Critical Point Theorem Research Articles

Existence and multiplicity results for elliptic problems with Nonlinear Boundary Conditions and variable exponents

By applaying the Ricceri's three critical points theorem, we show the existence of at least three solutions to the following elleptic problem:\begin{equation*}\begin{gathered}-div[a(x, \nabla u)]+|u|^{p(x)-2}u=\lambda f(x,u), \quad \text{in }\Omega, \\a(x, \nabla u).\nu=\mu g(x,u), \quad \text{on } \partial\Omega,\end{gathered}\end{equation*}where $\lambda$, $\mu \in \mathbb{R}^{+},$$\Omega\subset\mathbb{R}^N(N \geq 2)$ is a bounded domain ofsmooth boundary $\partial\Omega$ and $\nu$ is the outward normalvector on $\partial\Omega$. $p: \overline{\Omega} \mapsto\mathbb{R}$, $a: \overline{\Omega}\times \mathbb{R}^{N} \mapsto\mathbb{R}^{N},$ $f: \Omega\times\mathbb{R} \mapsto \mathbb{R}$and $g:\partial\Omega\times\mathbb{R} \mapsto \mathbb{R}$ arefulfilling appropriate conditions.

Multiplicity results for elliptic problems with variable exponent and nonhomogeneous Neumann conditions

Applying three critical point theorems, we prove the existence of at least three weak solutions for a class of differential equations with p(x)‐Laplacian and subject to small perturbations of nonhomogeneous Neumann conditions. Copyright © 2014 John Wiley & Sons, Ltd.

A note on gradient-type systems on fractals

The purpose of the paper is to point out that well-established methods to study the multiplicity of solutions of nonlinear elliptic problems defined on open sets in Euclidean spaces can be also used in the case of problems defined on fractals. More exactly, using variational methods and an abstract critical-point theorem by B. Ricceri, we prove the existence of three nonzero weak solutions of certain gradient-type systems defined on a famous fractal, the Sierpinski gasket. The paper emphasizes the way we overcame the difficulties arising from the major structural differences between the highly non-smooth Sierpinski gasket and the open subsets of Euclidean spaces on which gradient-type systems are usually considered.

Three periodic solutions for a class of ordinary p-Hamiltonian systems

We study the p-Hamiltonian systems , . Three periodic solutions are obtained by using a three critical points theorem.

Multiple solutions for nonlocal system of [formula omitted]-Kirchhoff type

In this paper, we study the existence of solutions for a Neumann elliptic system with variable exponents. Under some suitable conditions and by applying an equivalent variational approach to a recent Ricceri’s three critical points theorem, we established the existence of at least three weak solutions.

Existence of solutions of a second-order impulsive differential equation

Abstract This paper is concerned with the existence of solutions of a second-order impulsive differential equation with mixed boundary condition. We obtain sufficient conditions for the existence of a unique solution, at least one solution, at least two solutions and infinitely many solutions, respectively, by using critical point theorems. The main results are also demonstrated with examples. MSC:34B15, 34B18, 34B37, 58E30.

Existence of nonzero solutions for a class of damped vibration problems with impulsive effects

In this paper, a class of damped vibration problems with impulsive effects is considered. An existence result is obtained by using the variational method and the critical point theorem due to Brezis and Nirenberg. The obtained result is also valid and new for the corresponding second-order impulsive Hamiltonian system. Finally, an example is presented to illustrate the feasibility and effectiveness of the result.

Nontrivial solutions for singular semilinear elliptic equations on the Heisenberg group

Abstract. In this article, we prove the existence of nontrivial weak solutions to the singular boundary value problem - Δ ℍ n u = μ g ( ξ ) u ( | z | 4 + t 2 ) 1 2 + λ f ( ξ , u ) $-\Delta _{{\mathbb {H}}^{n}} u= \mu \frac{g(\xi ) u}{(|z|^{4}+ t^{2} )^{\frac{1}{2} }} +\lambda f(\xi , u)$ in Ω, u = 0 $ u =0$ on ∂ Ω $\partial \Omega $ on the Heisenberg group. We employ Bonanno's three critical point theorem to obtain the existence of weak solutions.

On a class of nonlocal elliptic systems of (p,q)-Kirchhoff type

Abstract In this paper, we establish the existence of at least three solutions for the following nonlocal elliptic system of ( p , q ) -Kirchhoff type - M 1 ∫ Ω | ∇ u | p dx p - 1 Δ p u = λ F u ( x , u , v ) + μ G u ( x , u , v ) in Ω , - M 2 ∫ Ω | ∇ v | q dx q - 1 Δ q v = λ F v ( x , u , v ) + μ G v ( x , u , v ) in Ω , u = v = 0 on ∂ Ω . Our technical approach is based on the three critical points theorem of Ricceri.

On a new critical point theorem and some applications to discrete equations

Using the Fenchel-Young duality we derive a new critical point theorem. We illustrate our results with solvability for certain discrete BVP. Multiple solutions are also considered.

Three solutions to discrete anisotropic problems with two parameters

AbstractIn this note we derive a type of a three critical point theorem which we further apply to investigate the multiplicity of solutions to discrete anisotropic problems with two parameters.

EXISTENCE OF THREE SOLUTIONS FOR A QUASILINEAR ELLIPTIC EQUATION INVOLVING THE p(x)–LAPLACE OPERATOR

In this paper, some existence results are obtained by us- ing a three critical point theorem based on variational principle. In that context, we verify that a quasilinear elliptic equation involving the p(x)-Laplace operator has at least three weak solutions under Neumann boundary condition. 1. Introduction and preliminaries In the present paper, we study the existence of solutions of the p(x)- Laplacian Neumann problem ( div jruj p(x) 2 ru +juj p(x) 2 u = f(x;u) +g (x;u) in ; @u @ = 0 on @ ; (P ) where R N (2 p(x) < N) is a bounded domain with smooth boundary

Multiple Solutions for the Discretep-Laplacian Boundary Value Problems

By employing a critical point theorem, established by Bonanno, we prove the existence of three distinct solutions to boundary value problems of nonlinear difference equations with a discretep-Laplacian operator. To demonstrate the applicability of our results, we also present an example.

On Homoclinic Solutions for First-Order Superquadratic Hamiltonian Systems with Spectrum Point Zero

The existence and multiplicity of homoclinic solutions for a class of first-order periodic Hamiltonian systems with spectrum point zero are obtained. The proof is based on two critical point theorems for strongly indefinite functionals. Some recent results are improved and extended.

Periodic Solutions for Semilinear Fourth-Order Differential Inclusions via Nonsmooth Critical Point Theory

Three periodic solutions with prescribed wavelength for a class of semilinear fourth-order differential inclusions are obtained by using a nonsmooth version critical point theorem. Some results of previous related literature are extended.

A Neumann boundary value problem for a class of gradient systems

In this paper we study a class of two-point boundary value systems. Using very recent critical points theorems, we establish the existence of one non-trivial solution and infinitely many solutions of this problem, respectively.

On the superlinear Steklov problem involving the p(x)-Laplacian

This paper is concerned with the existence and multiplicity of solutions for p(x)-Laplacian Steklov problem without the well-known Ambrosetti-Rabinowitz type growth conditions. By means of critical point theorems with Cerami condition, under weaker conditions, existence and multiplicity results of the solutions are proved.

The Solvability and Numerical Simulation for the Elastic Beam Problems with Nonlinear Boundary Conditions

We study the existence of multiple solutions for a fourth-order nonlinear boundary value problem. We give some new criteria for guaranteeing that the fourth-order elastic beam equation with a perturbed term has at least three solutions. The proof is based on some three critical points theorems of B. Ricceri. Furthermore, numerical simulations are also presented.

On sequences of solutions for discrete anisotropic equations

Taking advantage of a recent critical point theorem, the existence of infinitely many solutions for an anisotropic problem with a parameter is established. More precisely, a concrete interval of positive parameters, for which the treated problem admits infinitely many solutions, is determined without symmetry assumptions on the nonlinear data. Our goal was achieved by requiring an appropriate behavior of the nonlinear terms at zero, without any additional conditions.

Variational Approach to Fourth-Order Impulsive Differential Equations with Two Control Parameters

In this paper, we are concerned with the multiplicity of solutions for a fourth-order impulsive differential equation with Dirichlet boundary conditions and two control parameters. Using variational methods and a three critical points theorem, we give some new criteria to guarantee that the impulsive problem has at least three classical solutions. We also provide an example in order to illustrate the main abstract results of this paper.