We state our main result as follows.Theorem 1.
Assume that(S1)–(S3) hold and f(x,u) is odd in u. Then problem (1) possesses infinitely many solutions.Problem (1) was studied widely under various conditions on f(x,u); see, for example, [6–10]. In 2007, Rabinowitz et al. [6] studied the problem
(5)-Δu=λu+f(x,u)inΩ,u=0on∂Ω,
where Ω⊂RN is a bounded smooth domain, and assumed(f1)
f∈C1(Ω×R,R),(f2)
f(x,0)=0=fu(x,0),(f3)
|f(x,u)|≤C(1+|u|p-1), 2<p<2*,(f4)
∃μ>2, M>0, s.t.
(6)x∈Ω,|u|≥M⟹0<μF(x,u)≤uf(x,u),(f5)
F(x,u)≥0, for all x and u,  and uf(x,u)>0 for |u|>0 small.They got the existence of at least three nontrivial solutions.(f4) was given by Ambrosetti and Rabinowitz [11] to ensure that some compactness and the Mountain Pass setting hold.However, there are many functions which are superlinear but do not necessarily need to satisfy(f4). For example,
(7)F(x,u)=12u2ln(1+u)-12(u22-u+ln(1+u)).
It is easy to check that (f4) does not hold. On the other hand, in order to verify (f4), it is usually an annoying task to compute the primitive function of f and sometimes it is almost impossible, for example,
(8)f(x,u)=|u|u(1+e(1+|sinu|)α+|cosu|α),u∈R,α>0.
More examples are presented in Remark 2.We recall that(f4) implies a weaker condition
(9)F(x,u)≥c|u|θ-d,c,d>0,a.e.x∈Ω,u∈R,θ>2.
In [12], Willem and Zou gave one weaker condition
(10)c|u|θ≤uf(x,u)for|u|≥R0,a.e.x∈Ω,θ>2.
Note that (S2) is much weaker than the above conditions.In [13], Schechter and Zou proved that under the hypotheses that
(11)(S1)holdsandeitherlimu→-∞F(x,u)u2=+∞orlimu→∞F(x,u)u2=+∞,
problem (1) has a nontrivial weak solution.Recently, Miyagaki and Souto in [2] proved that problem (1) has a nontrivial solution via the Mountain Pass theorem under the following conditions:
(12)(S1)and(S2)hold,andlim|u|→0f(x,u)u=0,∃u0>0,s.t.f(x,u)uisincreasinginu≥u0anddecreasinginu≤-u0,∀x∈Ω,
and they adapted some monotonicity arguments used by Schechter and Zou [13]. This approach is interesting, but many powerful variational tools such as the Fountain theorem and Morse theory are not directly applicable. In addition, the monotonicity assumption on F(x,u)/u2 is weaker than the monotonicity assumption on f(x,u)/u.As to the case in the current paper, we make some concluding remarks as follows.Remark 2.
To show that our assumptions(S2) and (S3) are weaker than (f4), we give two examples:(1)
f(x,u)=2ulnu+u,(2)
f(x,u)=γ|u|γ-2u+(γ-1)|u|γ-3usin2u+|u|γ-1sin2u,u∈R∖{0},γ>2,
which do not satisfy(f4). Example (2) can be found in [3]. So the case considered here cannot be covered by the cases mentioned in [6, 11].Remark 3.
Compared with papers [11, 12], we do not assume any superlinear conditions near zero. Compared with paper [2], we do not impose any kind of monotonic conditions. In addition, although we do not assume (f4) holds, we are able to check the boundedness of P.S. (or P.S.*) sequences. So, our result is different from those in the literature.Our argument is variational and close to that in [2, 3, 13, 14]. The paper is arranged as follows. In Section 2 we formulate the variational setting and recall some critical point theorems required. We then in Section 3 complete the proof of Theorem 1.


## 2. Variational Setting

In this section, we will first recall some related preliminaries and establish the variational setting for our problem. Throughout this paper, we work on the spaceE=H01(Ω) equipped with the norm
(13)∥u∥=(∫Ω‍|∇u|2dx)1/2.Lemma 4.
E embeds continuously into Lp, for all 0<p≤2*, and compactly into Lp, for all 1≤p<2*; hence there exists τp>0 such that
(14)|u|p≤τp∥u∥,∀u∈E,
where |u|p=(∫Ω‍|u|pdx)1/p.Define the Euler-Lagrange functional associated to problem (1), given by
(15)I(u)=12∥u∥2-Ψ(u),u∈E,
where Ψ(u)=∫Ω‍F(x,u)dx. Note that (S1) implies that
(16)F(x,u)≤a1(|u|+|u|ν),∀(x,u)∈Ω×R.In view of (16) and Sobolev embedding theorem, Iμ and Ψ are well defined. Furthermore, we have the following.Lemma 5 (see [15] or [16]).
Suppose that(S1) is satisfied. Then Ψ∈C1(E,R) and Ψ′:E→E* is compact and hence I∈C1(E,R). Moreover
(17)Ψ′(u)v=∫Ω‍f(x,u)vdx,I′(u)v=∫Ω‍∇u∇vdx-Ψ′(u)v,
for all u,v∈E, and critical points of I on E are solutions of (1).Lemma 6 (see [17]).
Assume that|Ω|<∞,1≤p,r≤∞,f∈C(Ω-×R), and |f(x,u)|≤c(1+|u|p/r). Then for every u∈Lp(Ω),f(x,u)∈Lr(Ω), and the operator A:Lp(Ω)↦Lr(Ω):u↦f(x,u) is continuous.LetE be a Banach space equipped with the norm ∥·∥ and E=⨁j∈NXj¯, where dimXj<∞ for any j∈N. Set Yk=⨁j=1kXj and Zk=⨁j=k∞Xj¯. Consider the following C1 functional Φλ:E→R defined by
(18)Φλ(u)≔A(u)-λB(u),λ∈[1,2].
The following variant of the Fountain theorems was established in [3].Theorem 7 (see [3, Theorem 2.1]).
Assume that the functionalΦλ defined above satisfies the following: (F1)
Φλ maps bounded sets to bounded sets uniformly for λ∈[1,2]; furthermore, Φλ(-u)=Φλ(u) for all (λ,u)∈[1,2]×E;(F2)
B(u)≥0 for all u∈E; moreover, A(u)→∞ or B(u)→∞ as ∥u∥→∞;(F3)
there existsrk>ρk>0 such that
(19)αk(λ)≔infu∈Zk,∥u∥=ρkΦλ(u)>βk(λ)≔maxu∈Yk,∥u∥=rkΦλ(u),∀λ∈[1,2].Then(20)αk(λ)≤ζk(λ)≔infγ∈Γkmaxu∈BkΦλ(γ(u)),∀λ∈[1,2],whereBk={u∈Yk:∥u∥≤rk} and Γk:={γ∈C(Bk,E)∣γisodd,γ|∂Bk=id}. Moreover, for a.e. λ∈[1,2], there exists a sequence {umk(λ)}m=1∞ such that
(21)supm∥umk(λ)∥<∞,Φλ′(umk(λ))⟶0,Φλ(umk(λ))⟶ζk(λ)asm⟶∞.In order to apply the above theorem to prove our main results, we define the functionalsA, B, and Iλ on our working space E by
(22)A(u)=12∥u∥2,B(u)=∫Ω‍F(x,u)dx,(23)Iλ(u)=A(u)-λB(u)=12∥u∥2-λ∫Ω‍F(x,u)dx,
for all u∈E and λ∈[1,2]. Note that I1=I, where I is the functional defined in (15).From Lemma5, we know that Iλ∈C1(E,R), for all λ∈[1,2]. It is known that -Δ is a selfadjoint operator with a sequence of eigenvalues (counted with multiplicity)
(24)0<λ1<λ2≤λ3≤⋯≤λj≤⋯⟶∞,
and the corresponding system of eigenfunctions {ej:j∈N}(-Δej=λjej) forming an orthogonal basis in E. Let Xj=span{ej}, for all j∈N.


## 3. Proof of Theorem1

Lemma 8.
Assume that (S1)-(S2) hold. Then there exists a positive integer k1 and two sequences rk>ρk→∞ as k→∞ such that
(25)αk(λ)≔infu∈Zk,∥u∥=ρkIλ(u)>0,∀k≥k1,(26)βk(λ)≔maxu∈Yk,∥u∥=rkIλ(u)<0,∀k∈N,
where Yk=⨁j=1kXj=span{e1,…,ek} and Zk=⨁j=k∞Xj¯=span{ek,…},¯ for all k∈N.Proof
Step  1. We first prove (25).
By (16) and (23), for all λ∈[1,2] and u∈E, we have
(27)Iλ(u)≥12∥u∥2-2∫Ω‍a1(|u|+|u|ν)dx=12∥u∥2-2a1(|u|1+|u|νν),
where a1 is the constant in (16). Let
(28)σν(k)=supu∈Zk,∥u∥=1|u|ν,∀k∈N.
Then
(29)σν(k)⟶0ask⟶∞,
since E is compactly embedded into Lν. Combining (14), (27), and (28), we have
(30)Iλ≥12∥u∥2-2a1τ1∥u∥-2a1σνν(k)∥u∥ν,Iλ≥12∥u∥22-2∥u∥∀(λ,u)∈[1,2]×Zk.
By (29), there exists a positive integer k1>0 such that
(31)ρk≔(16a1σνν(k))1/(2-ν)>16a1τ1,∀k≥k1,
since ν>2. Evidently,
(32)ρk⟶∞ask⟶∞.
Combining (30) and (31), direct computation shows
(33)αk≔infu∈Zk,∥u∥=ρkIλ(u)≥ρk24>0,∀k≥k1.Step2. We then verify (26).
We claim that for any finite-dimensional subspaceF⊂E, there exists a constant ϵ>0 such that
(34)m({x∈Ω:|u(x)|≥ϵ∥u∥})≥ϵ,∀u∈F∖{0}.
Here and in the sequel, m(·) always denotes the Lebesgue measure in R.
If not, for anyn∈N, there exists un∈F∖{0} such that
(35)m({x∈Ω:|un(x)|≥1n∥un∥})<1n.
Let vn=un/∥un∥∈F, for all n∈N. Then ∥vn∥=1, for all n∈N, and
(36)m({x∈Ω:|vn(x)|≥1n})<1n,∀n∈N.
Passing to a subsequence if necessary, we may assume vn→v0inE, for some v0∈F, since F is of finite dimension. Evidently, ∥v0∥=1. In view of Lemma 4 and the equivalence of any two norms on F, we have
(37)∫Ω‍|vn-v0|dx⟶0asn⟶∞,
and |v0|∞>0.
By the definition of norm|·|∞, there exists a constant δ0>0 such that
(38)m({x∈Ω:|v0(x)|≥δ0})≥δ0.
For any n∈N, let
(39)Λn={x∈Ω:|vn(x)|<1n},Λnc=Ω∖Λn={x∈Ω:|vn(x)|≥1n}.
Set Λ0={x∈Ω:|v0(x)|≥δ0}. Then for n large enough, by (36) and (38), we have
(40)m(Λn∩Λ0)=m(Λ0∖Λnc)≥m(Λ0)-m(Λnc)≥δ0-1n≥δ02.
Consequently, for n large enough, there holds
(41)∫Ω‍|vn-v0|dx≥∫Λn∩Λ0‍|vn-v0|dx≥∫Λn∩Λ0‍(|v0|-|vn|)dx≥(δ0-1n)m(Λn∩Λ0)≥δ024>0.
This is in contradiction to (37). Therefore (34) holds.
Consequently, for anyk∈N, there exists a constant ϵk>0 such that
(42)m(Λuk)≥ϵk,∀u∈Yk∖{0},
where Λuk≔{x∈Ω:|u(x)|≥ϵk∥u∥}, for all k∈N, and u∈Yk∖{0}. By (S2), for any k∈N, there exists a constant Sk>0 such that
(43)F(x,u)≥|u|2ϵk3,∀|u|≥Sk.
Combining (23), (42), (43), and (S2), for any k∈N and λ∈[1,2], we have
(44)Iλ(u)≤12∥u∥2-∫Λuk‍|u|2ϵk3dx≤12∥u∥2-1ϵk3ϵk2∥u∥2m(Λuk)≤12∥u∥2-∥u∥2=-12∥u∥2
with ∥u∥≥Sk/ϵk. Now for any k∈N, if we choose
(45)rk>max{ρk,Skϵk},
then (44) implies
(46)βk(λ)≔maxu∈Yk,∥u∥=rkIλ(u)≤-rk22<0,∀k∈N,
ending the proof.Proof of Theorem1.
It follows from (16), (23), and Lemma 5 that Iλ maps bounded sets to bounded sets uniformly for λ∈[1,2]. In view of the evenness of F(x,u) in u, it holds that Iλ(-u)=Iλ(u) for all (λ,u)∈[1,2]×E. Thus the condition (F1) of Theorem 7 holds. Besides, A(u)=(1/2)∥u∥2→∞ as ∥u∥→∞ and B(u)≥0 since F(x,u)≥0. Thus the condition (F2) of Theorem 7 holds. And Lemma 8 shows that the condition (F3) holds for all k≥k1. Therefore, by Theorem 7, for any k≥k1 and a.e. λ∈[1,2], there exists a sequence {umk(λ)}m=1∞⊂E such that
(47)supm∥umk(λ)∥<∞,Iλ′(umk(λ))⟶0,Iλ(umk(λ))⟶ζk(λ)
as m→∞, where
(48)ζk(λ)≔infh∈Γkmaxu∈BkIλ(h(u)),∀λ∈[1,2],
with Bk={u∈Yk:∥u∥≤rk} and Γk≔{h∈C(Bk,E)∣hisodd,h|∂Bk=id}.
Furthermore, it follows from the proof of Lemma8 that
(49)ζk(λ)∈[α-k,ζ-k],∀k≥k1,
where ζ-k≔maxu∈BkIλ(u) and α-k≔ρk2/4→∞ask→∞ by (32).
Claim  1.  {umk(λ)}m=1∞⊂E possesses a strong convergent subsequence in E, for ∀λ∈[1,2] and k≥k1.
In fact, by the boundedness of the{umk(λ)}m=1∞,  passing to a subsequence, as m→∞,  we may assume
(50)umk(λ)⇀uk(λ)inE.
By the Sobolev embedding theorem,
(51)umk(λ)⟶uk(λ)inLν.
Lemma 6 implies that
(52)f(x,umk(λ))⟶f(x,uk(λ))inLν/(ν-1).
Observe that
(53)∥umk(λ)-uk(λ)∥2=(Iλ′(umk(λ))-Iλ′(uk(λ)),umk(λ)-uk(λ))+λ∫Ω‍(f(x,umk(λ))-f(x,uk(λ)))+λ∫Ω‍×(umk(λ)-uk(λ))dx.
By (47), it is clear that
(54)(Iλ′(umk(λ))-Iλ′(uk(λ)),umk(λ)-uk(λ))⟶0(Iλ′(umk(λ))5-Iλ′(uk(λ)),uk(λ))asm⟶∞.
It follows from the Hölder inequality, (51), and (52) that
(55)|∫Ω‍(f(x,umk(λ))-f(x,uk(λ)))(umk(λ)-uk(λ))dx|≤|f(x,umk(λ))-f(x,uk(λ))|ν/(ν-1)×|umk(λ)-uk(λ)|ν⟶0
as m→∞. Thus by (53), (54), and (55), we have proved that
(56)∥umk(λ)-uk(λ)∥⟶0asm⟶∞,
that is, umk(λ)→uk(λ) in E.
Thus, for eachk≥k1, we can choose λn→1 such that the sequence {umk(λn)}m=1∞ obtained a convergent subsequence; passing again to a subsequence, we may assume
(57)limm→∞umk(λn)=unkinE,∀n∈N,k≥k1.
This together with (47) and (49) yields
(58)Iλn′(unk)=0,Iλn(unk)∈[α-k,ζ-k],∀n∈N,k≥k1.
Claim2.   {unk}n=1∞ is bounded in E for all k≥k1.
For notational simplicity, we will setun=unk for all n∈N throughout this paragraph. If {un} is unbounded in E, we define vn=un/∥un∥. Since ∥vn∥=1, without loss of generality we suppose that there is v∈E such that
(59)vn⇀vinE,vn⟶vinLν(Ω),vn(x)⟶v(x)a.e.inΩ.
Let Ω≠={x∈Ω:v(x)≠0}. If x∈Ω≠, from (S2)  it follows that
(60)limn→∞F(x,un(x))un(x)2vn(x)2=∞.
On the other hand, after a simple calculation, we have
(61)limn→∞∫Ω‍F(x,un(x))un(x)2vn(x)2=12.
We conclude that Ω≠ has zero measure and v≡0a.e.inΩ.
Moreover, from (49) and (58)
(62)∫Ω‍(1/2)f(x,un)un-F(x,un)un2vn2dx=Iλn(un)λn∥un∥2>0.
By (S3),
(63)limsupn→∞(1/2)f(x,un)un-F(x,un)un2vn2<limsupn→∞b(un2+1)2un2vn2=0
which contradicts (62). Hence {un} is bounded.
Claim3.  {unk} possesses a convergent subsequence with the limit uk∈E for all k≥k1.
In fact, by  Claim 2, without loss of generality, we have assume(64)unk⇀ukasn⟶∞.
By virtue of the Riesz Representation theorem, Iλ′:E↦E* and Ψ′:E↦E* can be viewed as Iλ′:E↦E and Ψ′:E↦E, respectively, where E* is the dual space of E. Note that
(65)0=Iλn′(unk)=unk-λnΨ′(unk),
that is
(66)unk=λnΨ′(unk).
By Lemma 5, Ψ′:E↦E is also compact. Due to the compactness of Ψ′ and (64), the right-hand side of (66) converges strongly in E and hence unk→uk in E.
Now for eachk≥k1, by (58), the limit uk is just a critical point of I1=I with I(uk)∈[α-k,ζ-k]. Since α-k→∞  as k→∞ in (49), we get infinitely many nontrivial critical points of I. Therefore (1) possesses infinitely many nontrivial solutions by Lemma 5.



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