• Journal of applied mathematics & informatics
• /
• 제28권1_2호
• /
• pp.163-173
• /
• 2010

We consider the system $$\{{{-{\Delta}_pu\;=\;{\lambda}f(\upsilon),\;\;\;x\;{\in}\;{\Omega}, \atop -{\Delta}_q{\upsilon}\;=\;{\mu}g(u),\;\;\;x\;{\in}\;{\Omega},} \atop u\;=\;\upsilon\;=\;0,\;\;\;x\;{\in}\;{\partial\Omega},}$$ where ${\Delta}_pu\;=\;div(|{\nabla}_u|^{p-2}{\nabla}_u)$, ${\Delta}_{q{\upsilon}}\;=\;div(|{\nabla}_{\upsilon}|^{q-2}{\nabla}_{\upsilon})$, p, $q\;{\geq}\;2$, $\Omega$ is a ball in $\mathbf{R}^N$ with a smooth boundary $\partial\Omega$, $N\;{\geq}\;1$, $\lambda$, $\mu$ are positive parameters, and f, g are smooth functions that are negative at the origin and f(x) ~ $x^m$ g(x) ~ $x^n$ for x large for some m, $n\;{\geq}\;0$ with mn < (p - 1)(q - 1). We establish the existence and uniqueness of positive radial solutions when the parameters $\lambda$ and $\mu$ are large.

• Kyungpook Mathematical Journal
• /
• 제48권1호
• /
• pp.15-24
• /
• 2008

Let $N{\geq}1$ and p > 1. Let ${\Omega}$ be a domain of $\mathbb{R}^N$. In this article we shall establish Kato's inequalities for quasilinear degenerate elliptic operators of the form $A_pu$ = divA(x,$\nabla$u) for $u{\in}K_p({\Omega})$, ), where $K_p({\Omega})$ is an admissible class and $A(x,\xi)\;:\;{\Omega}{\times}\mathbb{R}^N{\rightarrow}\mathbb{R}^N$ is a mapping satisfying some structural conditions. If p = 2 for example, then we have $K_2({\Omega})\;= \;\{u\;{\in}\;L_{loc}^1({\Omega})\;:\;\partial_ju,\;\partial_{j,k}^2u\;{\in}\;L_{loc}^1({\Omega})\;for\;j,k\;=\;1,2,{\cdots},N\}$. Then we shall prove that $A_p{\mid}u{\mid}\;\geq$ (sgn u) $A_pu$ and $A_pu^+\;\geq\;(sgn^+u)^{p-1}\;A_pu$ in D'(${\Omega}$) with $u\;\in\;K_p({\Omega})$. These inequalities are called Kato's inequalities provided that p = 2. The class of operators $A_p$ contains the so-called p-harmonic operators $L_p\;=\;div(\mid{{\nabla}u{\mid}^{p-2}{\nabla}u)$ for $A(x,\xi)={\mid}\xi{\mid}^{p-2}\xi$.

• 대한수학회보
• /
• 제50권1호
• /
• pp.57-71
• /
• 2013

In this paper, we establish the existence of at least three solutions to a Navier boundary problem involving the ($p_1$, ${\cdots}$, $p_n$)-biharmonic systems. We use a variational approach based on a three critical points theorem due to Ricceri [B. Ricceri, A three critical points theorem revisited, Nonlinear Anal. 70 (2009), 3084-3089].

• Journal of applied mathematics & informatics
• /
• 제29권5_6호
• /
• pp.1229-1243
• /
• 2011

In this paper we consider a system of N-Laplacian elliptic equations with critical exponential growth. The existence and multiplicity results of solutions are obtained by a limit index method and Trudinger-Moser inequality.

• 대한수학회보
• /
• 제43권2호
• /
• pp.245-252
• /
• 2006

In this paper we study the uniform decay estimates of the energy for the nonlinear wave equation of Kirchhoff type $$y'(t)-M({\mid}{\nabla}y(t){\mid}^2){\triangle}y(t)\;+\;{\delta}y'(t)=f(t)$$ with the damping constant ${\delta} > 0$ in a bounded domain ${\Omega}\;{\subset}\;\mathbb{R}^n$.

• 대한수학회보
• /
• 제50권6호
• /
• pp.1817-1826
• /
• 2013

The existence of at least three weak solutions is established for a class of quasilinear elliptic equations involving the p(x)-biharmonic operators with Navier boundary value conditions. The technical approach is mainly based on a three critical points theorem due to Ricceri [11].

• 대한수학회지
• /
• 제31권3호
• /
• pp.401-416
• /
• 1994

We study the existence of an intermediate solution of nonlinear elliptic boundary value problems (BVP) of the form $$ (BVP) {\Delta u = f(x,u,\Delta u), in \Omega {Bu(x) = \phi(x), on \partial\Omega, $$ where $\Omega$ is a smooth bounded domain in $R^n, n \geq 1, and \partial\Omega \in C^{2,\alpha}, (0 < \alpha < 1), \Delta$ is the Laplacian operator, $\nabla u = (D_1u, D_2u, \cdots, D_nu)$ denotes the gradient of u and $$ Bu(x) = p(x)u(x) + q(x)\frac{d\nu}{du} (x), $$ where $\frac{d\nu}{du} denotes the outward normal derivative of u on $\partial\Omega$.

• 대한수학회지
• /
• 제32권2호
• /
• pp.329-339
• /
• 1995

Let us consider the Neumann problem for a quasilinear equation $$ (I_\varepsilon) {\varepsilon^m div($\mid$\nabla_u$\mid$^{m-2}\nabla_u) - u$\mid$u$\mid$^{m-2} + f(u) = 0 in \Omega {\frac{\partial\nu}{\partial u} = 0 on \partial\Omega. $$ where $1 < m < N, N \geq 2, \varepsilon > 0, \Omega$ is a smooth bounded domain in $R^n$ and $\nu$ is the unit outer normal vector to $\partial\Omega$.

• Journal of applied mathematics & informatics
• /
• 제11권1_2호
• /
• pp.185-199
• /
• 2003

In this paper, we apply finite element Galerkin method to a single-phase quasi-linear Stefan problem with a forcing term. We consider the existence and uniqueness of a semidiscrete approximation and optimal error estimates in $L_2$, $L_{\infty}$, $H_1$ and $H_2$ norms for semidiscrete Galerkin approximations we derived.

• Kyungpook Mathematical Journal
• /
• 제46권4호
• /
• pp.477-488
• /
• 2006

In this paper, the authors first classify all nonoscillatory solutions of equation (1) $${\Delta}^2|{\Delta}^2{_{y_n}}|^{{\alpha}-1}{\Delta}^2{_{y_n}}+q_n|y_{{\sigma}(n)}|^{{\beta}-1}y_{{\sigma}(n)}=o,\;n{\in}\mathbb{N}$$ into six disjoint classes according to their asymptotic behavior, and then they obtain necessary and sufficient conditions for the existence of solutions in these classes. Examples are inserted to illustrate the results.