Critical Point Theorem Research Articles

Existence of non-trivial solutions for fractional Schrödinger-Poisson systems with subcritical growth

In this paper, we are concerned with the following fractional Schrodinger-Poisson system:    (−∆s)u + u + λφu = µf(u) +|u|p−2|u|, x ∈R3 (−∆t)φ = u2, x ∈R3 where λ,µ are two parameters, s,t ∈ (0,1] ,2t + 4s > 3 ,1 < p ≤ 2∗ s and f : R → R is continuous function. Using some critical point theorems and truncation technique, we obtain the existence and multiplicity of non-trivial solutions with the help of the variational methods.

Multiple solutions for a Choquard system with periodic potential

In this paper we prove the existence of infinitely many geometrically distinct solutions of a Choquard system with periodic potential. For this purpose, we prove a new global critical point theorem without the τ-upper semi-continuity condition on the variational functional.

Existence and multiplicity of solutions for fractional p(x,.)-Kirchhoff-type problems in ℝ N

The purpose of this paper is to investigate the existence and multiplicity of solutions for the following class of fractional -Kirchhoff-type problems in We first establish the compact embedding of the space into . Using this embedding result and some critical-points theorems, we obtain the existence and multiplicity results.

Three solutions for a nonlocal fractional p-Kirchhoff type elliptic system

In this paper, we study the existence of three weak solutions for a Kirchhoff-type elliptic system involving nonlocal fractional p-Laplacian with homogeneous Dirichlet boundary conditions. The approach is based on the three critical points theorem and some variational methods.

Three solutions for a Kirchhoff-type problem involving nonlocal fractional $p$-Laplacian

‎‎Using the three critical points theorem‎, ‎we obtain the existence of three weak solutions for a Kirchhoff-type problem involving the nonlocal fractional $p$-Laplacian operator in a fractional Sobolev space‎, ‎with homogeneous Dirichlet boundary conditions‎.

Multiple brake orbits of even Hamiltonian systems on torus

In this paper, we establish a relationship between the Morse index at rest points in the saddle point reduction and the brake-orbit-type Maslov index at corresponding brake orbits. As an application, we give a criterion to find brake orbits which are contractible and start at $$\left\{0 \right\} \times \mathbb{T}{^n} \subset {\mathbb{T}^{2n}}$$ for even Hamiltonian on $${\mathbb{T}^{2n}}$$ by the methods of the Maslov-index theory and a critical point theorem formulated by Bartsch and Wang (1997). Explicitly, if all trivial solutions of a Hamiltonian are nondegenerate in the brake orbit boundary case, there are at least max{iL0 (z0)} pairs of nontrivial 1-periodic brake orbits if iL0 (z0) > 0 or at least max{−iL0 (z0) − n} pairs of nontrivial 1-periodic brake orbits if iL0 (z0) < −n. In the end, we give an example to find brake orbits for certain Hamiltonian via this criterion.

Triple solutions for a damped impulsive differential equation

In this paper, we aim to obtain the existence of at least three classical solutions for nonlinear impulsive problems neither adding superlinear or local sublinear conditions to nonlinear term at zero nor adding superlinear assumptions to nonlinear term at infinity. Our approach relies on variational methods, more specifically, on a three critical points theorem.

Existence results for a fourth-order discrete boundary value problem with a parameter

The discrete boundary value problem for a fourth-order difference equation is discussed in this paper. The existence result of multiple solutions is established by restricting the algebraic assumptions on the nonlinear term and confirming the parameter μ in an interval related to eigenvalues. The main methods adopted are the critical point theorems applicable for the finite dimensional Banach space recently proposed by Bonanno et al. (2014).

Critical Point Approaches to Generalized Yamabe Equations on Riemannian Manifolds and Applications to Emden–Fowler Problems

In this work, we continue the study of the multiplicity results for generalized Yamabe equations on Riemannian manifolds arising from conformal Riemannian geometry, astrophysics, and in the theories of thermionic emission, isothermal stationary gas sphere, and gas combustion. As applications, we consider the Emden–Fowler equations involving sublinear terms at infinity. In fact, employing two critical point theorems, we guarantee the existence of two and three solutions for our problem.

Multiple positive solutions for perturbed nonlinear fractional differential system with two control parameters

In this paper, we study a class of perturbed nonlinear fractional differential systems with two control parameters. Under some appropriate conditions on the primitive function of nonlinear terms F_{u} and F_{v}, we prove the multiplicity of weak solutions by using the variational method and Ricceri’s critical points theorems.

Energy–based localization and multiplicity of radially symmetric states for the stationary p–Laplace diffusion

ABSTRACTIt is investigated the role of the state–dependent source–term for the localization by means of the kinetic energy of radially symmetric states for the stationary p–Laplace diffusion. It is shown that the oscillatory behavior of the source–term, with respect to the state amplitude, yields multiple possible states, located in disjoint energy bands. The mathematical analysis makes use of critical point theory in conical shells and of a version of Pucci–Serrin three–critical point theorem for the intersection of a cone with a ball. A key ingredient is a Harnack type inequality in terms of the energetic norm.

The Multiplicity of Solutions for a Class of Nonlinear Fractional Dirichlet Boundary Value Problems with p-Laplacian Type via Variational Approach

Abstract In this paper, by using variational methods and a critical point theorem due to Bonanno and Marano, the existence of at least three weak solutions is obtained for a class of p-Laplacian type nonlinear fractional coupled systems depending on two parameters. Two examples are given to illustrate the applications of our main results.

Critical Points Theorems via the Generalized Ekeland Variational Principle and its Application to Equations of $p(x)$-Laplace Type in $\mathbb{R}^{N}$

In this paper, we investigate abstract critical point theorems for continuously Gâteaux differentiable functionals satisfying the Cerami condition via the generalized Ekeland variational principle developed by C.-K. Zhong. As applications of our results, under certain assumptions, we show the existence of at least one or two weak solutions for nonlinear elliptic equations with variable exponents \[ -\operatorname{div} (\varphi(x, \nabla u)) + V(x) |u|^{p(x)-2} u = \lambda f(x,u) \quad \textrm{in } \mathbb{R}^{N}, \] where the function $\varphi(x,v)$ is of type $|v|^{p(x)-2}v$ with a continuous function $p \colon \mathbb{R}^{N} \to (1,\infty)$, $V \colon \mathbb{R}^{N} \to (0,\infty)$ is a continuous potential function, $\lambda$ is a real parameter, and $f \colon \mathbb{R}^{N} \times \mathbb{R} \to \mathbb{R}$ is a Caratheodory function. Especially, we localize precisely the intervals of $\lambda$ for which the above equation admits at least one or two nontrivial weak solutions by applying our critical points results.

Infinitely many solutions for a class of fourth-order impulsive differential equations

AbstractIn this paper, by employing a critical point theorem, we establish the existence of infinitely many solutions for fourth-order impulsive differential equations depending on two real parameters.

Two non-zero solutions for Sturm–Liouville equations with mixed boundary conditions

In this paper, we establish the existence of two non-zero solutions for a mixed boundary value problem with the Sturm–Liouville equation. The approach is based on a recent two critical point theorem.

Infinitely many solutions to boundary value problem for fractional differential equations

Abstract Variational methods and critical point theorems are used to discuss existence of infinitely many solutions to boundary value problem for fractional order differential equations where Riemann-Liouville fractional derivatives and Caputo fractional derivatives are used. An example is given to illustrate our result.

Discrete fourth-order boundary value problems with four parameters

The theory of nonlinear difference equations and discrete boundary value problems has been widely used to study discrete models in many fields such as computer science, economics, mechanical engineering, control systems, artificial or biological neural networks, ecology, cybernetics, and so on. Fourth-order difference equations derived from various discrete elastic beam problems. In this paper, we seek further study of the multiplicity results for discrete fourth-order boundary value problems with four parameters. In fact, using a consequence of the local minimum theorem due Bonanno we look for the existence one solution under algebraic conditions on the nonlinear term and two solutions for the problem under algebraic conditions with the classical Ambrosetti–Rabinowitz (AR) condition on the nonlinear term. Furthermore, by employing two critical point theorems, one due Averna and Bonanno, and another one due Bonanno we guarantee the existence of two and three solutions for our problem in a special case.

A boundary value problem associated to non-homogeneous operators in Orlicz-Sobolev spaces

ABSTRACTWe apply a three critical points theorem of B. Ricceri to establish the existence of at least three weak solutions for a class of non-homogeneous Neumann problems. Furthermore, by using another theorem of him, we prove that an appropriate oscillating behaviour of the nonlinear term ensures the existence of infinitely many weak solutions. Our analysis is based on recent variational methods for smooth functionals defined on Orlicz-Sobolev spaces.

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.

Infinitely many nonnegative solutions for a fractional differential inclusion with oscillatory potential

In the present paper we study a fractional differential inclusion with Sturm–Liouville boundary value conditions. By means of nonsmooth critical point theorem, the existence of infinitely many nonnegative solutions is established when the potential F is oscillatory at infinity and origin, respectively.