Critical Point Theorem Research Articles

ON A NEW MULTIPLE CRITICAL POINT THEOREM AND SOME APPLICATIONS TO ANISOTROPIC PROBLEMS

Using the Fenchel-Young duality and mountain pass geometry we derive a new multiple critical point theorem. In a finite dimensional setting it becomes three critical point theorem while in an infinite dimensional case we obtain the existence of at least two critical points. The applications to anisotropic problems show that one can obtain easily that all critical points are nontrivial.

Bifurcation and multiplicity results for critical fractional p‐Laplacian problems

We prove a bifurcation and multiplicity result for a critical fractional p‐Laplacian problem that is the analog of the Brézis‐Nirenberg problem for the nonlocal quasilinear case. This extends a result in the literature for the semilinear case to all , in particular, it gives a new existence result. When , the nonlinear operator , has no linear eigenspaces, so our extension is nontrivial and requires a new abstract critical point theorem that is not based on linear subspaces. We prove a new abstract result based on a pseudo‐index related to the ‐cohomological index that is applicable here.

Multiple non semi-trivial solutions of systems of Kirchhoff-type equations with discontinuous nonlinearities inRN

In the present paper, we study the problem of multiple non semi-trivial solutions for the following systems of Kirchhoff-type equations with discontinuous nonlinearities (1.1) where F∈C1(RN×R+×R+,R),V∈C(RN,R), and By establishing a new index theory, we obtain some multiple critical point theorems on product spaces, and as applications, three multiplicity results of non semi-trivial solutions for (1.1) are obtained. Copyright © 2015 John Wiley & Sons, Ltd.

Homoclinic orbits and critical points of barrier functions

We interpret the close link between the critical points of Mather's barrier functions and minimal homoclinic orbits with respect to the Aubry sets on . We also prove a critical point theorem for barrier functions and the existence of such homoclinic orbits on as an application.

Periodic solutions for nonautonomous first order delay differential systems via Hamiltonian systems

The existence of the nontrivial periodic solutions for the nonautonomous first order delay differential equation $x'(t)=-[f(t, x(t-1))+f(t, x(t-2))+\cdots+f(t, x(t-(2N-1)))]$ is investigated, where $f\in C({\mathbf{R}}\times{\mathbf{R}}, {\mathbf{R}})$ is 2N-periodic in t and odd in x, N is a positive integer. We prove several new existence results by some recent critical point theorems.

Existence of solutions for a class of second‐order differential equations with impulsive effects

In this paper, by using the variational method and the critical point theorem because of Brezis and Nirenberg, we investigate the existence of solutions to a class of second‐order impulsive Hamiltonian system. Copyright © 2015 John Wiley & Sons, Ltd.

Infinitely many solutions for fractional differential system via variational method

In this paper we investigate a boundary value problem for a coupled nonlinear differential system of fractional order. Under appropriate hypotheses and by applying the critical point theorem, we obtain some new criteria to guarantee that the fractional differential system has infinitely many weak solutions. In addition, an example is given to illustrate the main results.

Three solutions to impulsive differential equations involving p-Laplacian

This paper is concerned with the existence of three solutions of Neumann boundary value problems for impulsive differential equations depending on a parameter λ. We find the range of the control parameter in which the system admits at least three solutions by using an existing three critical points theorem. An example is also given to illustrate our results.

Existence of positive ground state solutions for Kirchhoff type problems

In this paper, we study the existence of positive ground state solutions for the nonlinear Kirchhoff type problem {−(a+b∫R3|∇u|2)△u+V(x)u=f(u)in R3,u∈H1(R3),u>0in R3, where a,b>0 are constants, f∈C(R,R) is subcritical near infinity and superlinear near zero and satisfies the Berestycki–Lions condition. By using an abstract critical point theorem established by Jeanjean and a new global compactness lemma, we show that the above problem has at least a positive ground state solution. Our result generalizes the results of Li and Ye (2014) concerning the nonlinearity f(u)=|u|p−1u with p∈(2,5).

ON EXISTENCE OF THREE SOLUTIONS FOR $p(x)$-KIRCHHOFF TYPE DIFFERENTIAL INCLUSION PROBLEM VIA NONSMOOTH CRITICAL POINT THEORY

In this paper, we study a class of differential inclusion problems driven by the $p(x)$-Kirchhoff with non-standard growth depending on a real parameter. Working within the framework of variable exponent spaces, a new existence result of at least three solutions for the considered problem is established by using the nonsmooth version three critical points theorem.

Multiple solutions for discrete periodic nonlinear Schrödinger equations

In this paper, we obtain infinitely many geometrically distinct solutions with exponential decay at infinity of the discrete periodic nonlinear Schrödinger equation Lun − ωun = ϱgn(un), n ∈ ℤ, where ω belongs to a spectral gap of the linear operator L, ϱ = ± 1, and the potential gn(s) is symmetric in s, asymptotically or super linear with more general hypotheses as s→∞ for all n ∈ ℤ. Our arguments are based on some abstract critical point theorems about strongly indefinite functional developed recently.

Discrete boundary value problem based on the fractional Gâteaux derivative

This article aims to organize positive solutions to discrete fractional boundary value problems for continuous Gâteaux differentiable functions. A generalized Gâteaux derivative is introduced using a fractional discrete operator for a Jumarie fractional operator. The method of finding solutions is based on critical point theorems of finite dimensional Banach spaces.

Multiple solutions to elliptic inclusions via critical point theory on closed convex sets

The existence of multiple solutions $u\in H^1_0(\Omega)$ to a differential inclusion of the type $-\Delta u\in \partial J(x,u)$ in $\Omega$, where $\partial J(x,\cdot)$ denotes the generalized sub-differential of $J(x,\cdot)$, is investigated through critical point theorems for locally Lipschitz continuous functionals on closed convex sets of a Hilbert space.

Three Nontrivial Solutions for Kirchhoff-Type Variational–Hemivariational Inequalities

We establish the existence of three distinct nontrivial solutions for a nonlocal perturbed Kirchhoff-type variational–hemivariational inequalities. The proof is based on three critical point theorems for non-smooth functional due to Bonanno and Winkert.

New results for perturbed second-order impulsive differential equation on the half-line

Abstract By using a variational method and some critical points theorems, we establish some results on the multiplicity of solutions for second-order impulsive differential equation depending on two real parameters on the half-line. In addition, two examples to illustrate our results are given.

A Brezis–Nirenberg splitting approach for nonlocal fractional equations

In this paper we consider problems modeled by the following nonlocal fractional equation {(−Δ)su+a(x)u=μf(u)in Ωu=0in Rn∖Ω, where s∈(0,1) is fixed, Ω is an open bounded subset of Rn, n>2s, with Lipschitz boundary, (−Δ)s is the fractional Laplace operator and μ is a real parameter.Under two different types of conditions on the functions a and f, by using a famous critical point theorem in the presence of splitting established by Brezis and Nirenberg, we obtain the existence of at least two nontrivial weak solutions for our problem.

SIGN-CHANGING SOLUTIONS FOR A CLASS OF DAMPED VIBRATION PROBLEMS WITH IMPULSIVE EFFECTS

In this paper, some sufficient conditions are obtained for theexistence and multiplicity of sign-changing solutions for the dampedvibration problem with impulsive effects \begin{eqnarray*} \left\{ \begin{array}{ll} -u''(t)+g(t)u'(t)=f(t,u(t)), & \hbox{a.e. $t\in [0,T]$;} u(0)=u(T)=0, & \hbox{} \Delta u'(t_{j})=u'(t_{j}^{+})-u'(t_{j}^{-})=I_{j}(u(t_{j})), & \hbox{$j=1,2,\ldots,p,$} \end{array} \right. \end{eqnarray*} where $t_{0}=0<t_{1}<t_{2}<\ldots<t_{p}<t_{p+1}=T,g\in L^{1}(0,T;\mathbb{R}),I_{j}:\mathbb{R}\rightarrow\mathbb{R},j=1,2,\ldots,p$ are continuous, $f:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carath e odory function with subcritical growth condition: $(A) |f(t, u)| ≤ C(1 + |u|^{s-1}), \forall t \in[0,T], u\in \mathbb{R}, s\in [2,+\infty)$. The sign-changing solutions are sought by means of some sign-changing critical point theorems and two examples are presented to illustrate the feasibility and effectiveness of our results.

Existence of Multiple Solutions for a Quasilinear Biharmonic Equation.

Using three critical points theorems, we prove the existence of at least three solutions for a quasilinear biharmonic equation.

Minimax solutions for a problem with sign changing nonlinearity and lack of strict convexity

A result of existence of a nonnegative and a nontrivial solution is proved via critical point theorems for non smooth functionals. The equation considered presents a convex part and a nonlinearity which changes sign.

A nonlinear boundary value problem for fourth-order elastic beam equations

By using an infinitely many critical points theorem, we study the existence of infinitely many solutions for a fourth-order nonlinear boundary value problem, depending on two real parameters. No symmetric condition on the nonlinear term is assumed. Some recent results are improved and extended.