• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.11 no.4
• /
• pp.89-99
• /
• 2007

In this paper, we define the upper and lower solutions for a p-Laplacian system with singular nonlinearity at the boundaries. And we prove the theorem for the upper and power solutions method.

• East Asian mathematical journal
• /
• v.40 no.1
• /
• pp.85-94
• /
• 2024

In this paper we study the existence and multiplicity of positive solutions for p-Laplacian problems with a singular weight. Proofs mainly make use of Global Continuation Theorem and Fixed Point Index argument.

• East Asian mathematical journal
• /
• v.31 no.5
• /
• pp.727-735
• /
• 2015

We introduce the fundamental theorem of upper and lower solutions for a class of singular ($p_1,\;p_2$)-Laplacian systems and give the proof by using the Schauder fixed point theorem. It will play an important role to study the existence of solutions.

• Journal of applied mathematics & informatics
• /
• v.36 no.3_4
• /
• pp.173-179
• /
• 2018

In this paper, we investigate the stability of positive weak solution for the singular p-Laplacian nonlinear system $-div[{\mid}x{\mid}^{-ap}{\mid}{\nabla}u{\mid}^{p-2}{\nabla}u]+m(x){\mid}u{\mid}^{p-2}u={\lambda}{\mid}x{\mid}^{-(a+1)p+c}b(x)f(u)$ in ${\Omega}$, Bu = 0 on ${\partial}{\Omega}$, where ${\Omega}{\subset}R^n$ is a bounded domain with smooth boundary $Bu={\delta}h(x)u+(1-{\delta})\frac{{\partial}u}{{\partial}n}$ where ${\delta}{\in}[0,1]$, $h:{\partial}{\Omega}{\rightarrow}R^+$ with h = 1 when ${\delta}=1$, $0{\in}{\Omega}$, 1 < p < n, 0 ${\leq}$ a < ${\frac{n-p}{p}}$, m(x) is a weight function, the continuous function $b(x):{\Omega}{\rightarrow}R$ satisfies either b(x) > 0 or b(x) < 0 for all $x{\in}{\Omega}$, ${\lambda}$ is a positive parameter and $f:[0,{\infty}){\rightarrow}R$ is a continuous function. We provide a simple proof to establish that every positive solution is unstable under certain conditions.

• Journal of applied mathematics & informatics
• /
• v.25 no.1_2
• /
• pp.329-337
• /
• 2007

In the paper, we obtain the existence of positive solutions and establish a corresponding iterative scheme for BVPs $$\{^{\;(\phi_p(u'))'\;+\;q(t)f(t,u)=0,\;0\;<\;t\;<\;1,}_{\;u(0)\;-\;B(u'({\eta}))\;=\;0,\;u'(1)\;=\;0}$$ and $$\{^{\;(\phi_p(u'))'\;+\;q(t)f(t,u)=0,\;0\;<\;t\;<\;1,}_{\;u'(0)\;=\;0,\;u(1)+B(u'(\eta))\;=\;0.}$$. The main tool is the monotone iterative technique. Here, the coefficient q(t) may be singular at t = 0, 1.

• Communications of the Korean Mathematical Society
• /
• v.10 no.1
• /
• pp.49-55
• /
• 1995

This paper is the third of a series in which smoothness properties of function in several variables are discussed. The germ of the whole theory was laid in the works by Folland and Stein [4]. On nilpotent Lie groups, they difined analogues of the classical $L^p$ Sobolev or potential spaces in terms of fractional powers of sub-Laplacian, L and extended several basic theorems from the Euclidean theory of differentaiability to these spaces: interpolation properties, boundedness of singular integrals,..., and imbeding theorems. In this paper we study the analogue to the extension theorem for the Folland-Stein spaces. The analogue to Stein's restriction theorem were studied by M. Mekias [5] and Y.M. Kim [6]. First, we have the space of Bessel potentials on the Heisenberg group introduced by Folland [4].