Distinct Weak Solutions Research Articles

Three weak solutions for a class of fourth order p(x)-Kirchhoff type problem with Leray-Lions operators

In this work, we study the multiplicity of a weak solution for a fourth order p(x)-Kirchhoff type problem involving the Leray-Lions type operators with no flux boundary condition. By using variational approach and critical point theory, we determine an open interval of parameters for which our problem admits at least three distinct weak solutions.

Solvability for the ψ-Caputo-Type Fractional Differential System with the Generalized p-Laplacian Operator

In this article, by combining a recent critical point theorem and several theories of the ψ-Caputo fractional operator, the multiplicity results of at least three distinct weak solutions are obtained for a new ψ-Caputo-type fractional differential system including the generalized p-Laplacian operator. It is noted that the nonlinear functions do not need to adapt certain asymptotic conditions in the paper, but, instead, are replaced by some simple algebraic conditions. Moreover, an evaluation criterion of the equation without solutions is also provided. Finally, two examples are given to demonstrate that the ψ-Caputo fractional operator is more accurate and can adapt to deal with complex system modeling problems by changing different weight functions.

Existence of triple solutions for elliptic equations driven by p-Laplacian-like operators with Hardy potential under Dirichlet–Neumann boundary conditions

In this article, we focus on triple weak solutions for some p-Laplacian-type elliptic equations with Hardy potential, two parameters, and mixed boundary conditions. We show the existence of at least three distinct weak solutions by using variational methods, the Hardy inequality, and the Bonanno–Marano-type three critical points theorem under suitable assumptions, and the existence of solutions to some particular cases of this type of elliptic equations are also obtained.

On the multiplicity of solutions for a kind of fourth-order equation depending on two real parameters

In this paper, by suitable assumptions on nonlinear boundary term, we establish the existence of three distinct weak solutions for a kind of fourth-order boundary value problem depending on two parameters.

Multiplicity Solutions of Fractional Impulsive p -Laplacian Systems: New Result

In this paper, the existence of multiplicity distinct weak solutions is proved for differentiable functionals for perturbed systems of impulsive nonlinear fractional differential equations. Further, examples are given to show the feasibility and efficacy of the key findings. This work is an extension of the previous works to Banach space.

To study existence of at least three weak solutions to a system of over-determined Fredholm fractional integro-differential equations

In this article, the weak solutions of a class of nonlinear system of fractional boundary value problems including integral equations is investigated. Each equation in the system is either a nonlinear fractional partial integro-differential equation containing a gradient of a nonlinear source term or a Fredholm integral equation of the second kind. It is proved that there exist at least three distinct weak solutions by applying the critical point theory and the variational structure. In order to achieve this purpose, we use the well-known theorem on the construction of the critical set of functionals with a weak compactness condition. Furthermore, our main result is demonstrated by an illustrative example to show its legitimacy and applicability.

Triple solutions for elliptic Dirichlet problems with a parameter

In this article, we focus on triple weak solutions for nonlinear elliptic problems with Dirichlet boundary condition. We show the existence of at least three distinct weak solutions by using variational methods and a recent obtained three critical points theorem under suitable assumptions.

Multiple solutions for nonlocal elliptic problems driven by $ p(x) $-biharmonic operator

<abstract> In this article, we study the existence of at least three distinct weak solutions for nonlocal elliptic problems involving $ p(x) $-biharmonic operator. The results are obtained by means of variational methods. We also provide an example in order to illustrate our main abstract results. We extend and improve some recent results. </abstract>

Existence of at Least Three Distinct Weak Solutions for a Class of Nonlinear System of Fractional Differential Equations

In this article, it is concerned existence of solutions to a class of nonlinear system of fractional differential equations. It is proved the existence of at least three different weak solutions through the critical point theory and the variational method. In this way, it is applied well-known theorem proved by Bonanno and Marano on the construction of the critical set of functionals with a weak compactness condition. Furthermore, the main result is demonstrated by some examples to show its validity and efficiency.

Weak Solutions to a System of Nonlinear Fractional Boundary Value Problems Via Variational Form

In this paper, the weak solutions to a class of nonlinear system of fractional boundary value problems are studied. Each equation of the system does have two kinds of nonlinear terms; one of them is in terms of gradient, and the other one is a nonlinear source term with respect to dependent variable. It is proved that there exists at least three distinct weak solutions by using the critical point theory and the variational method. To this aim, we apply the well-known theorem on the construction of the critical set of functionals with a weak compactness condition. Moreover, the main result is demonstrated by some examples to show its validity.

Perturbed fourth-order Kirchho-type problems

We establish the existence of at least three distinct weak solutions for a perturbed nonlocal fourth-order Kirchhoff-type problem with Navier boundary conditions under appropriate hypotheses on nonlinear terms. Our main tools are based on variational methods and some critical points theorems. We give some examples to illustrate the obtained results.

A variational approach to perturbed impulsive fractional differential equations

In this paper, perturbed systems of impulsive nonlinear fractional differential equations, including Lipschitz continuous nonlinear terms, are studied. The existence of at least three distinct weak solutions is obtained based on a recent three critical points theorem for differentiable functionals. In addition, examples are presented to illustrate the feasibility and effectiveness of the main results.

Local discontinuous Galerkin schemes for a nonlinear variational wave equation modelling liquid crystals

We consider a nonlinear variational wave equation that models the dynamics of nematic liquid crystals. Discontinuous Galerkin schemes that either conserve or dissipate a discrete version of the energy associated with these equations are designed. Numerical experiments illustrating the stability and efficiency of the schemes are presented. An interesting feature of these schemes is their ability to approximate two distinct weak solutions of the underlying system.

Multiplicity results for nonlinear Neumann boundary value problems involving p-Laplace type operators

We consider the existence of at least two or three distinct weak solutions for the nonlinear elliptic equations $$ \textstyle\begin{cases} {-}\operatorname{div}(\varphi(x,\nabla u))+{|u|}^{p-2}u= \lambda f(x,u) &\mbox{in } \Omega, \varphi(x,\nabla u) \frac{\partial u}{\partial n}= \lambda g(x,u) & \mbox{on }\partial\Omega. \end{cases} $$ Here the function $\varphi(x,v)$ is of type $|v|^{p-2}v$ and the functions f, g satisfy a Caratheodory condition. To do this, we give some critical point theorems for continuously differentiable functions with the Cerami condition which are extensions of the recent results in Bonanno (Adv. Nonlinear Anal. 1:205-220, 2012) and Bonanno and Marano (Appl. Anal. 89:1-10, 2010) by applying Zhong’s Ekeland variational principle.

Existence of two weak solutions for some singular elliptic problems

In this paper, we establish the existence of at least two distinct weak solutions for some singular elliptic problems involving a p-Laplace operator, subject to Dirichlet boundary conditions in a smooth bounded domain in \(\mathbb {R}^N.\) A critical point result for differentiable functionals is exploited, in order to prove that the problem admits at least two distinct non-trivial weak solutions.

Multiple solutions for a coupled system of nonlinear fractional differential equations via variational methods

In this paper, a coupled system of nonlinear fractional differential equations is considered. The existence of at least three distinct weak solutions is obtained by means of the variational methods and the critical point theory due to Bonanno and Marano. In addition, an example is presented to illustrate the feasibility and effectiveness of the main results.

Multiple solutions to singular fourth order elliptic equations on compact manifolds

Using the method of Nehari manifolds, we prove the existence of at least two distinct weak solutions to elliptic equation of four order with singularities and with critical Sobolev growth.

Multiple solutions for a perturbed fourth-order Kirchhoff type elliptic problem

The existence of three distinct weak solutions for a perturbed fourth-order Kirchhoff-type elliptic problem is investigated. Our approach is based on variational methods.

Multiple solutions for a perturbed mixed boundary value problem involving the one-dimensionalp-Laplacian

The existence of three distinct weak solutions for a perturbed mixed boundary value problem involving the one-dimensional p-Laplacian operator is established under suitable assumptions on the nonlinear term. Our approach is based on recent variational methods for smooth functionals defined on reflexive Banach spaces.

Multiple solutions for [formula omitted]-harmonic type equations

In this paper, we establish the existence of multiple solutions for an equation involving a p -harmonic operator, subject to Dirichlet boundary conditions in a bounded smooth open domain of R N , while the nonlinearity has a ( p − 1 ) -sublinear growth at infinity. Using a three critical points theorem, we prove the existence of at least three distinct weak solutions in W 0 2 , p ( Ω ) to this problem.