In this article, we consider a class of nonlinear Dirichlet problems driven by a Leray-Lions type... more In this article, we consider a class of nonlinear Dirichlet problems driven by a Leray-Lions type operator with variable exponent. The main result establishes an existence property by means of nonvariational arguments, that is, nonlinear monotone operator theory and approximation method. Under some natural conditions, we show that a weak limit of approximate solutions is a solution of the given quasilinear elliptic partial differential equation involving variable exponent.
In the present paper, by using variational method, the existence of non-trivial solutions to an a... more In the present paper, by using variational method, the existence of non-trivial solutions to an anisotropic discrete non-linear problem involving p(k)-Laplacian operator with Dirichlet boundary condition is investigated. The main technical tools applied here are the two local minimum theorems for differentiable functionals given by Bonanno.
In the present paper, we deal with the existence of solutions to a class of an elliptic equation ... more In the present paper, we deal with the existence of solutions to a class of an elliptic equation with Robin boundary condition. The problem is settled in Orlicz-Sobolev spaces and the main tool used is Ekeland's variational principle.
In this paper, using theory of monotone operators, we study the existence of weak solutions for a... more In this paper, using theory of monotone operators, we study the existence of weak solutions for a class of nonlocal nonvariational problems in the Orlicz-Sobolev spaces.
In this article, we prove the existence of nontrivial weak solutions for a class of discrete boun... more In this article, we prove the existence of nontrivial weak solutions for a class of discrete boundary value problems. The main tools used here are the variational principle and critical point theory.
Electronic Journal of Differential Equations, 2013
In the present paper, using the three critical points theorem and variational method, we study th... more In the present paper, using the three critical points theorem and variational method, we study the existence and multiplicity of solutions for a Dirichlet problem involving the discrete p(x)−Laplacian operator.
The present paper deals with a Kirchhoff problem under ho mogeneous Dirichlet boundary conditions... more The present paper deals with a Kirchhoff problem under ho mogeneous Dirichlet boundary conditions, set in a bounded smooth domain Ω o f ℝ^{N}. The problem studied is a stationary version of the orig inal Kirchhoff equation, involving the p(x)-Lap lacian operator, in the framework of the variable exponent Lebesgue and Sobolev spaces. The question of the existence of weak solutions is treated. Applying the Mountain Pass Theorem of A mbrosetti and Rabinowitz, the existence of a nontrivial weak solution is obtained in the variable exponent Sobolev space W₀^{1,p(x)}(Ω).
Ee introduce necessary and sufficient conditions for the weight pairs (υ, ω) that provide the bou... more Ee introduce necessary and sufficient conditions for the weight pairs (υ, ω) that provide the boundedness of the Riemann-Liouville operator R α and Weyl operator W α from L p ω (0, ∞) to L q υ (0, ∞) when 1 < p ≤ q < ∞, and 1/q < α < 1 or α > 1.
This paper deals with the existence of weak solutions for some nonlocal problem involving the p (... more This paper deals with the existence of weak solutions for some nonlocal problem involving the p (x)-Laplace operator. Using a direct variational method and the theory of the variable exponent Sobolev spaces, we set some conditions that ensures the existence of nontrivial weak solutions. Keywords p(x)-Laplace operator, p (x)-Kirchhoff-type equations, variable exponent Sobolev spaces, variational method, mountain pass theorem, Ekeland variational principle
In the present paper, using direct variational approach, the exis-tence and uniqueness of solutio... more In the present paper, using direct variational approach, the exis-tence and uniqueness of solutions for a quasilinear elliptic equation involving the p-Laplacian is obtained.
Ee introduce necessary and sufficient conditions for the weight pairs (υ, ω) that provide the bou... more Ee introduce necessary and sufficient conditions for the weight pairs (υ, ω) that provide the boundedness of the Riemann-Liouville operator R α and Weyl operator W α from L p ω (0, ∞) to L q υ (0, ∞) when 1 < p ≤ q < ∞, and 1/q < α < 1 or α > 1.
In the present paper, we deal with two different existence results of solutions for a nonlocal el... more In the present paper, we deal with two different existence results of solutions for a nonlocal elliptic Dirichlet boundary value problem involving p(x)-Laplacian. The first one is based on the Brouwer fixed point theorem and the Galerkin method which gives a priori estimate of a nontrivial weak soltion. The second one is based on the variational methods. By using Mountain-Pass theorem, we obtain at least one nontrivial weak soltion.
British Journal of Mathematics & Computer Science, 2015
ABSTRACT In the present paper, using direct variational approach, and the monotone operator metho... more ABSTRACT In the present paper, using direct variational approach, and the monotone operator method, the existence of nontrivial solutions for a quasilinear elliptic equation involving the p-Laplace operator is obtained.
The object of this paper is to study a nonlocal problem involving the p(x)-Laplacian where nonlin... more The object of this paper is to study a nonlocal problem involving the p(x)-Laplacian where nonlinearities f do not necessarily satisfy the classical conditions, such as Ambrosetti-Rabinowitz condition, but are limited by functions that do satisfy some specific conditions. By using the direct variational approach and the theory of the variable exponent Sobolev spaces, the existence and uniqueness of solutions is obtained. 2010 Mathematics Subject Classification. Primary 60J05; Secondary 60J20. Key words and phrases. p(x)-Laplacian, variational method, nonlocal problems, variable exponent Sobolev spaces.
International Journal of Partial Differential Equations, 2013
In view of variational approach we discuss a nonlocal problem, that is, a Kirchhoff-type equation... more In view of variational approach we discuss a nonlocal problem, that is, a Kirchhoff-type equation involving ( 1 ( ), 2 ( ))-Laplace operator. Establishing some suitable conditions, we prove the existence and multiplicity of solutions.
International Journal of Pure and Apllied Mathematics, 2013
This paper deals with the existence of solutions for some elliptic equations with nonstandard gro... more This paper deals with the existence of solutions for some elliptic equations with nonstandard growth under zero Dirichlet boundary condition. Using a direct variational method and the theory of the variable exponent Sobolev spaces, we set some conditions that ensures the existence of nontrivial weak solutions.
• The paper deals with the existence of solutions of a fourth order stationary Kirchhoff equation... more • The paper deals with the existence of solutions of a fourth order stationary Kirchhoff equation depending on one parameter λ. • It is possible to weaken the assumption on b in the Kirchhoff function M(t) = a + bt, and require b ≥ 0. • The techniques used are quite unusual. a b s t r a c t This paper is concerned with the existence of nontrivial solutions for a class of fourth order elliptic equations of Kirchhoff type ∆ 2 u − λ a + b Ω |∇u| 2 dx ∆u = f (x, u), in Ω, u = 0, ∆u = 0, on ∂Ω, (1) where a > 0, b ≥ 0 are constants, and λ > 0 is a parameter. We will show that there exists a λ * such that (1) has nontrivial solutions for 0 < λ < λ * by using the mountain pass techniques and the truncation method.
In the present paper, in view of the variational approach, we discuss a Ni-Serrin type equation i... more In the present paper, in view of the variational approach, we discuss a Ni-Serrin type equation involving non-standard growth condition and arising from the capillarity phenomena. Establishing some suitable conditions, we prove the existence and multiplicity of solutions. MSC: 35D05; 35J60; 35J70
In the present paper, we show the existence of ground state solution of a discrete p(n)-Laplacian... more In the present paper, we show the existence of ground state solution of a discrete p(n)-Laplacian type equation involving unbounded potential by using the Mountain-Pass theorem and Nehari manifold.
In this article, we consider a class of nonlinear Dirichlet problems driven by a Leray-Lions type... more In this article, we consider a class of nonlinear Dirichlet problems driven by a Leray-Lions type operator with variable exponent. The main result establishes an existence property by means of nonvariational arguments, that is, nonlinear monotone operator theory and approximation method. Under some natural conditions, we show that a weak limit of approximate solutions is a solution of the given quasilinear elliptic partial differential equation involving variable exponent.
In the present paper, by using variational method, the existence of non-trivial solutions to an a... more In the present paper, by using variational method, the existence of non-trivial solutions to an anisotropic discrete non-linear problem involving p(k)-Laplacian operator with Dirichlet boundary condition is investigated. The main technical tools applied here are the two local minimum theorems for differentiable functionals given by Bonanno.
In the present paper, we deal with the existence of solutions to a class of an elliptic equation ... more In the present paper, we deal with the existence of solutions to a class of an elliptic equation with Robin boundary condition. The problem is settled in Orlicz-Sobolev spaces and the main tool used is Ekeland's variational principle.
In this paper, using theory of monotone operators, we study the existence of weak solutions for a... more In this paper, using theory of monotone operators, we study the existence of weak solutions for a class of nonlocal nonvariational problems in the Orlicz-Sobolev spaces.
In this article, we prove the existence of nontrivial weak solutions for a class of discrete boun... more In this article, we prove the existence of nontrivial weak solutions for a class of discrete boundary value problems. The main tools used here are the variational principle and critical point theory.
Electronic Journal of Differential Equations, 2013
In the present paper, using the three critical points theorem and variational method, we study th... more In the present paper, using the three critical points theorem and variational method, we study the existence and multiplicity of solutions for a Dirichlet problem involving the discrete p(x)−Laplacian operator.
The present paper deals with a Kirchhoff problem under ho mogeneous Dirichlet boundary conditions... more The present paper deals with a Kirchhoff problem under ho mogeneous Dirichlet boundary conditions, set in a bounded smooth domain Ω o f ℝ^{N}. The problem studied is a stationary version of the orig inal Kirchhoff equation, involving the p(x)-Lap lacian operator, in the framework of the variable exponent Lebesgue and Sobolev spaces. The question of the existence of weak solutions is treated. Applying the Mountain Pass Theorem of A mbrosetti and Rabinowitz, the existence of a nontrivial weak solution is obtained in the variable exponent Sobolev space W₀^{1,p(x)}(Ω).
Ee introduce necessary and sufficient conditions for the weight pairs (υ, ω) that provide the bou... more Ee introduce necessary and sufficient conditions for the weight pairs (υ, ω) that provide the boundedness of the Riemann-Liouville operator R α and Weyl operator W α from L p ω (0, ∞) to L q υ (0, ∞) when 1 < p ≤ q < ∞, and 1/q < α < 1 or α > 1.
This paper deals with the existence of weak solutions for some nonlocal problem involving the p (... more This paper deals with the existence of weak solutions for some nonlocal problem involving the p (x)-Laplace operator. Using a direct variational method and the theory of the variable exponent Sobolev spaces, we set some conditions that ensures the existence of nontrivial weak solutions. Keywords p(x)-Laplace operator, p (x)-Kirchhoff-type equations, variable exponent Sobolev spaces, variational method, mountain pass theorem, Ekeland variational principle
In the present paper, using direct variational approach, the exis-tence and uniqueness of solutio... more In the present paper, using direct variational approach, the exis-tence and uniqueness of solutions for a quasilinear elliptic equation involving the p-Laplacian is obtained.
Ee introduce necessary and sufficient conditions for the weight pairs (υ, ω) that provide the bou... more Ee introduce necessary and sufficient conditions for the weight pairs (υ, ω) that provide the boundedness of the Riemann-Liouville operator R α and Weyl operator W α from L p ω (0, ∞) to L q υ (0, ∞) when 1 < p ≤ q < ∞, and 1/q < α < 1 or α > 1.
In the present paper, we deal with two different existence results of solutions for a nonlocal el... more In the present paper, we deal with two different existence results of solutions for a nonlocal elliptic Dirichlet boundary value problem involving p(x)-Laplacian. The first one is based on the Brouwer fixed point theorem and the Galerkin method which gives a priori estimate of a nontrivial weak soltion. The second one is based on the variational methods. By using Mountain-Pass theorem, we obtain at least one nontrivial weak soltion.
British Journal of Mathematics & Computer Science, 2015
ABSTRACT In the present paper, using direct variational approach, and the monotone operator metho... more ABSTRACT In the present paper, using direct variational approach, and the monotone operator method, the existence of nontrivial solutions for a quasilinear elliptic equation involving the p-Laplace operator is obtained.
The object of this paper is to study a nonlocal problem involving the p(x)-Laplacian where nonlin... more The object of this paper is to study a nonlocal problem involving the p(x)-Laplacian where nonlinearities f do not necessarily satisfy the classical conditions, such as Ambrosetti-Rabinowitz condition, but are limited by functions that do satisfy some specific conditions. By using the direct variational approach and the theory of the variable exponent Sobolev spaces, the existence and uniqueness of solutions is obtained. 2010 Mathematics Subject Classification. Primary 60J05; Secondary 60J20. Key words and phrases. p(x)-Laplacian, variational method, nonlocal problems, variable exponent Sobolev spaces.
International Journal of Partial Differential Equations, 2013
In view of variational approach we discuss a nonlocal problem, that is, a Kirchhoff-type equation... more In view of variational approach we discuss a nonlocal problem, that is, a Kirchhoff-type equation involving ( 1 ( ), 2 ( ))-Laplace operator. Establishing some suitable conditions, we prove the existence and multiplicity of solutions.
International Journal of Pure and Apllied Mathematics, 2013
This paper deals with the existence of solutions for some elliptic equations with nonstandard gro... more This paper deals with the existence of solutions for some elliptic equations with nonstandard growth under zero Dirichlet boundary condition. Using a direct variational method and the theory of the variable exponent Sobolev spaces, we set some conditions that ensures the existence of nontrivial weak solutions.
• The paper deals with the existence of solutions of a fourth order stationary Kirchhoff equation... more • The paper deals with the existence of solutions of a fourth order stationary Kirchhoff equation depending on one parameter λ. • It is possible to weaken the assumption on b in the Kirchhoff function M(t) = a + bt, and require b ≥ 0. • The techniques used are quite unusual. a b s t r a c t This paper is concerned with the existence of nontrivial solutions for a class of fourth order elliptic equations of Kirchhoff type ∆ 2 u − λ a + b Ω |∇u| 2 dx ∆u = f (x, u), in Ω, u = 0, ∆u = 0, on ∂Ω, (1) where a > 0, b ≥ 0 are constants, and λ > 0 is a parameter. We will show that there exists a λ * such that (1) has nontrivial solutions for 0 < λ < λ * by using the mountain pass techniques and the truncation method.
In the present paper, in view of the variational approach, we discuss a Ni-Serrin type equation i... more In the present paper, in view of the variational approach, we discuss a Ni-Serrin type equation involving non-standard growth condition and arising from the capillarity phenomena. Establishing some suitable conditions, we prove the existence and multiplicity of solutions. MSC: 35D05; 35J60; 35J70
In the present paper, we show the existence of ground state solution of a discrete p(n)-Laplacian... more In the present paper, we show the existence of ground state solution of a discrete p(n)-Laplacian type equation involving unbounded potential by using the Mountain-Pass theorem and Nehari manifold.
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