scholarly journals Du Bois–Reymond Type Lemma and Its Application to Dirichlet Problem with the p(t)–Laplacian on a Bounded Time Scale

Entropy ◽  
2021 ◽  
Vol 23 (10) ◽  
pp. 1352
Author(s):  
Jean Mawhin ◽  
Ewa Skrzypek ◽  
Katarzyna Szymańska-Dȩbowska
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Time Scale ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Mountain Pass ◽  
Du Bois

This paper is devoted to study the existence of solutions and their regularity in the p(t)– Laplacian Dirichlet problem on a bounded time scale. First, we prove a lemma of du Bois–Reymond type in time-scale settings. Then, using direct variational methods and the mountain pass methodology, we present several sufficient conditions for the existence of solutions to the Dirichlet problem.

Solutions with boundary layer and positive peak for an elliptic Dirichlet problem

2004 ◽  
Vol 134 (3) ◽  
pp. 515-536 ◽  
Author(s):  
Gongbao Li ◽  
Jianfu Yang ◽  
Shusen Yan
Keyword(s):  
Boundary Layer ◽  
Dirichlet Problem ◽  
Elliptic Problem ◽  
Existence Of Solutions ◽  
Mountain Pass ◽  
Single Peak ◽  
Type Solution ◽  
Positive Peak

In this paper, we study existence of solutions with boundary layer and peaks for an elliptic problem. We prove that the problem has a mountain-pass-type solution, which has a boundary layer and a single peak near the boundary. Moreover, we also study the existence of solutions with interior peak.

scholarly journals Boundary Value Problems for Fourth Order Nonlinearp-Laplacian Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Qinqin Zhang
Keyword(s):  
Difference Equation ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Difference Equations ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Mountain Pass ◽  
Mountain Pass Lemma

We consider the boundary value problem for a fourth order nonlinearp-Laplacian difference equation containing both advance and retardation. By using Mountain pass lemma and some established inequalities, sufficient conditions of the existence of solutions of the boundary value problem are obtained. And an illustrative example is given in the last part of the paper.

Solutions for a Class of Elliptic Equation

2013 ◽  
Vol 734-737 ◽  
pp. 3007-3010
Author(s):  
Zong Hu Xiu
Keyword(s):  
Elliptic Equation ◽  
Variational Methods ◽  
Existence Of Solutions ◽  
Sufficient Conditions

In this paper, we study the existence of solutions for a class of elliptic equation. By the variational methods, we give the sufficient conditions that insure the existence of solutions.

scholarly journals Variational Methods for a Fractional Dirichlet Problem Involving Jumarie’s Derivative

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Rafał Kamocki
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Existence And Uniqueness ◽  
Fundamental Lemma ◽  
Du Bois ◽  
Uniqueness Of A Solution

We investigate a fractional Dirichlet problem involving Jumarie’s derivative. Using some variational methods a theorem on the existence and uniqueness of a solution to such problem is proved. In the proof of the main result we use a fractional counterpart of the du Bois-Reymond fundamental lemma.

scholarly journals Existence of Solutions for Choquard Type Elliptic Problems with Doubly Critical Nonlinearities

2019 ◽  
Vol 21 (1) ◽  
pp. 77-93
Author(s):  
Yansheng Shen
Keyword(s):  
Variational Methods ◽  
Minimization Problem ◽  
Existence Of Solutions ◽  
Elliptic Problems ◽  
Type Equation ◽  
Nontrivial Solutions ◽  
Critical Nonlinearities

Abstract In this article, we first study the existence of nontrivial solutions to the nonlocal elliptic problems in ℝ N {\mathbb{R}^{N}} involving fractional Laplacians and the Hardy–Sobolev–Maz’ya potential. Using variational methods, we investigate the attainability of the corresponding minimization problem, and then obtain the existence of solutions. We also consider another Choquard type equation involving the p-Laplacian and critical nonlinearities in ℝ N {\mathbb{R}^{N}} .

scholarly journals Existence of Solutions of Nonlinear Stochastic Volterra Fredholm Integral Equations of Mixed Type

2010 ◽  
Vol 2010 ◽  
pp. 1-16 ◽  
Author(s):  
K. Balachandran ◽  
J.-H. Kim
Keyword(s):  
Integral Equations ◽  
Existence And Uniqueness ◽  
Mixed Type ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Stochastic Integral ◽  
Sufficient Conditions ◽  
Fredholm Integral Equations ◽  
Equations Of Mixed Type ◽  
Stochastic Integral Equations

We establish sufficient conditions for the existence and uniqueness of random solutions of nonlinear Volterra-Fredholm stochastic integral equations of mixed type by using admissibility theory and fixed point theorems. The results obtained in this paper generalize the results of several papers.

scholarly journals Antiperiodic Solutions forp-Laplacian Systems via Variational Approach

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Lifang Niu ◽  
Kaimin Teng
Keyword(s):  
Boundary Conditions ◽  
Variational Methods ◽  
Variational Approach ◽  
Existence Of Solutions

We establish the existence of solutions forp-Laplacian systems with antiperiodic boundary conditions through using variational methods.

Existence of solutions for p(x)-Laplacian Dirichlet problem

2003 ◽  
Vol 52 (8) ◽  
pp. 1843-1852 ◽  
Author(s):  
Xian-Ling Fan ◽  
Qi-Hu Zhang
Keyword(s):  
Dirichlet Problem ◽  
Existence Of Solutions

scholarly journals Existence of Solutions for Anti-Periodic Fractional Differential Inclusions Involving ψ-Riesz-Caputo Fractional Derivative

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 630
Author(s):  
Dandan Yang ◽  
Chuanzhi Bai
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Differential Inclusions ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Fractional Differential Inclusions ◽  
Nonlinear Alternative ◽  
Contractive Map ◽  
Fractional Differential ◽  
Definition Of

In this paper, we investigate the existence of solutions for a class of anti-periodic fractional differential inclusions with ψ -Riesz-Caputo fractional derivative. A new definition of ψ -Riesz-Caputo fractional derivative of order α is proposed. By means of Contractive map theorem and nonlinear alternative for Kakutani maps, sufficient conditions for the existence of solutions to the fractional differential inclusions are given. We present two examples to illustrate our main results.

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