Critical Point Theorem Research Articles

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.

Homoclinic orbits for damped vibration systems with asymptotically quadratic or subquadratic potentials

In this paper, we study a class of damped vibration systems, $$ \ddot{u}(t)+B\dot{u}(t)-L(t)u(t)+\nabla W\bigl(t,u(t)\bigr)=0, \quad \forall t \in \mathbb{R}, $$ where $W(t,u)$ is of indefinite sign. By using a critical point theorem of Ding, we establish a new criterion to guarantee that the above system has infinitely many nontrivial homoclinic orbits under the assumption that $W(t,u)$ is asymptotically quadratic or subquadratic as $|u|\rightarrow\infty$ . Recent results in the literature are generalized and significantly improved.

MULTIPLICITY RESULTS FOR NONLINEAR SCHRÖDINGER-POISSON SYSTEMS WITH SUBCRITICAL OR CRITICAL GROWTH

In this paper, we consider the following Schrodinger-Poisson system: 8 < : −△u + u + ��u = µf(u) + |u|p−2u, in , −△� = u2, in , � = u = 0, on @, where is a smooth and bounded domain in R 3 , p ∈ (1,6), �,µ are two parameters and f : R → R is a continuous function. Using some critical point theorems and truncation technique, we obtain three multiplicity results for such a problem with subcritical or critical growth.

A Schechter type critical point result in annular conical domains of a Banach space and applications

Using Ekeland's variational principle we obtain a critical point theorem of Schechter type for extrema of a functional in an annular conical domain of a Banach space. The result can be seen as a variational analogue of Krasnoselskii's fixed point theorem in cones and can be applied for the existence, localization and multiplicity of the positive solutions of variational problems. The result is then applied to $p$-Laplace equations, where the geometric condition on the boundary of the annular conical domain is established via a weak Harnack type inequality given in terms of the energetic norm. This method can be applied also to other homogeneous operators in order to obtain existence, multiplicity or infinitely many solutions for certain classes of quasilinear equations.

Variational approaches to impulsive elastic beam equations of Kirchhoff type

In this paper, we study the existence of multiple solutions for impulsive fourth-order differential equations of Kirchhoff type. Using a variational method and some critical points theorems, we obtain some new criteria for guaranteeing that impulsive fourth-order differential equations of Kirchhoff type have three and infinitely many solutions. Some recent results are extended and improved. Some examples are presented to demonstrate the applications of our main results.

(N,q)-Laplacian problems with critical Trudinger–Moser nonlinearities

We obtain nontrivial solutions of a $(N,q)$-Laplacian problem with a critical Trudinger-Moser nonlinearity in a bounded domain. In addition to the usual difficulty of the loss of compactness associated with problems involving critical nonlinearities, this problem lacks a direct sum decomposition suitable for applying the classical linking theorem. We show that every Palais-Smale sequence at a level below a certain energy threshold admits a subsequence that converges weakly to a nontrivial critical point of the variational functional. Then we prove an abstract critical point theorem based on a cohomological index and use it to construct a minimax level below this threshold.

Existence and multiplicity of solutions for nonlocal systems involving fractional Laplacian with non-differentiable terms

In this paper, we devote to the study of the existence and multiplicity of solutions of nonlocal systems involving fractional Laplacian with non-differentiable terms using some extended critical point theorems for locally Lipschitz function on product spaces.

ON A NEW FIXED POINT THEOREM ON HILBERT SPACES VIA THE MOUNTAIN PASS LEMMA AND APPLICATIONS TO ELLIPTIC BOUNDARY VALUE PROBLEMS

In this paper we present a new fixed point theorem in Hilbert spaces. Using a suitable critical point theorem we provide conditions under which potential operators have a fixed point. An application is given to illustrate the theory.

On the Existence and Multiplicity of Solutions for Dirichlet’s problem for Fractional Differential equations

Abstract In this paper, by using variational methods and critical point theorems, we prove the existence and multiplicity of solutions for boundary value problem for fractional order differential equations where Riemann-Liouville fractional derivatives and Caputo fractional derivatives are used. Our results extend the second order boundary value problem to the non integer case. Moreover, some conditions to determinate nonnegative solutions are presented and examples are given to illustrate our results.

Existence and Multiplicity of Solutions for Some Damped Dirichlet Nonlinear Impulsive Differential Equations

In this paper, we consider the existence and multiplicity of solutions for some damped Dirichlet nonlinear impulsive differential equations. By using the critical point theorem we give some new criteria to guarantee that the impulsive differential equation has at leat one solution or infinitely many solutions under the assumptions that the nonlinear term satisfies sublinear, superlinear or asymptotically linear. Some recent results are extended and improved.

Nonsmooth generalization of some critical point theorems for $\bm{C^2}$ functionals

In this paper, we combine the authors recent results on splitting theorems, with metric critical point theory and critical point theory for locally Lipschitz functionals, to generalize some critical point theorems for $C^2$ functionals obtained via Morse theoretic methods to a class of both Frechet differentiable and continuously directional differential functionals. As a key, if a functional on an open subset of a Banach space is Frechet differentiable, and strictly Hadamard differentiable (which is stronger than locally Lipschitz continuity, but weaker than continuously directional differentiability), we observe that it has the same critical set as a continuous functional on a metric space, a Frechet differentiable one, and a locally Lipschitz continuous one.

Existence of Infinitely Many Periodic Solutions for Perturbed Semilinear Fourth-Order Impulsive Differential Inclusions

This paper discusses the existence of infinitely many periodic solutions for a semilinear fourth-order impulsive differential inclusion with a perturbed nonlinearity and two parameters. The approach is based on a critical point theorem for nonsmooth functionals.

Solutions for Impulsive Fractional Differential Equations via Variational Methods

We investigate the boundary value problems of impulsive fractional order differential equations. First, we obtain the existence of at least one solution by the minimization result of Mawhin and Willem. Then by the variational methods and a very recent critical points theorem of Bonanno and Marano, the existence results of at least triple solutions are established. At last, two examples are offered to demonstrate the application of our main results.

Nonsmooth critical point theorems and its applications to quasilinear schrödinger equations

In this paper, the existence and nonexistence of solutions to a class of quasilinear elliptic equations with nonsmooth functionals are discussed, and the results obtained are applied to quasilinear Schrödinger equations with negative parameter which arose from the study of self-channeling of high-power ultrashort laser in matter.

Bifurcation and multiplicity results for critical p-Laplacian problems

We prove a bifurcation and multiplicity result that is independent of the dimension N for a critical p-Laplacian problem that is the analog of the Brezis-Nirenberg problem for the quasilinear case. This extends a result in the literature for the semilinear case p = 2 to all p in (1;infty). In particular, it gives a new existence result when N \le p^2. When p \neq 2 the nonlinear operator -\Delta_p 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 pseudoindex related to the Z^2-cohomological index that is applicable here.

Three solutions for second-order impulsive differential inclusions with Sturm-Liouville boundary conditions via nonsmooth critical point theory

A second-order impulsive differential inclusion with Sturm-Liouville boundary conditions is studied. By using a nonsmooth version of a three critical point theorem of Ricceri, the existence of three solutions is obtained.

A three critical point theorem for non-smooth functionals with application in differential inclusions

A variety of three-critical-point theorems have been established for non-smooth functionals, based on a minimax inequality. In this paper, a generalized form of a recent result due to Ricceri is introduced for non-smooth functionals and by a few hypotheses, without any minimax inequality, the existence of at least three critical points with a uniform bound on the norms of solutions, is obtained. Also, as an application, our main theorem is used to obtain at least three anti-periodic solutions for a second order differential inclusion.

A coincidence point theorem for sequentially continuous mappings

The aim of this paper is to present a coincidence point theorem for sequentially weakly continuous maps. Moreover, as a consequence, a critical point theorem for functionals possibly containing a nonsmooth part is obtained. Finally, as an application, existence results for nonlinear differential problems depending also on the derivative of the solution are established.

On new critical point theorems without the Palais–Smale condition

In this paper we prove new theorems on critical point theory based on the weak Ekeland's variational principle.

On a class of critical (p, q)-Laplacian problems

We obtain nontrivial solutions of a critical (p, q)-Laplacian problem in a bounded domain. In addition to the usual difficulty of the loss of compactness associated with problems involving critical Sobolev exponents, this problem lacks a direct sum decomposition suitable for applying the classical linking theorem. We show that every Palais–Smale sequence at a level below a certain energy threshold admits a subsequence that converges weakly to a nontrivial critical point of the variational functional. Then we prove an abstract critical point theorem based on a cohomological index and use it to construct a minimax level below this threshold.