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Start Over Subject $p$-Laplacian

1. Existence and iteration of positive solutions for a singular two-point boundary value problem with a $p$-Laplacian operator

Creator:
Ma, De-xiang, Ge, Wei-Gao, and Gui, Zhan-Ji
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
iteration, symmetric and monotone positive solution, nonlinear boundary value problem, and $p$-Laplacian
Language:
English
Description:
In the paper, we obtain the existence of symmetric or monotone positive solutions and establish a corresponding iterative scheme for the equation $(\phi _p(u^{\prime }))^{\prime }+q(t)f(u)=0$, $0<t<1$, where $\phi _p(s):=|s|^{p-2}s$, $p>1$, subject to nonlinear boundary condition. The main tool is the monotone iterative technique. Here, the coefficient $q(t)$ may be singular at $t=0,1$.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

2. Existence of positive solutions for singular four-point boundary value problem with a $p$-Laplacian

Creator:
Miao, Chunmei, Zhao, Junfang, and Ge, Weigao
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
singular, four-point, positive solution, and $p$-Laplacian
Language:
English
Description:
In this paper we deal with the four-point singular boundary value problem $$ \begin {cases} (\phi _p(u'(t)))'+q(t)f(t,u(t),u'(t))=0,& t\in (0,1),\\ u'(0)-\alpha u(\xi )=0, \quad u'(1)+\beta u(\eta )=0, \end {cases} $$ where $\phi _p(s)=|s|^{p-2}s$, $p>1$, $0<\xi <\eta <1$, $\alpha ,\beta >0$, $q\in C[0,1]$, $q(t)>0$, $t\in (0,1)$, and $f\in C([0,1]\times (0,+\infty )\times \mathbb R,(0,+\infty ))$ may be singular at $u = 0$. By using the well-known theory of the Leray-Schauder degree, sufficient conditions are given for the existence of positive solutions.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

3. Existence of solutions and of multiple solutions for nonlinear nonsmooth periodic systems

Creator:
Papageorgiou, Evgenia H. and Papageorgiou, Nikolaos S.
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
$p$-Laplacian, nonsmooth critical point theory, Clarke subdifferential, saddle point theorem, periodic solution, Poincare-Wirtinger inequality, Sobolev inequality, and nonsmooth Palais-Smale condition
Language:
English
Description:
In this paper we examine nonlinear periodic systems driven by the vectorial $p$-Laplacian and with a nondifferentiable, locally Lipschitz nonlinearity. Our approach is based on the nonsmooth critical point theory and uses the subdifferential theory for locally Lipschitz functions. We prove existence and multiplicity results for the “sublinear” problem. For the semilinear problem (i.e. $p = 2$) using a nonsmooth multidimensional version of the Ambrosetti-Rabinowitz condition, we prove an existence theorem for the “superlinear” problem. Our work generalizes some recent results of Tang (PAMS 126(1998)).
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public