• Title/Summary/Keyword: p-Laplacian equations

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THE p-LAPLACIAN OPERATORS WITH POTENTIAL TERMS

  • Chung, Soon-Yeong;Lee, Hee-Soo
    • Communications of the Korean Mathematical Society
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    • v.26 no.4
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    • pp.591-601
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    • 2011
  • In this paper, we deal with the discrete p-Laplacian operators with a potential term having the smallest nonnegative eigenvalue. Such operators are classified as its smallest eigenvalue is positive or zero. We discuss differences between them such as an existence of solutions of p-Laplacian equations on networks and properties of the energy functional. Also, we give some examples of Poisson equations which suggest a difference between linear types and nonlinear types. Finally, we study characteristics of the set of a potential those involving operator has the smallest positive eigenvalue.

FRACTIONAL HYBRID DIFFERENTIAL EQUATIONS WITH P-LAPLACIAN OPERATOR

  • CHOUKRI DERBAZI;ABDELKRIM SALIM;HADDA HAMMOUCHE;MOUFFAK BENCHOHRA
    • Journal of Applied and Pure Mathematics
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    • v.6 no.1_2
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    • pp.21-36
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    • 2024
  • In this paper, we study the existence of solutions for hybrid fractional differential equations with p-Laplacian operator involving fractional Caputo derivative of arbitrary order. This work can be seen as an extension of earlier research conducted on hybrid differential equations. Notably, the extension encompasses both the fractional aspect and the inclusion of the p-Laplacian operator. We build our analysis on a hybrid fixed point theorem originally established by Dhage. In addition, an example is provided to demonstrate the effectiveness of the main results.

INFINITELY MANY SOLUTIONS FOR (p(x), q(x))-LAPLACIAN-LIKE SYSTEMS

  • Heidari, Samira;Razani, Abdolrahman
    • Communications of the Korean Mathematical Society
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    • v.36 no.1
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    • pp.51-62
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    • 2021
  • Variational method has played an important role in solving problems of uniqueness and existence of the nonlinear works as well as analysis. It will also be extremely useful for researchers in all branches of natural sciences and engineers working with non-linear equations economy, optimization, game theory and medicine. Recently, the existence of infinitely many weak solutions for some non-local problems of Kirchhoff type with Dirichlet boundary condition are studied [14]. Here, a suitable method is presented to treat the elliptic partial derivative equations, especially (p(x), q(x))-Laplacian-like systems. This kind of equations are used in the study of fluid flow, diffusive transport akin to diffusion, rheology, probability, electrical networks, etc. Here, the existence of infinitely many weak solutions for some boundary value problems involving the (p(x), q(x))-Laplacian-like operators is proved. The method is based on variational methods and critical point theory.

ON PERIODIC BOUNDARY VALUE PROBLEMS OF HIGHER ORDER NONLINEAR FUNCTIONAL DIFFERENCE EQUATIONS WITH p-LAPLACIAN

  • Liu, Yuji;Liu, Xingyuan
    • Communications of the Korean Mathematical Society
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    • v.24 no.1
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    • pp.29-40
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    • 2009
  • Motivated by [Linear Algebra and its Appl. 420(2007), 218-227] and [Linear Algebra and its Appl. 425(2007), 171-183], we, in this paper, study the solvability of periodic boundary value problems of higher order nonlinear functional difference equations with p-Laplacian. Sufficient conditions for the existence of at least one solution of this problem are established.

EXISTENCE AND UNIQUENESS OF PERIODIC SOLUTIONS FOR A CLASS OF p-LAPLACIAN EQUATIONS

  • Kim, Yong-In
    • The Pure and Applied Mathematics
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    • v.19 no.2
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    • pp.103-109
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    • 2012
  • The existence and uniqueness of T-periodic solutions for the following p-Laplacian equations: $$({\phi}_p(x^{\prime}))^{\prime}+{\alpha}(t)x^{\prime}+g(t,x)=e(t),\;x(0)=x(T),x^{\prime}(0)=x^{\prime}(T)$$ are investigated, where ${\phi}_p(u)={\mid}u{\mid}^{p-2}u$ with $p$ > 1 and ${\alpha}{\in}C^1$, $e{\in}C$ are T-periodic and $g$ is continuous and T-periodic in $t$. By using coincidence degree theory, some existence and uniqueness results are obtained.

EXISTENCE OF WEAK SOLUTIONS TO A CLASS OF SCHRÖDINGER TYPE EQUATIONS INVOLVING THE FRACTIONAL p-LAPLACIAN IN ℝN

  • Kim, Jae-Myoung;Kim, Yun-Ho;Lee, Jongrak
    • Journal of the Korean Mathematical Society
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    • v.56 no.6
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    • pp.1529-1560
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    • 2019
  • We are concerned with the following elliptic equations: $$(-{\Delta})^s_pu+V (x){\mid}u{\mid}^{p-2}u={\lambda}g(x,u){\text{ in }}{\mathbb{R}}^N$$, where $(-{\Delta})_p^s$ is the fractional p-Laplacian operator with 0 < s < 1 < p < $+{\infty}$, sp < N, the potential function $V:{\mathbb{R}}^N{\rightarrow}(0,{\infty})$ is a continuous potential function, and $g:{\mathbb{R}}^N{\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ satisfies a $Carath{\acute{e}}odory$ condition. We show the existence of at least one weak solution for the problem above without the Ambrosetti and Rabinowitz condition. Moreover, we give a positive interval of the parameter ${\lambda}$ for which the problem admits at least one nontrivial weak solution when the nonlinearity g has the subcritical growth condition.

INTERVAL CRITERIA FOR FORCED OSCILLATION OF DIFFERENTIAL EQUATIONS WITH p-LAPLACIAN AND NONLINEARITIES GIVEN BY RIEMANN-STIELTJES INTEGRALS

  • Hassan, Taher S.;Kong, Qingkai
    • Journal of the Korean Mathematical Society
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    • v.49 no.5
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    • pp.1017-1030
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    • 2012
  • We consider forced second order differential equation with $p$-Laplacian and nonlinearities given by a Riemann-Stieltjes integrals in the form of $$(p(t){\phi}_{\gamma}(x^{\prime}(t)))^{\prime}+q_0(t){\phi}_{\gamma}(x(t))+{\int}^b_0q(t,s){\phi}_{{\alpha}(s)}(x(t))d{\zeta}(s)=e(t)$$, where ${\phi}_{\alpha}(u):={\mid}u{\mid}^{\alpha}\;sgn\;u$, ${\gamma}$, $b{\in}(0,{\infty})$, ${\alpha}{\in}C[0,b)$ is strictly increasing such that $0{\leq}{\alpha}(0)<{\gamma}<{\alpha}(b-)$, $p$, $q_0$, $e{\in}C([t_0,{\infty}),{\mathbb{R}})$ with $p(t)>0$ on $[t_0,{\infty})$, $q{\in}C([0,{\infty}){\times}[0,b))$, and ${\zeta}:[0,b){\rightarrow}{\mathbb{R}}$ is nondecreasing. Interval oscillation criteria of the El-Sayed type and the Kong type are obtained. These criteria are further extended to equations with deviating arguments. As special cases, our work generalizes, unifies, and improves many existing results in the literature.