• Journal of the Korean Mathematical Society
• /
• v.33 no.2
• /
• pp.291-307
• /
• 1996

In this paper, we shall show the existence of solutions of the following nonlinear partial differential equation $$ {^{divA(-\Delta u) = f(x, u, \Delta u) in \Omega}^{u = 0 on \partial\Omega} $$ where $f(x, u, \Delta u) = -u$\mid$\Delta u$\mid$^{p-2} + h, p \geq 2, h \in L^\infty$.

• Bulletin of the Korean Mathematical Society
• /
• v.49 no.6
• /
• pp.1179-1192
• /
• 2012

In this paper, we consider a doubly nonlinear parabolic partial differential equation $\frac{{\partial}{\beta}(u)}{{\partial}t}-{\Delta}_pu+f(x,t,u)=0$ in ${\Omega}{\times}[0,T]$, with Dirichlet boundary condition and initial data given. We prove the existence of a discrete approximate solution by means of the Rothe discretization in time method under some conditions on ${\beta}$, $f$ and $p$.

• Bulletin of the Korean Mathematical Society
• /
• v.48 no.6
• /
• pp.1169-1182
• /
• 2011

In this paper we study the existence of non-trivial weak solutions of the Neumann problem for quasilinear elliptic equations in the form $$-div(h(x){\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)+b(x){\mid}u{\mid}^{p-2}u=f(x,\;u),\;p{\geq}2$$ in an unbounded domain ${\Omega}{\subset}\mathbb{R}^N$, $N{\geq}3$, with sufficiently smooth bounded boundary ${\partial}{\Omega}$, where $h(x){\in}L_{loc}^1(\overline{\Omega})$, $\overline{\Omega}={\Omega}{\cup}{\partial}{\Omega}$, $h(x){\geq}1$ for all $x{\in}{\Omega}$. The proof of main results rely essentially on the arguments of variational method.

• Journal of the Chungcheong Mathematical Society
• /
• v.24 no.4
• /
• pp.639-663
• /
• 2011

Motivated by Agarwal and O'Regan ( Boundary value problems for general discrete systems on infinite intervals, Comput. Math. Appl. 33(1997)85-99), this article deals with the discrete type BVP of the infinite difference equations. The sufficient conditions to guarantee the existence of at least three positive solutions are established. An example is presented to illustrate the main results. It is the purpose of this paper to show that the approach to get positive solutions of BVPs by using multi-fixed-point theorems can be extended to treat BVPs for infinite difference equations. The strong Caratheodory (S-Caratheodory) function is defined in this paper.

• Journal of the Korean Mathematical Society
• /
• v.57 no.6
• /
• pp.1451-1470
• /
• 2020

We are concerned with the following elliptic equations: $$\{(-{\Delta})^s_pu={\lambda}f(x,u)\;{\text{in {\Omega}}},\\u=0\;{\text{on {\mathbb{R}}^N{\backslash}{\Omega}},$$ where λ are real parameters, (-∆)^s_p is the fractional p-Laplacian operator, 0 < s < 1 < p < + ∞, sp < N, and f : Ω × ℝ → ℝ satisfies a Carathéodory condition. By applying abstract critical point results, we establish an estimate of the positive interval of the parameters λ for which our problem admits at least one or two nontrivial weak solutions when the nonlinearity f has the subcritical growth condition. In addition, under adequate conditions, we establish an apriori estimate in L^∞(Ω) of any possible weak solution by applying the bootstrap argument.

• Journal of applied mathematics & informatics
• /
• v.24 no.1_2
• /
• pp.209-220
• /
• 2007

In this paper, the existence and multiplicity of solutions of three-point p-Laplacian boundary value problems at resonance with one-sided Nagumo condition are studied by using degree theory and upper and lower solutions method. Some known results are improved.

• Communications of the Korean Mathematical Society
• /
• v.36 no.3
• /
• pp.447-463
• /
• 2021

In this paper we study the existence and long-time behavior of weak solutions to a class of quasilinear degenerate parabolic equations involving weighted p-Laplacian operators with a new class of nonlinearities. First, we prove the existence and uniqueness of weak solutions by combining the compactness and monotone methods and the weak convergence techniques in Orlicz spaces. Then, we prove the existence of global attractors by using the asymptotic a priori estimates method.

• Bulletin of the Korean Mathematical Society
• /
• v.40 no.2
• /
• pp.195-205
• /
• 2003

For the boundary value problem (BVP) of second order functional differential equations for the one-dimensional $\rho$-Laplaclan: ($\Phi$$_{\rho}$(y'))'(t)+m(t)f(t, $y^{t}$ )=0 for t$\in$[0,1], y(t)=η(t) for t$\in$[-$\sigma$,0], y'(t)=ξ(t) for t$\in$[1,d], suitable conditions are imposed on f(t, $y^{t}$ ) which yield the existence of at least two positive solutions. Our result generalizes the main result of Avery, Chyan and Henderson.

• Journal of the Korean Mathematical Society
• /
• v.47 no.1
• /
• pp.135-163
• /
• 2010

Motivated by [Science in China (Ser. A Mathematics) 36 (2006), no. 7, 721?732], this article deals with the following discrete type BVP $\LARGE\left\{{{\;{\Delta}[{\phi}({\Delta}x(n))]\;+\;f(n,\;x(n\;+\;1),{\Delta}x(n),{\Delta}x(n + 1))\;=\;0,\;n\;{\in}\;[0,N],}}\\{\;{x(0)-{\sum}^m_{i=1}{\alpha}_ix(n_i) = A,}}\\{\;{x(N+2)-\;{\sum}^m_{i=1}{\beta}_ix(n_i)\;=\;B.}}\right.$ The sufficient conditions to guarantee the existence of at least three positive solutions of the above multi-point boundary value problem are established by using a new fixed point theorem obtained in [5]. An example is presented to illustrate the main result. It is the purpose of this paper to show that the approach to get positive solutions of BVPs by using multifixed-point theorems can be extended to treat nonhomogeneous BVPs. The emphasis is put on the nonlinear term f involved with the first order delta operator ${\Delta}$x(n).

• Bulletin of the Korean Mathematical Society
• /
• v.46 no.5
• /
• pp.999-1011
• /
• 2009

In this paper, we are concerned with the following eigenvalue problems of m-point boundary value problem for p-Laplacian dynamic equation on time scales $(\varphi_p(u^{\Delta}(t)))^\nabla+{\lambda}h(t)f(u(t))=0,\;t\in(0,T)$, $u(0)=0,\varphi_p(u^{\Delta}(T))=\sum\limits_{i=1}^{m-2}a_i\varphi_p(u^{\Delta}(\xi_i))$, where $\varphi_p(u)=|u|^{p-2}$u, p > 1 and $\lambda$ > 0 is a real parameter. Under certain assumptions, some new results on existence of one or two positive solution and nonexistence are obtained for $\lambda$ evaluated in different intervals. Our work develop and improve many known results in the literature even for the continual case. In doing so the usual restriction that $f_0=lim_{u{\rightarrow}0}+f(u)/\varphi_p(u)$ and $f_\infty = lim_{u{\rightarrow}{\infty}}f(u)/\varphi_p({u})$ exist is removed. As an applications, an example is given to illustrate the main results obtained.