Critical Point Theorem Research Articles

New critical point theorem and infinitely many small-magnitude solutions of a nonlinear Iwatsuka model

In this study, the following nonlinear magnetic Schrödinger equation is considered−ΔAu+V(x)u=Eu+|u|p−2u,x=(x1,x2)∈R2, where 2<p<+∞, A(x)=(0,−b(x1)), b(x1)=∫0x1B(t)dt, B∈C∞(R,R) is an increasing function with different limits at +∞ and −∞. Under these assumptions, the magnetic Laplacian operator −ΔA is known as an Iwatsuka model, and its spectrum has a band structure. In addition, E is an edge of a spectral gap of operator −ΔA, and V is a power-like decay potential. It is proved that this equation has a sequence of non-zero solutions, whose L∞ norms tend to zero along this sequence. The proof is given by establishing a new critical point theorem without the usual (PS)⁎ condition. This study is a subsequent work to a recent one (Chen, 2022 [11]).

Multiplicity of solutions for nonlinear coercive problems

We are concerned in this paper with problems that involve nonlinear potential mappings satisfying condition (S) and whose potentials are coercive. We first provide mild sufficient conditions for the minimizing sequence in the Weierstrass-Tonelli theorem in order to have strongly convergent subsequences. Next, we establish a three critical point theorem which is based on the Pucci-Serrin type mountain pass lemma and which is an infinite dimensional counterpart of the Courant theorem. Ricceri-type three critical point results then follow. Some applications to Dirichlet boundary value problems driven by the perturbed Laplacian are given in the final part of this paper.

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.

Variational approach of p-Laplacian impulsive differential equations with periodic conditions

Abstract In this paper, we study a boundary value problem for a second-order impulsive p-Laplacian dynamic equation on a time scale. The approach is based on a critical point theorem. Finally, some examples are given to illustrate the results.

Variational Approaches to a Discrete Elliptic Problem with a Weight

In this article, we are concerned with the existence of at least one, two and three distinct solutions for discrete boundary value problems driven by the Laplacian. The proof of the main result depends on variational methods. By using a consequence of the local minimum theorem due Bonanno we investigate the existence of at least one solution and two solutions for the problem with the weight. Furthermore, by using two critical point theorems, one due Averna and Bonanno, and another due Bonanno we explore the existence of two and three solutions for the problem. We also provide two examples in order to illustrate the main results.

Infinitely Many Sign-Changing Solutions for the Nonlinear Schrödinger-Poisson System with Super 2-linear Growth at Infinity

In this paper, we investigate the sign-changing solutions to the following Schrödinger-Poisson system $$\begin{aligned} \qquad \left\{ \begin{array}{ll} -\Delta u+V(x)u+\lambda \phi (x) u =f(u),\ \ \ &{}\ x \in {\mathbb {R}}^{3},\\ -\Delta \phi =u^2, \ \ \ &{}\ x \in {\mathbb {R}}^{3}, \\ \end{array} \right. \end{aligned}$$ where $$\lambda >0$$ is a parameter and f is super 2-linear at infinity. By using the method of invariant sets of descending flow and a multiple critical points theorem, we prove that this system possesses infinitely many sign-changing solutions for any $$\lambda >0$$ .

Instantaneous and Non-Instantaneous Impulsive Boundary Value Problem Involving the Generalized ψ-Caputo Fractional Derivative

This paper studies a new class of instantaneous and non-instantaneous impulsive boundary value problem involving the generalized ψ-Caputo fractional derivative with a weight. Depending on critical point theorems and some properties of ψ-Caputo-type fractional integration and differentiation, the variational construction and multiplicity result of solutions are established.

Study on a new [formula omitted]-Laplacian fractional differential model generated by instantaneous and non-instantaneous impulsive effects

In this paper, we discuss a new p-Laplacian fractional differential equation involving instantaneous and non-instantaneous impulses, supplemented with Sturm–Liouville boundary conditions. To study the stated problem, we introduce a new norm for fractional space, denoted by Eα,p. Under this framework, using the critical point theorem proposed by Bonanno and Marano, we prove the existence result of multiple solutions. Finally, an illustrative example is presented.

Bifurcation type phenomena for positive solutions of a class of impulsive differential equations

In this paper, we are concerned with the existence, nonexistence, and multiplicity of positive solutions for a impulsive periodic boundary value problem with positive parameter . Using the critical point theorem, we obtain a bifurcation type result about positive solutions; that is, there exists such that our problem has at least two positive solutions for all , one positive solution for , and no positive solution for under the appropriate conditions.

VARIATIONAL FORMULATION FOR THE STURM-LIOUVILLE PROBLEM OF FRACTIONAL DIFFERENTIAL EQUATION WITH GENERALIZED (&lt;i&gt;P&lt;/i&gt;, &lt;i&gt;Q&lt;/i&gt;)-LAPLACIAN OPERATOR

In this paper, the Sturm-Liouville boundary value problem is studied for fractional differential equation with generalized (<i>p</i>, <i>q</i>)-Laplacian operator. By imposing mild assumptions on nonlinearity <i>f</i>, several new existence results of at least one or two nontrivial weak solutions are established through variational methods and critical point theorems. Furthermore, the criteria is also investigated for the nonexistence result.

Dirichlet problems of fractional p-Laplacian equation with impulsive effects.

The purpose of the article is to investigate Dirichlet boundary-value problems of the fractional p-Laplacian equation with impulsive effects. By using the Nehari manifold method, mountain pass theorem and three critical points theorem, some new results are achieved under more general growth conditions. In addition, this paper weakens the commonly used p-suplinear and p-sublinear growth conditions.

Non-trivial solutions for a partial discrete Dirichlet nonlinear problem with $ p $-Laplacian

&lt;abstract&gt;&lt;p&gt;We investigate the non-trivial solutions for a partial discrete Dirichlet nonlinear problem with $ p $-Laplacian by applying Ricceri's variational principle and a two non-zero critical points theorem. In addition, we identify open intervals of the parameter $ \lambda $ under appropriate constraints imposed on the nonlinear term. This allows us to ensure that the nonlinear problem has at least one or two non-trivial solutions.&lt;/p&gt;&lt;/abstract&gt;

The existence of one solution for impulsive differential equations via variational methods

We prove the existence of at least one non-trivial weak solution for a nonlinear Dirichlet boundary value problem subject to perturbations of impulsive terms via employing a critical point theorem for differentiable functionals.

Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods

Abstract While it is known that one can consider the existence of solutions to boundary-value problems for fractional differential equations with derivative terms, the situations for the multiplicity of weak solutions for the p-Laplacian fractional differential equations with derivative terms are less considered. In this article, we propose a new class of p-Laplacian fractional differential equations with the Caputo derivatives. The multiplicity of weak solutions is proved by the variational method and critical point theorem. At the end of the article, two examples are given to illustrate the validity and practicality of our main results.

A Class of Fourth-Order Symmetrical Kirchhoff Type Systems

This paper deals with the existence and multiplicity of solutions for a perturbed nonlocal fourth-order class of p(·)&amp;q(·)-Kirchhoff elliptic systems under Navier boundary conditions. By using the variational method and Ricceri’s critical point theorem, we can find a proper conditions to ensure that the perturbed fourth-order of (p(x),q(x))-Kirchhoff systems has at least three weak solutions. We have extended and improved some recent results. The complexity of the combination of variable exponent theory and fourth-order Kirchhoff systems makes the results of this work novel and new contribution. Finally, a very concrete example is given to illustrate the applications of our results.

New multiplicity results for critical p-Laplacian problems

We prove new multiplicity results for the Brézis-Nirenberg problem for the p-Laplacian. Our proofs are based on a new abstract critical point theorem involving the Z2-cohomological index that requires less compactness than the (PS) condition.

Ground state and least positive energy solutions of elliptic problems involving mixed fractional $p$-Laplacians

We establish the existence of ground state and least positive energy solutions to some problems involving mixed fractional $p$-Laplacians using some abstract critical point theorems based on the $\mathbb Z_2$-cohomological index.

Existence results for perturbed fourth-order Kirchhoff type elliptic problems with singular term

Under appropriate growth conditions on the nonlinearity, the existence of multiple solutions for a perturbednonlocal fourth-order Kirchhoff-type problem involving the Hardy term:$$\Delta_p ^2 u-\big[M(\int_{\Omega}|\nabla u|^{p}dx)\big]^{p-1}\Delta_{p}u-\mu\frac{|u|^{p-2}u}{|x|^{2p}}= \lambda f(x,u),$$is established. Our main tools are based on variational methods and some critical pointstheorems. We give some examples to illustrate the obtained results.

Existence and multiplicity results for a Steklov problem involving (p(x), q(x))-Laplacian operator

Abstract In this work, we are concerned with a generalized Steklov problem with (p(x), q(x))-Laplacian operator. Under some appropriate conditions on the data involved in the elliptic problem, we prove the existence of at least three solutions using Ricceri’s three critical points theorem.

Existence and multiplicity of solutions for a p(x)-biharmonic problem with Neumann boundary conditions

In this paper, we study the p(x)-biharmonique problem with Neumannboundary conditions. Using the three critical point Theorem, we establish the existence of at least threesolutions of this problem.