scholarly journals Infinitely Many Solutions for a Class of Fractional Impulsive Coupled Systems with (p,q)-Laplacian

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Junping Xie ◽  
Xingyong Zhang
Keyword(s):  
Mixed Type ◽  
Coupled Systems ◽  
Mountain Pass ◽  
Mountain Pass Lemma ◽  
Infinitely Many Solutions ◽  
Symmetric Mountain Pass Lemma

By using the symmetric mountain pass lemma, we investigate the problem of existence of infinitely many solutions for a class of fractional impulsive coupled systems with (p,q)-Laplacian, which possesses mixed type nonlinearities, and the nonlinearities do not need to satisfy the well-known Ambrosetti-Rabinowitz condition.

Infinitely Many Solutions for a Non-homogeneous Differential Inclusion with Lack of Compactness

2019 ◽  
Vol 19 (3) ◽  
pp. 625-637 ◽  
Author(s):  
Bin Ge ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Variational Principle ◽  
Differential Inclusion ◽  
Energy Functional ◽  
Mountain Pass Lemma ◽  
Infinitely Many Solutions ◽  
Locally Lipschitz Functions ◽  
Fountain Theorem ◽  
Inclusion Problems ◽  
Locally Lipschitz ◽  
Symmetric Mountain Pass Lemma

Abstract In this paper, we consider the following class of differential inclusion problems in {\mathbb{R}^{N}} involving the {p(x)} -Laplacian: -\Delta_{p(x)}u+V(x)\lvert u\rvert^{p(x)-2}u\in a(x)\partial F(x,u)\quad\text{% in}\ \mathbb{R}^{N}. We are concerned with a multiplicity property, and our arguments combine the variational principle for locally Lipschitz functions with the properties of the generalized Lebesgue–Sobolev space. Applying the nonsmooth symmetric mountain pass lemma and the fountain theorem, we establish conditions such that the associated energy functional possesses infinitely many critical points, and then we obtain infinitely many solutions.

scholarly journals Multiple Solutions for Generalized Asymptotical Linear Hamiltonian Systems Satisfying Bolza Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Yuan Shan ◽  
Baoqing Liu
Keyword(s):  
Boundary Conditions ◽  
Hamiltonian Systems ◽  
Multiple Solutions ◽  
Index Theory ◽  
Mountain Pass ◽  
Mountain Pass Lemma ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Linear Hamiltonian Systems ◽  
Symmetric Mountain Pass Lemma

This paper is devoted to multiple solutions of generalized asymptotical linear Hamiltonian systems satisfying Bolza boundary conditions. We classify the linear Hamiltonian systems by the index theory and obtain the existence and multiplicity of solutions for the Hamiltonian systems, based on an application of the classical symmetric mountain pass lemma.

scholarly journals Infinitely many solutions for a class of hemivariational inequalities involving p(x)-Laplacian

2017 ◽  
Vol 25 (2) ◽  
pp. 65-83
Author(s):  
Fariba Fattahi ◽  
Mohsen Alimohammady
Keyword(s):  
Main Tool ◽  
Neumann Boundary ◽  
Hemivariational Inequalities ◽  
Mountain Pass Lemma ◽  
Infinitely Many Solutions ◽  
Smooth Bounded Domain ◽  
Laplacian Equation ◽  
Locally Lipschitz Functional ◽  
Locally Lipschitz ◽  
Symmetric Mountain Pass Lemma

AbstractIn this paper hemivariational inequality with nonhomogeneous Neumann boundary condition is investigated. The existence of infinitely many small solutions involving a class of p(x) - Laplacian equation in a smooth bounded domain is established. Our main tool is based on a version of the symmetric mountain pass lemma due to Kajikiya and the principle of symmetric criticality for a locally Lipschitz functional.

scholarly journals Infinitely many solutions for the p-fractional Kirchhoff equations with electromagnetic fields and critical nonlinearity

2018 ◽  
Vol 23 (4) ◽  
pp. 599-618
Author(s):  
Sihua Liang ◽  
Jihui Zhang
Keyword(s):  
Electromagnetic Fields ◽  
Mountain Pass Lemma ◽  
Infinitely Many Solutions ◽  
Concentration Compactness ◽  
Critical Nonlinearity ◽  
Kirchhoff Equations ◽  
Fractional Sobolev Space ◽  
Concentration Compactness Principle ◽  
Symmetric Mountain Pass Lemma ◽  
Compactness Principle

In this paper, we consider the fractional Kirchhoff equations with electromagnetic fields and critical nonlinearity. By means of the concentration-compactness principle in fractional Sobolev space and the Kajikiya's new version of the symmetric mountain pass lemma, we obtain the existence of infinitely many solutions, which tend to zero for suitable positive parameters.

scholarly journals Periodic Solutions of a Class of Fourth-Order Superlinear Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Yanyan Li ◽  
Yuhua Long
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Nonlinear Term ◽  
Single Equation ◽  
Mountain Pass ◽  
Mountain Pass Lemma ◽  
Variational Techniques ◽  
Superlinear Growth ◽  
Symmetric Mountain Pass Lemma

This paper deals with the periodic solutions of a class of fourth-order superlinear differential equations. By using the classical variational techniques and symmetric mountain pass lemma, the periodic solutions of a single equation in literature are extended to that of equations, and also, the cubic growth of nonlinear term is extended to a general form of superlinear growth.

scholarly journals Existence Results for apx-Kirchhoff-Type Equation without Ambrosetti-Rabinowitz Condition

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Libo Wang ◽  
Minghe Pei
Keyword(s):  
Existence Of Solutions ◽  
Type Equation ◽  
Existence Results ◽  
Mountain Pass ◽  
Mountain Pass Lemma ◽  
Infinitely Many Solutions ◽  
Fountain Theorem ◽  
Multiplicity Of Solutions ◽  
Kirchhoff Type ◽  
Existence And Multiplicity

We consider the existence and multiplicity of solutions for thepx-Kirchhoff-type equations without Ambrosetti-Rabinowitz condition. Using the Mountain Pass Lemma, the Fountain Theorem, and its dual, the existence of solutions and infinitely many solutions were obtained, respectively.

scholarly journals On the Existence of Nontrivial Solutions for Nonlocal Elliptic Kirchhoff Type Problems with Nonlinear Boundary Conditions

2015 ◽  
Vol 55 (1) ◽  
pp. 183-188
Author(s):  
S. H. Rasouli ◽  
B. Salehi
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Type Equation ◽  
Nonlinear Boundary Conditions ◽  
Mountain Pass ◽  
Mountain Pass Lemma ◽  
Nontrivial Solutions ◽  
Kirchhoff Type ◽  
Kirchhoff Type Equation ◽  
Kirchhoff Type Problems

Abstract In this paper, by using the Mountain Pass Lemma, we study the existence of nontrivial solutions for a nonlocal elliptic Kirchhoff type equation together with nonlinear boundary conditions.

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