• Title, Summary, Keyword: p-Laplacian

Search Result 96, Processing Time 0.046 seconds

ON DISCONTINUOUS ELLIPTIC PROBLEMS INVOLVING THE FRACTIONAL p-LAPLACIAN IN ℝN

  • Kim, In Hyoun;Kim, Yun-Ho;Park, Kisoeb
    • Bulletin of the Korean Mathematical Society
    • /
    • v.55 no.6
    • /
    • pp.1869-1889
    • /
    • 2018
  • We are concerned with the following fractional p-Laplacian inclusion: $$(-{\Delta})^s_pu+V(x){\mid}u{\mid}^{p-2}u{\in}{\lambda}[{\underline{f}}(x,u(x)),\;{\bar{f}}(s,u(x))]$$ in ${\mathbb{R}}^N$, where $(-{\Delta})^s_p$ is the fractional p-Laplacian operator, 0 < s < 1 < p < $+{\infty}$, sp < N, and $f:{\mathbb{R}}^N{\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ is measurable with respect to each variable separately. We show that our problem with the discontinuous nonlinearity f admits at least one or two nontrivial weak solutions. In order to do this, the main tool is the Berkovits-Tienari degree theory for weakly upper semicontinuous set-valued operators. In addition, our main assertions continue to hold when $(-{\Delta})^s_pu$ is replaced by any non-local integro-differential operator.

EXISTENCE OF A POSITIVE INFIMUM EIGENVALUE FOR THE p(x)-LAPLACIAN NEUMANN PROBLEMS WITH WEIGHTED FUNCTIONS

  • Kim, Yun-Ho
    • Korean Journal of Mathematics
    • /
    • v.22 no.3
    • /
    • pp.395-406
    • /
    • 2014
  • We study the following nonlinear problem $-div(w(x){\mid}{\nabla}u{\mid}^{p(x)-2}{\nabla}u)+{\mid}u{\mid}^{p(x)-2}u={\lambda}f(x,u)$ in ${\Omega}$ which is subject to Neumann boundary condition. Under suitable conditions on w and f, we give the existence of a positive infimum eigenvalue for the p(x)-Laplacian Neumann problem.

MULTIPLE SOLUTIONS TO DISCRETE BOUNDARY VALUE PROBLEMS FOR THE p-LAPLACIAN WITH POTENTIAL TERMS ON FINITE GRAPHS

  • CHUNG, SOON-YEONG;PARK, JEA-HYUN
    • Bulletin of the Korean Mathematical Society
    • /
    • v.52 no.5
    • /
    • pp.1517-1533
    • /
    • 2015
  • In this paper, we prove the existence of at least three nontrivial solutions to nonlinear discrete boundary value problems $$\{^{-{\Delta}_{p,{\omega}}u(x)+V(x){\mid}u(x){\mid}^{q-2}u(x)=f(x,u(x)),x{\in}S,}_{u(x)=0,\;x{\in}{\partial}S}$$, involving the discrete p-Laplacian on simple, nite and connected graphs $\bar{S}(S{\cup}{\partial}S,E)$ with weight ${\omega}$, where 1 < q < p < ${\infty}$. The approach is based on a suitable combine of variational and truncations methods.

ONE-DIMENSIONAL JUMPING PROBLEM INVOLVING p-LAPLACIAN

  • Jung, Tacksun;Choi, Q-Heing
    • Korean Journal of Mathematics
    • /
    • v.26 no.4
    • /
    • pp.683-700
    • /
    • 2018
  • We get one theorem which shows existence of solutions for one-dimensional jumping problem involving p-Laplacian and Dirichlet boundary condition. This theorem is that there exists at least one solution when nonlinearities crossing finite number of eigenvalues, exactly one solutions and no solution depending on the source term. We obtain these results by the eigenvalues and the corresponding normalized eigenfunctions of the p-Laplacian eigenvalue problem when 1 < p < ${\infty}$, variational reduction method and Leray-Schauder degree theory when $2{\leq}$ p < ${\infty}$.

ANTI-PERIODIC SOLUTIONS FOR HIGHER-ORDER LIÉENARD TYPE DIFFERENTIAL EQUATION WITH p-LAPLACIAN OPERATOR

  • Chen, Taiyong;Liu, Wenbin
    • Bulletin of the Korean Mathematical Society
    • /
    • v.49 no.3
    • /
    • pp.455-463
    • /
    • 2012
  • In this paper, by using degree theory, we consider a kind of higher-order Li$\acute{e}$enard type $p$-Laplacian differential equation as follows $$({\phi}_p(x^{(m)}))^{(m)}+f(x)x^{\prime}+g(t,x)=e(t)$$. Some new results on the existence of anti-periodic solutions for above equation are obtained.

STABILITY FOR A VISCOELASTIC PLATE EQUATION WITH p-LAPLACIAN

  • Park, Sun Hye
    • Bulletin of the Korean Mathematical Society
    • /
    • v.52 no.3
    • /
    • pp.907-914
    • /
    • 2015
  • In this paper, we consider a viscoelastic plate equation with p-Laplacian $u^{{\prime}{\prime}}+{\Delta}^2u-div({\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)+{\sigma}(t){\int}_{0}^{t}g(t-s){\Delta}u(s)ds-{\Delta}u^{\prime}=0$. By introducing suitable energy and Lyapunov functionals, we establish a general decay estimate for the energy, which depends on the behavior of both ${\sigma}$ and g.

PERIODIC SOLUTIONS FOR DUFFING TYPE p-LAPLACIAN EQUATION WITH MULTIPLE DEVIATING ARGUMENTS

  • Jiang, Ani
    • Journal of applied mathematics & informatics
    • /
    • v.31 no.1_2
    • /
    • pp.27-34
    • /
    • 2013
  • In this paper, we consider the Duffing type p-Laplacian equation with multiple deviating arguments of the form $$({\varphi}_p(x^{\prime}(t)))^{\prime}+Cx^{\prime}(t)+go(t,x(t))+\sum_{k=1}^ngk(t,x(t-{\tau}_k(t)))=e(t)$$. By using the coincidence degree theory, we establish new results on the existence and uniqueness of periodic solutions for the above equation. Moreover, an example is given to illustrate the effectiveness of our results.

CONSTANT-SIGN SOLUTIONS OF p-LAPLACIAN TYPE OPERATORS ON TIME SCALES VIA VARIATIONAL METHODS

  • Zhang, Li;Ge, Weigao
    • Bulletin of the Korean Mathematical Society
    • /
    • v.49 no.6
    • /
    • pp.1131-1145
    • /
    • 2012
  • The purpose of this paper is to use an appropriate variational framework to discuss the boundary value problem with p-Laplacian type operators $$\{({\alpha}(t,x^{\Delta}(t)))^{\Delta}-a(t){\phi}_p(x^{\sigma}(t))+f({\sigma}(t),x^{\sigma}(t))=0,\;{\Delta}-a.e.\;t{\in}I\\x^{\sigma}(0)=0,\\{\beta}_1x^{\sigma}(1)+{\beta}_2x^{\Delta}({\sigma}(1))=0,$$ where ${\beta}_1$, ${\beta}_2$ > 0, $I=[0,1]^{k^2}$, ${\alpha}({\cdot},x({\cdot}))$ is an operator of $p$-Laplacian type, $\mathbb{T}$ is a time scale. Some sufficient conditions for the existence of constant-sign solutions are obtained.

MULTIPLE PERIODIC SOLUTIONS FOR EIGENVALUE PROBLEMS WITH A p-LAPLACIAN AND NON-SMOOTH POTENTIAL

  • Zhang, Guoqing;Liu, Sanyang
    • Bulletin of the Korean Mathematical Society
    • /
    • v.48 no.1
    • /
    • pp.213-221
    • /
    • 2011
  • In this paper, we establish a multiple critical points theorem for a one-parameter family of non-smooth functionals. The obtained result is then exploited to prove a multiplicity result for a class of periodic eigenvalue problems driven by the p-Laplacian and with a non-smooth potential. Under suitable assumptions, we locate an open subinterval of the eigenvalue.