Being informed that one of our articles is cited in the paper mentioned in the title, we downloaded it, and we were surprised to see that, practically, all the results from our paper were reproduced ...in Section 3 of Meghea and Stamin’s article. Having in view the title of the article, one is tempted to think that the remarks mentioned in the paper are original and there are examples given as to where and how (at least) some of the reviewed results are effectively applied. Unfortunately, a closer look shows that most of those remarks in Section 3 are, in fact, extracted from our article, and it is not shown how a specific result is used in a certain application. So, our aim in the present note is to discuss the content of Section 3 of Meghea and Stamin’s paper, emphasizing their Remark 8, in which it is asserted that the proof of Lemma 7 in our article is “full of errors.”
Metric regularity is an important concept in variational analysis. Perturbation analysis of metric regularity is studied in this paper. An improved stability result on the metric regularity under ...Lipschitz set-valued perturbations is established. Compared to the known results, the conditions are relaxed and the proof is provided more concisely. In other words, we allow the diameter of the image of the perturbation mapping at the reference point to vary in a larger domain, and give a simpler argument. An example is presented in which our result can be applied but not the known results.
On eigenvalue problems for the p(x)-Laplacian Marcos, Aboubacar; Soninhekpon, Janvier
Journal of mathematical analysis and applications,
07/2024, Letnik:
535, Številka:
2
Journal Article
Recenzirano
This paper studies nonlinear eigenvalue problems with weight governed by the non-homogeneous p(x)-Laplacian operator, under the Dirichlet boundary condition on a bounded domain of RN(N≥2). The ...nonlinear part is also non-homogeneous and depending on its features (sublinear or superlinear), we show that the spectrum includes a continuous family of eigenvalues and in some contexts, is exactly the whole set R+⁎. Moreover, we show that the smallest eigenvalue obtained from the Lagrange multipliers is exactly the first eigenvalue in the Ljusternik-Schnirelman eigenvalues sequence and we also provide sufficient conditions for multiplicity results.
In this paper, we deal with the existence of solutions to the nonuniformly elliptic equation of the form
(0.1)
−
div
(
a
(
x
,
∇
u
)
)
+
V
(
x
)
|
u
|
N
−
2
u
=
f
(
x
,
u
)
|
x
|
β
+
ε
h
(
x
)
in
R
N
...when
f
:
R
N
×
R
→
R
behaves like
exp
(
α
|
u
|
N
/
(
N
−
1
)
)
when
|
u
|
→
∞
and satisfies the Ambrosetti–Rabinowitz condition. In particular, in the case of
N-Laplacian, i.e.,
a
(
x
,
∇
u
)
=
|
∇
u
|
N
−
2
∇
u
, we obtain multiplicity of weak solutions of
(0.1). Moreover, we can get the nontriviality of the solution in this case when
ε
=
0
. Finally, we show that the main results remain true if one replaces the Ambrosetti–Rabinowitz condition on the nonlinearity by weaker assumptions and thus we establish the existence and multiplicity results for a wider class of nonlinearity, see Section 7 for more details.
The present paper is concerned with the Ekeland Variational Principle (EkVP) and its equivalents (Caristi-Kirk fixed point theorem, Takahashi minimization principle, Oettli-Théra equilibrium version ...of EkVP) in quasi-uniform spaces. These extend some results proved by Hamel and Löhne A minimal point theorem in uniform spaces. In: Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday. Vols 1, 2. Dordrecht: Kluwer Academic Publishers; 2003. p. 577-593 and Hamel Equivalents to Ekeland's variational principle in uniform spaces. Nonlinear Anal. 2005;62:913-924 in uniform spaces, as well as those proved in quasi-metric spaces by various authors. The case of F-quasi-gauge spaces, a non-symmetric version of F-gauge spaces introduced by Fang The variational principle and fixed point theorems in certain topological spaces. J Math Anal Appl. 1996;202:398-412, is also considered. The paper ends with the quasi-uniform versions of some minimization principles proved by Arutyunov and Gel'man The minimum of a functional in a metric space, and fixed points. Zh Vychisl Mat Mat Fiz. 2009;49:1167-1174 and Arutyunov Caristi's condition and existence of a minimum of a lower bounded function in a metric space. Applications to the theory of coincidence points. Proc Steklov Inst Math. 2015;291(1):24-37 in complete metric spaces.
In this paper, the operator approach based on the fixed point principles of Banach and Schaefer is used to establish the existence of solutions to stationary Kirchhoff equations with reaction terms. ...Next, for a coupled system of Kirchhoff equations, it is proved that under suitable assumptions, there exists a unique solution which is a Nash equilibrium with respect to the energy functionals associated to the equations of the system. Both global Nash equilibrium, in the whole space, and local Nash equilibrium, in balls are established. The solution is obtained by using an iterative process based on Ekeland's variational principle and whose development simulates a noncooperative game.
This paper deals with the following critical nonlocal Choquard equation on the Heisenberg group: −(a−b∫Ω|∇Hu|2dξ)ΔHu=μ|u|q−2u+∫Ω|u(η)|Qλ∗|η−1ξ|λdη|u|Qλ∗−2uinΩ,u=0on∂Ω,where Ω⊂HN is a smooth bounded ...domain, ΔH is the Kohn-Laplacian on the Heisenberg group HN, 1<q<2 or 2<q<Qλ∗, a,b>0, μ>0, 0<λ<4, and Qλ∗=2Q−λQ−2 is the critical exponent. Existence results are obtained by using the Ekeland variational principle, Clark critical point theorem, mountain pass theorem, and Krasnoselskii genus theorem, respectively. Due to critical nonlinearities as well as the presence of the double non-local teams, there are some difficulties on the Heisenberg group’s framework. Our results are new even in the Euclidean case.
•The background of the research for critical problem on the Heisenberg group is rather outstanding. Nowadays, Geometric Analysis in the Heisenberg group is a fast-growing research field, because of its important applications in quantum mechanics, partial differential equations and other fields, as well as the great interest in pure mathematics. The study of critical problems is deeply connected to the concentration phenomena taking place when considering sequences of approximated solutions.•The study of critical nonlocal Choquard equation is more meaningful. One of the key points of this paper is that the method of proving the compactness condition is reasonable and natural. It is difficult to prove the boundedness of Palais–Smale sequences. Moreover, from the point of view of mathematical research, the occurrence of a double local terms makes the study of this problem more complicated and interesting.•The conclusions obtained extend previous results not only from the Euclidean space to the Heisenberg group, but also in the Euclidean case. Even if some properties are similar between the Kohn Laplacian ΔH and the classical Laplacian Δ, the similarities may be deceitful. However, in order to obtain the existence of multiple solutions by using the Ekeland variational principle, Clark critical point theorem, mountain pass theorem, and Krasnoselskii genus theorem, we need to establish some new strategies to prove the compactness condition and the mountain-pass structure.
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