ON ASYMPTOTICALLY LINEAR ELLIPTIC PROBLEMS JOSE R. S. NASCIMENTO MARCOS T. O. PIMENTA AND JO AO R. SANTOS J UNIOR Abstract. In this paper we prove the existence of a signed ground state solution in the

2025-04-24 0 0 502.03KB 22 页 10玖币
侵权投诉
ON ASYMPTOTICALLY LINEAR ELLIPTIC PROBLEMS
JOS´
E R. S. NASCIMENTO, MARCOS T. O. PIMENTA, AND JO ˜
AO R. SANTOS J ´
UNIOR
Abstract. In this paper we prove the existence of a signed ground state solution in the
mountain pass level for a class of asymptotically linear elliptic problems, even when the
nonlinearity is just continuous in the second variable. The (strongly) resonant and non-
resonant cases are discussed. A multiplicity result is also proved when fis odd with respect
to the second variable.
1. Introduction
We are interested in the existence of ground state and other nontrivial solutions to the
following class of semilinear problems
(P) −∆u=f(x, u) in Ω,
u= 0 on ∂Ω,
where Ω ⊂IRNis a bounded smooth domain, N≥1, f: Ω ×IR →IR is a Carath´eodory
function which is asymptotically linear at the origin and at infinity, that is,
(1.1) α(x) = lim
t→02F(x, t)/t2and η(x) = lim
|t|→∞ 2F(x, t)/t2uniformly in x∈Ω
are functions in L∞(Ω).
It is well known in the literature that asymptotically linear problems can be classified as
resonant at infinity (if λm(η) = 1, for some m∈IN) or non-resonant at infinity (if λm(η)6= 1,
for all m∈IN), where, throughout this paper, λm(θ) denotes the m-th eigenvalue of the problem
(1.2) −∆u=λθ(x)uin Ω,
u= 0 on ∂Ω.
In particular, observe that if θ(x) = θis a nontrivial constant, then λm(θ) = λm/θ, where λm
denotes the m-th eigenvalue of Laplacian operator with Dirichlet boundary condition. In fact,
the resonant case is subdivided depending on how small at infinity is the function
(1.3) g(x, t) = η(x)t−f(x, t).
As observed in [5], the smaller gis at infinity, the stronger resonance is. The worst situation
is when, for a.e. x∈Ω,
(1.4) lim
|t|→∞ g(x, t) = 0 and lim
|t|→∞ Zt
0
g(x, s)ds =β(x)6≡ ∞.
In this case, we say that problem (P) is strongly resonant. One of the very first works dealing
with this situation is [5] where, Bartolo, Benci and Fortunato show the existence of multiple
2010 Mathematics Subject Classification. 35J15, 35J25, 35J61.
Key words and phrases. strongly resonant problems, ground state solution, genus theory.
Marcos T.O. Pimenta is partially supported by FAPESP 2021/04158-4, CNPq 303788/2018-6 and FAPDF,
Brazil. Joao R. Santos J´unior is partially supported by FAPESP 2021/10791-1 and CNPq 313766/2021-5, Brazil.
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
1
arXiv:2210.05452v1 [math.AP] 11 Oct 2022
2 J. R. S. NASCIMENTO, M. T. O. PIMENTA, AND J. R. SANTOS JR.
solutions for strongly resonant problems in the presence of some symmetry in the autonomous
nonlinearity. Their proofs are based on a deformation theorem and pseudo-index theory.
Besides [5], there exist so many other works dealing with problem (P). Without any intention
to be complete, we refer the reader to some papers and references therein. The existence of
solution for problem (P) was investigated under different conditions, for instance, by Ahmad [1],
Amann and Zehnder [4], Ambrosetti [2], Dancer [6], De Figueiredo and Miyagaki [8], Li and
Willem [12], Liu and Zou [14] and Struwe [16]. Multiplicity results for problem (P) were also
investigated by Li and Willem [13], Liu and Zhou [11] , Su [18] and Su and Zhao [19]. It is
essential to point out that in the great majority of previous references, the nonlinearity fis
assumed to be differentiable (or even C1) in the second variable, being this assumption crucial
in their arguments.
P. Bartolo, V. Benci and D. Fortunato [5] studied (P) when f(x, t) = f(t) is a smooth
function, satisfying:
(BBF1) lim|t|→∞(λmt−f(t))t= 0;
(BBF2) limt→∞ Rt
−∞(λms−f(s))ds = 0;
(BBF3) Rt
−∞(λms−f(s))ds ≥0, for all t∈IR.
Under these assumptions, the authors were able to prove the existence of solution to (P).
Observe that (BBF1) implies that f(t)/t →λm“faster” than t2→ ∞ as |t|→∞. More
recently, Gongbao Li and Huan-Song Zhou [11] relaxed the differentiability of fby considering
problem (P) under the following assumptions:
(LZ1) f∈C(Ω ×IR,IR);
(LZ2) limt→0f(x, t)/t = 0 uniformly in x∈Ω;
(LZ3) lim|t|→∞ f(x, t)/t =luniformly in x∈Ω, where l∈(0,∞) is a constant, or l=∞and
|f(x, t)| ≤ c1+c2|t|q−1, for some positive constants c1, c2and q∈(2,2∗);
(LZ4) f(x, t)/t is nondecreasing in t≥0 for almost every x∈Ω;
(LZ5) f(x, −t) = −f(x, t) for all (x, t)∈Ω×IR.
By using a symmetric version of the mountain pass theorem for C1-functionals, the authors
were able to prove the following result:
Theorem 1.1. Assume that f(x, t)satisfies conditions (LZ1)−(LZ3) and (LZ5), and l > λm,
then the following hold:
(i)If l∈(λm,∞)is not an eigenvalue of −∆with zero Dirichlet data, then problem (P)
has at least kpairs of nontrivial solutions in H1
0(Ω).
(ii)Suppose that condition
(fF) lim
|t|→∞ [(1/2)f(x, t)t−F(x, t)] = ∞
is satisfied, where Fis the primitive of f. Then the conclusion of (i)holds even if
l=λmis an eigenvalue of −∆with zero Dirichlet data.
(iii)If l=∞in condition (LZ3) and (LZ4) holds, then problem (P)has infinitely many
nontrivial solutions.
Let gbe given in (1.3) and Gits primitive, since
(1/2)f(x, t)t−F(x, t) = G(x, t)−(1/2)tg(x, t),
ON ASYMPTOTICALLY LINEAR ELLIPTIC PROBLEMS 3
it follows that if tg(x, t) = t2[η(x)−f(x, t)/t]→0 as |t| → ∞, then
(1.5) lim
|t|→∞ [(1/2)f(x, t)t−F(x, t)] = lim
|t|→∞ G(x, t) = β(x).
Consequently, in the resonant case (that is λm(η) = 1 for some m), the limit in (1.5) determines
the degree of resonance of problem (P). For instance, in [5] we have β(x) = β∈IR, thus the
problem is strongly resonant. On the other hand, the result for the resonant situation in [11]
does not cover the strongly resonant case when f(x, t)/t →η(x) = l=λm“faster” than
t2→ ∞ as |t|→∞. In fact, since (f F ) is assumed, it follows from (1.5) that β(x)≡ ∞.
In this paper we complement previous results. In fact, in the sequel, we suppose that f
satisfies the following hypotheses:
(f1)t7→ f(x, t)/|t|is increasing (a. e. in Ω) and α+, η+6= 0;
(f2)λm(η)<1< λ1(α), for some m≥1,
where α+and η+are the positive parts of functions αand βdefined in (1.1). It is important
to point out that, by α+, η+6= 0 and [7], there exist the positive eigenvalues λm(η) and λ1(α).
Assuming (f1)−(f2) we have provided existence of ground state solution and multiplicity
for (P) in both cases: the non-resonant case (NRC) and resonant case (RC) (see Theorem
4.5), that is:
(NRC)λm+k(η)6= 1 for all k∈IN,
(RC)λm+k(η) = 1 for some k∈IN.
In particular, since in the case (RC) we have not assumed hypothesis (fF ), our result cover
strongly resonant nonlinearities, even when f(x, t)/t →λm“faster” than t2→ ∞ as |t|→∞,
see Section 5.
Finally, replacing assumptions (f1)−(f2) by
(f0
1) supt∈A|F(., t)/t2|∞<∞, for each bounded set A⊂IR, and α+, η+6= 0;
(f0
2)λm(α)<1< λ1(η), for some m≥1,
we still have been able to prove the existence of multiple solution for (P), see Theorem 4.6.
In Theorems 4.5 and 4.6, some progresses are obtained regarding the previous works. In
what follows, we enumerate the main contributions: (1) Condition (fF) is not required. Instead,
we are imposing condition (β), which is certainly weaker than (fF). In our best knowledge,
condition (β) has not appeared in previous papers and allows us to study the existence of
ground state solutions for some classes of strongly resonant problems which had not been
treated. In a first moment, due to the presence of the number τm, whose dependence on fis
not so explicit, assumption (β) may seem difficult to be checked. In order to ensure its viability,
we provide in the section 5a concrete problem (P) for which (β) holds and (fF) is not verified;
(2) Since α,ηand βdepend on xand fis not differentiable, our assumptions are more general
than a large part of previous papers (which usually require these functions be constants); (3)
We provide an unified approach to deal concurrently the non-resonant case, the strong and the
non-strong resonant cases.
Our approach is based on the Nehari method, which consists in minimizing the energy
functional Iover the so called Nehari manifold N, a set which contains all the nontrivial
solutions of the problem. Although this method has been carefully treated by A. Szulkin
and T. Weth [20] for the case of nonlinearities which satisfies superquadraticity conditions
at infinity, it is not a trivial task to apply it for problems involving asymptotically linear
nonlinearities. In order to cite the main difficulties, we point out that the method consists
4 J. R. S. NASCIMENTO, M. T. O. PIMENTA, AND J. R. SANTOS JR.
in proving the existence of a homeomorphism γbetween Nand a submanifold Mof H1
0(Ω).
Despite the absence of a differentiable structure in N, such a homeomorphism allows us to define
aC1-functional Ψ on Mwith very useful properties. However, differently of the problem for
superquadract nonlinearities, in which Mis the unit sphere Sof H1
0(Ω), for asymptotically
linear nonlinearities, it is not exactly clear who is the suitable manifold M. In fact, after a
careful study (see Lemma 2.1, Propositions 3.1 and 3.2) we are able to prove that M=SA:=
S∩A is a noncomplete submanifold of H1
0(Ω), where A:= u∈H1
0(Ω) : kuk2<RΩη(x)u2dx.
This fact brings additional problems. Indeed, it is important to assure that minimizing
sequences {un}for Ψ are not near the boundary of SA. In [20], this step is strongly based in
the fact that fhas a superquadratic growth at infinity, what implies that {Ψ(un)}tends to
infinity as the distance from {un}to the boundary tends to zero. In our case, the behaviour of
{Ψ(un)}at infinity, as dist(un, ∂SA)→0, is indefinite. This fact makes difficult, for example,
to know how to extend Ψ to SAin order to apply the Ekeland variational principle, which is
crucial to prove that {un}can be seen as Palais-Smale sequence.
The paper is organized as follows. In Section 2we present the variational background. In
Section 3we deeply study the Nehari manifold and some of its topological features. In Section
4we state and prove our main results. Finally, in Section 5, hypothesis (β) is discussed in a
concrete problem.
2. Preliminaries
Our main goal in this section is to introduce some variational background for (P). We start
denoting by I:H1
0(Ω) →IR the energy functional associated to problem (P), given by
I(u) = 1
2kuk2−ZΩ
F(x, u)dx,
where kuk2=ZΩ|∇u|2dx and F(x, t) = Zt
0
f(x, s)ds. It is well known that I∈C1(H1
0(Ω),IR)
and
I0(u)ϕ=ZΩ∇u∇ϕdx −ZΩ
f(x, u)ϕdx.
Thus, critical points of Iare weak solutions of (P).
The Nehari manifold associated to the functional Iis the set
N={u∈H1
0(Ω)\{0}:kuk2=ZΩ
f(x, u)udx}.
Since fis just a Carath´eodory function, we cannot ensure that Nis a smooth manifold.
Moreover, Sdenotes the unit sphere in H1
0(Ω) and
A:= u∈H1
0(Ω) : kuk2<ZΩ
η(x)u2dx.
Now, it is important to fix some notation. Throughout this paper we denote by eja
normalized (in H1
0(Ω) norm) eigenfunction associated to λj(η) and use the symbol [u6= 0] to
denote the set {x∈Ω : u(x)6= 0}. Moreover, |A|will always denote the Lebesgue measure of a
measurable set A⊂IRN,S(Ω) and |θ|∞denote, respectively, the best constant of the continuous
embedding from H1
0(Ω) into L2∗(Ω) and the L∞-norm of a function θ,χ(θ) = Pm
j=1 dimVλj(θ)
is the sum of the dimensions of the first meigenspaces Vλj(θ)associated to (1.2). Finally, Sχ(θ)
denotes the unit sphere of ⊕m
k=1Vλk(θ)and θ+denotes the positive part of a function θ.
ON ASYMPTOTICALLY LINEAR ELLIPTIC PROBLEMS 5
Lemma 2.1. Suppose that fsatisfies (f1)−(f2). Then, the following claims hold:
(i)The set Ais open and nonempty;
(ii)∂A={u∈H1
0(Ω) : kuk2=RΩη(x)u2dx};
(iii)Ac={u∈H1
0(Ω) : kuk2≥RΩη(x)u2dx};
(iv)N ⊂ A;
(v)S ∩ A 6=∅.
Proof. (i) By (f2), if uis an eigenfunction associated to λj(η), for some j∈ {1, . . . , m}, then
ubelongs to A. Moreover, A=ϕ−1(−∞,0), where ϕ:H1
0(Ω) →Ris the continuous function
ϕ(u) = kuk2−RΩη(x)u2dx. Items (ii)-(iii) are immediate consequences of the definition of A.
(iv) If u∈ N then,
kuk2=Z[u6=0] f(x, u)
uu2dx.
By (f1), we conclude
kuk2<ZΩ
η(x)u2dx.
Showing that u∈ A.
(v) It is enough to choose an eigenfunction ejassociated to λj(η) and normalized in H1
0(Ω),
whatever j∈ {1, . . . , m}. For sure, we have ej∈ S ∩ A.
In what follows, we will denote SA:= S ∩ A. Since Sis a C1-manifold of H1
0(Ω) and, by
Lemma 2.1,Ais open set of H1
0(Ω) whose boundary has intersection with S, it follows that
SAis a noncomplete C1-manifold of H1
0(Ω). Moreover, from (ii) and (iii), it is clear that
∂SA={u∈ S :1=RΩη(x)u2dx}and Sc
A={u∈ S : 1 ≥RΩη(x)u2dx}.
The following property of functions in ∂SAplays an important role in the existence of
solution for (P), see Lemmas 3.6 and 4.3.
Lemma 2.2. The following inequality holds true:
inf
u∈∂SA|[u6= 0]| ≥ (S(Ω)/|η|∞)N/2.
Proof. By using H¨older inequality, it follows that, for each u∈∂SA
1≤ |η|∞Z[u6=0]
u2dx ≤ |η|∞|u|2
2∗|[u6= 0]|2/N .
By continuous Sobolev embedding from H1
0(Ω) into L2∗(Ω),
1≤ |η|∞(1/S(Ω))|[u6= 0]|2/N .
Therefore,
|[u6= 0]| ≥ (S(Ω)/|η|∞)N/2,∀u∈∂SA.
The result is proved.
Next lemma provides some consequences of hypothesis (f1) which will be useful later on.
Lemma 2.3. Suppose that (f1)holds. Then, a.e. in Ω,
(i)t7→ (1/2)f(x, t)t−F(x, t)is increasing in (0,∞)and decreasing in (−∞,0);
(ii)t7→ F(x, t)/t2is increasing in (0,∞)and decreasing in (−∞,0);
(iii)f(x, t)/t > 2F(x, t)/t2for all t∈IR\{0}.
摘要：

ONASYMPTOTICALLYLINEARELLIPTICPROBLEMSJOSER.S.NASCIMENTO,MARCOST.O.PIMENTA,ANDJO~AOR.SANTOSJUNIORAbstract.Inthispaperweprovetheexistenceofasignedgroundstatesolutioninthemountainpasslevelforaclassofasymptoticallylinearellipticproblems,evenwhenthenonlinearityisjustcontinuousinthesecondvariable.The(s...

展开>> 收起<<
ON ASYMPTOTICALLY LINEAR ELLIPTIC PROBLEMS JOSE R. S. NASCIMENTO MARCOS T. O. PIMENTA AND JO AO R. SANTOS J UNIOR Abstract. In this paper we prove the existence of a signed ground state solution in the.pdf

共22页,预览5页

还剩页未读， 继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

相关推荐

更多
立即下载
分类：图书资源 价格：10玖币 属性：22 页 大小：502.03KB 格式：PDF 时间：2025-04-24

开通VIP享超值会员特权

  • 多端同步记录
  • 高速下载文档
  • 免费文档工具
  • 分享文档赚钱
  • 每日登录抽奖
  • 优质衍生服务

作者详情

相关内容

更多

热门标签

人际关系 配电装置 动力学连接体 力的合成 高考理综 全宋诗作者索引 公务员考试
/ 22
评分 收藏
分享
立即下载
客服
关注