Hichem Ounaies - Academia.edu (original) (raw)

Papers by Hichem Ounaies

$Research paper thumbnail of Basic results of fractional Orlicz-Sobolev space and applications to non-local problems$

arXiv (Cornell University), Jan 3, 2019

In this paper, we study the interplay between Orlicz-Sobolev spaces L M and W 1,M and fractional ... more In this paper, we study the interplay between Orlicz-Sobolev spaces L M and W 1,M and fractional Sobolev spaces W s,p. More precisely, we give some qualitative properties of the new fractional Orlicz-Sobolev space W s,M , where s ∈ (0, 1) and M is a Young function. We also study a related non-local operator, which is a fractional version of the nonhomogeneous M-Laplace operator. As an application, we prove existence of weak solution for a non-local problem involving the new fractional M −Laplacian operator.

$Research paper thumbnail of Strauss and Lions type theorems for the fractional Sobolev spaces with variable exponent and applications to nonlocal Kirchhoff-Choquard problem$

arXiv (Cornell University), May 1, 2020

This paper deals with Strauss and Lions-type theorems for fractional Sobolev spaces with variable... more This paper deals with Strauss and Lions-type theorems for fractional Sobolev spaces with variable exponent W s,p(.),p(.,.) (Ω), when p andp satisfies some conditions. As application, we study the existence of solutions for a class of Kirchhoff-Choquard problem in R N .

$Research paper thumbnail of Existence of solutions for a perturbation sublinear elliptic equation in <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow></mrow><annotation encoding="application/x-tex"></annotation></semantics></math>{\mathbb {R}^N}$$$

Nonlinear Differential Equations And Applications Nodea, May 5, 2010

In this paper we consider the following problem −Δu = u − |u| −2θ u + f u ∈ H 1 (R N) ∩ L 2(1−θ) ... more In this paper we consider the following problem −Δu = u − |u| −2θ u + f u ∈ H 1 (R N) ∩ L 2(1−θ) (R N) f ∈ L 2 (R N)∩L 2(1−θ) 1−2θ (R N), N ≥ 3, f ≥ 0, f = 0 and 0 < θ < 1 2 We prove that this problem has at least two solutions via variational methods, one of them is nonnegative. Also, we study the continuity of the nonnegative solution in the perturbation parameter f at 0.

$Research paper thumbnail of Orlicz-Sobolev versus H\"older local minimizer for nonlinear Robin problems$

arXiv (Cornell University), Dec 9, 2021

We establish a regularity results for weak solutions of Robin problems driven by the wellknown Or... more We establish a regularity results for weak solutions of Robin problems driven by the wellknown Orlicz g-Laplacian operator given by    −△ g u = f (x, u), x ∈ Ω a(|∇u|) ∂u dν + b(x)|u| p−2 u = 0, x ∈ ∂Ω, (P) where △ g u := div(a(|∇u|)∇u), Ω ⊂ R N , N ≥ 3, is a bounded domain with C 2-boundary ∂Ω, ∂u dν = ∇u.ν, ν is the unit exterior vector on ∂Ω, p > 0, b ∈ C 1,γ (∂Ω) with γ ∈ (0, 1) and inf x∈∂Ω b(x) > 0. Precisely, by using a suitable variation of the Moser iteration technique, we prove that every weak solution of problem (P) is bounded. Moreover, we combine this result with the Lieberman regularity theorem, to show that every C 1 (Ω)-local minimizer is also a W 1,G (Ω)-local minimizer for the corresponding energy functional of problem (P).

Applied Mathematics Letters, Oct 1, 2018

We establish the existence of entire compactly supported solutions for a class of Schrödinger equ... more We establish the existence of entire compactly supported solutions for a class of Schrödinger equations with competing terms and indefinite potentials. The analysis developed in this paper corresponds to the case of small perturbations of the reaction term.

arXiv (Cornell University), Jun 23, 2023

$Research paper thumbnail of Variational eigenvalues of the fractional g-Laplacian$

Complex Variables and Elliptic Equations, Feb 21, 2022

In the present work we study existence of sequences of variational eigenvalues to non-local non-s... more In the present work we study existence of sequences of variational eigenvalues to non-local non-standard growth problems ruled by the fractional g−Laplacian operator with different boundary conditions (Dirichlet, Neumann and Robin). Due to the non-homogeneous nature of the operator several drawbacks must be overcome, leading to some results that contrast with the case of power functions. 12 4.1. The Dirichlet case 13 4.2. The Neumann/Robin case 15 5. Minimax eigenvalues 16 Acknowledgements. 19 References 19 B ′ (u) = λA ′ (u), A(u) = c, has been a challenging labor whose beginning dates back to the mid-20th century (here A ′ and B ′ denote the Fréchet derivatives of the functionals). The study on Hilbert spaces was addressed by Krasnoselskij in [37]; for Banach spaces, it can

Nonlinear Analysis-theory Methods & Applications, 2003

Differential and Integral Equations, 2000

$Research paper thumbnail of Basic results of fractional Orlicz-Sobolev space and applications to non-local problems$

Topological Methods in Nonlinear Analysis, Jun 7, 2020

Proceedings, Jun 1, 2015

In this paper we are concerned with qualitative properties of entire solutions to a Schrödinger e... more In this paper we are concerned with qualitative properties of entire solutions to a Schrödinger equation with sublinear nonlinearity and sign-changing potentials. Our analysis considers three distinct cases and we establish sufficient conditions for the existence of infinitely many solutions.

$Research paper thumbnail of Least-energy nodal solutions of nonlinear equations with fractional Orlicz–Sobolev spaces$

Asymptotic Analysis, Jan 11, 2023

In this paper, we study the existence of least-energy nodal (sign-changing) weak solutions for a ... more In this paper, we study the existence of least-energy nodal (sign-changing) weak solutions for a class of fractional Orlicz equations given by (−△ g) α u + g(u) = K(x)f (u), in R N , where N ≥ 3, (−△ g) α is the fractional Orlicz g-Laplace operator, while f ∈ C 1 (R) and K is a positive and continuous function. Under a suitable conditions on f and K, we prove a compact embeddings result for weighted fractional Orlicz-Sobolev spaces. Next, by a minimization argument on Nehari manifold and a quantitative deformation lemma, we show the existence of at least one nodal (sign-changing) weak solution.

Nonlinear Analysis-theory Methods & Applications, Sep 1, 2020

In this paper, we deal with the following general elliptic equation involving the weigh p-Laplace... more In this paper, we deal with the following general elliptic equation involving the weigh p-Laplace operator : −∆pu + V (x)u = a(x)|u| q−1 u, x ∈ R N , where N ≥ 3, p ≥ 2, 0 < q, and a(x), V (x) change sign in R N. The main result of this work, establishes some qualitative properties of the solutions of the above equation. More precisely, in a first part we study the compactness of support of classical solutions. In the second part, we study the behavior and the qualitative properties of the radial classical solutions and we give a classification of these solutions.

$Research paper thumbnail of Ground state and nodal solutions for fractional Orlicz problems with lack of regularity and without the Ambrosetti-Rabinowitz condition$

arXiv (Cornell University), Apr 4, 2022

We consider a non-local Shrödinger problem driven by the fractional Orlicz g-Laplace operator as ... more We consider a non-local Shrödinger problem driven by the fractional Orlicz g-Laplace operator as follows (−△g) α u + g(u) = K(x)f (x, u), in R d , (P) where d ≥ 3, (−△g) α is the fractional Orlicz g-Laplace operator, f : R d × R → R is a measurable function and K is a positive continuous function. Employing the Nehari manifold method and without assuming the well-known Ambrosetti-Rabinowitz and differentiability conditions on the non-linear term f , we prove that the problem (P) has a ground state of fixed sign and a nodal (or sign-changing) solutions.

$Research paper thumbnail of Infinitely many solutions for a class of fractional Orlicz-Sobolev Schr\"odinger equations$

arXiv (Cornell University), Jan 5, 2019

In the present paper, we deal with a new compact embedding theorem for a subspace of the new frac... more In the present paper, we deal with a new compact embedding theorem for a subspace of the new fractional Orlicz-Sobolev spaces. We also establish some useful inequalities which yields to apply the variational methods. Using these abstract results, we study the existence of infinitely many nontrivial solutions for a class of fractional Orlicz-Sobolev Schrödinger equations whose simplest prototype is (−△) s m u + V (x)m(u)u = f (x, u), x ∈ R N , where 0 < s < 1, N ≥ 2, (−△) s m is fractional M-Laplace operator and the nonlinearity f is sublinear as |u| → ∞. The proof is based on the variant Fountain theorem established by Zou.

Journal of Mathematical Analysis and Applications, Oct 1, 2009

In this paper, we study the existence and the uniqueness of positive solution for the sublinear e... more In this paper, we study the existence and the uniqueness of positive solution for the sublinear elliptic equation, − u + u = |u| p sgn(u) + f in R N , N 3, 0 0 a.e. in R N. We show by applying a minimizing method on the Nehari manifold that this problem has a unique positive solution in H 1 (R N) ∩ L p+1 (R N). We study its continuity in the perturbation parameter f at 0.

Dans ce travail, on s'interesse a l'etude des solutions periodiques de l'equation de ... more Dans ce travail, on s'interesse a l'etude des solutions periodiques de l'equation de Hill: q+k q/q#32j q3#2aq=0. On montre que pour periode t>0 fixee, il existe aux moins deux solutions anti-t/2 periodiques de l'equation de Hill au voisinage d'une variete des solutions circulaires de l'equation de Kepler, et ceci pour des petites valeurs de. Ensuite, on relie ces solutions anti-t/2 periodiques aux solutions circulaires de l'equation de Kepler par l'application du theoreme des fonctions implicites. Ceci nous conduit a un developpement de Taylor de solutions anti-t/2 periodiques de l'equation de Hill en fonction du parametre et de la periode t. On se sert de ce developpement pour faire une experience numerique dont le but est de verifier les resultats theoriques. On termine par une etude des multiplicateurs de Floquet des solutions anti-t/2 periodiques de l'equation de Hill, en particulier on montre que les multiplicateurs de Floquet des solutions circulaires de l'equation Kepler sont tous egaux a 1

$Research paper thumbnail of Problems involving the fractional g-Laplacian with lack of compactness$

Journal of Mathematical Physics, 2023

In this paper we prove compact embedding of a subspace of the fractional Orlicz-Sobolev space W s... more In this paper we prove compact embedding of a subspace of the fractional Orlicz-Sobolev space W s,G R N consisting of radial functions, our target embedding spaces are of Orlicz type. Also, we prove a Lions and Lieb type results for W s,G R N that works together in a particular way to get a sequence whose the weak limit is non trivial. As an application, we study the existence of solutions to Quasilinear elliptic problems in the whole space R N involving the fractional g−Laplacian operator, where the conjugated function G of G doesn't satisfy the ∆ 2-condition.

$Research paper thumbnail of Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems$

arXiv (Cornell University), Sep 14, 2019

In the present paper, we deal with a new continuous and compact embedding theorems for the fracti... more In the present paper, we deal with a new continuous and compact embedding theorems for the fractional Orlicz-Sobolev spaces, also, we study the existence of infinitely many nontrivial solutions for a class of non-local fractional Orlicz-Sobolev Schrödinger equations whose simplest prototype is (−△) s m u + V (x)m(u) = f (x, u), x ∈ R d , where 0 < s < 1, d ≥ 2 and (−△) s m is the fractional M-Laplace operator. The proof is based on the variant Fountain theorem established by Zou.

$Research paper thumbnail of Problems involving the fractional <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>g-Laplacian with Lack of Compactness$