• Korean Journal of Mathematics
• /
• v.19 no.4
• /
• pp.423-436
• /
• 2011

We obtain a theorem that shows the existence of multiple solutions for the mixed type nonlinear elliptic equation with Dirichlet boundary condition. Here the nonlinear part contain the jumping nonlinearity and the subcritical growth nonlinearity. We first show the existence of a positive solution and next find the second nontrivial solution by applying the variational method and the mountain pass method in the critical point theory. By investigating that the functional I satisfies the mountain pass geometry we show the existence of at least two nontrivial solutions for the equation.

• Journal of the Korean Mathematical Society
• /
• v.59 no.5
• /
• pp.911-926
• /
• 2022

Let u be a function on a locally finite graph G = (V, E) and Ω be a bounded subset of V. Let 𝜀 > 0, p > 2 and 0 ≤ λ < λ₁(Ω) be constants, where λ₁(Ω) is the first eigenvalue of the discrete Laplacian, and h : V → ℝ be a function satisfying h ≥ 0 and $h{\not\equiv}0$. We consider a perturbed Yamabe equation, say $$\{\begin{array}{lll}-{\Delta}u-{\lambda}u={\mid}u{\mid}^{p-2}u+{\varepsilon}h,&&\text{ in }{\Omega},\\u=0,&&\text{ on }{\partial}{\Omega},\end{array}$$ where Ω and ∂Ω denote the interior and the boundary of Ω, respectively. Using variational methods, we prove that there exists some positive constant 𝜀₀ > 0 such that for all 𝜀 ∈ (0, 𝜀₀), the above equation has two distinct solutions. Moreover, we consider a more general nonlinear equation $$\{\begin{array}{lll}-{\Delta}u=f(u)+{\varepsilon}h,&&\text{ in }{\Omega},\\u=0,&&\text{ on }{\partial}{\Omega},\end{array}$$ and prove similar result for certain nonlinear term f(u).

• Journal of the Chungcheong Mathematical Society
• /
• v.20 no.4
• /
• pp.551-560
• /
• 2007

We prove the existence of solutions for the system of the nonlinear biharmonic equations with Dirichlet boundary condition $$\{^{-{\Delta}^2u-c{\Delta}u+{\gamma}(bu^+-av^-)=s{\phi}_1\;in\;{\Omega},\;}_{-{\Delta}^2u-c{\Delta}u+{\delta}(bu^+-av^-)=s{\phi}_1\;in\;{\Omega}}$$, where $u^+$ = max{u, 0}, ${\Delta}^2$ denotes the biharmonic operator and ${\phi}_1$ is the positive eigenfunction of the eigenvalue problem $-{\Delta}$ with Dirichlet boundary condition.

• Korean Journal of Mathematics
• /
• v.17 no.4
• /
• pp.495-505
• /
• 2009

By means of Green function and fixed point theorem related with cone theoretic method we show that there exist multiple nonnegative solutions of a Dirichlet problem $$\array{-[p(t)x^{\prime}(t)]^{\prime}={\lambda}q(t)f(x(t)),\;t{\in}I=[0,\;T]\\x(0)=0=x(T)}$$, and a mixed problem $$\array{-[p(t)x^{\prime}(t)]^{\prime}={\mu}q(t)f(x(t)),\;t{\in}I=[0,\;T]\\x^{\prime}(0)=0=x(T)}$$, where ${\lambda}$ and ${\mu}$ are positive parameters.

• Korean Journal of Mathematics
• /
• v.16 no.3
• /
• pp.389-402
• /
• 2008

We show the existence of at least two solutions for a class of systems of the critical growth nonlinear suspension bridge equations with Dirichlet boundary condition and periodic condition. We first show that the system has a positive solution under suitable conditions, and next show that the system has another solution under the same conditions by the linking arguments.

• Journal of the Chungcheong Mathematical Society
• /
• v.29 no.2
• /
• pp.187-220
• /
• 2016

This paper is concerned with a kind of non-homogeneous boundary value problems for singular second order differential systems with Laplacian operators. Using multiple fixed point theorems, sufficient conditions to guarantee the existence of at least three solutions of this kind of boundary value problems are established. An example is presented to illustrate the main results.

• 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 applied mathematics & informatics
• /
• v.35 no.1_2
• /
• pp.121-130
• /
• 2017

This study is concerned with the existence of positive solution for the following nonlinear elliptic system $$\{-M_1(\int_{\Omega}{\mid}x{\mid}^{-ap}{\mid}{\nabla}u{\mid}^pdx)div({\mid}x{\mid}^{-ap}{\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)\\{\hfill{120}}={\mid}x{\mid}^{-(a+1)p+c_1}\({\alpha}_1A_1(x)f(v)+{\beta}_1B_1(x)h(u)\),\;x{\in}{\Omega},\\-M_2(\int_{\Omega}{\mid}x{\mid}^{-bq}{\mid}{\nabla}v{\mid}^qdx)div({\mid}x{\mid}^{-bq}{\mid}{\nabla}v{\mid}^{q-2}{\nabla}v)\\{\hfill{120}}={\mid}x{\mid}^{-(b+1)q+c_2}\({\alpha}_2A_2(x)g(u)+{\beta}_2B_2(x)k(v)\),\;x{\in}{\Omega},\\{u=v=0,\;x{\in}{\partial}{\Omega},$$ where ${\Omega}$ is a bounded smooth domain of ${\mathbb{R}}^N$ with $0{\in}{\Omega}$, 1 < p, q < N, $0{\leq}a$ < $\frac{N-p}{p}$, $0{\leq}b$ < $\frac{N-q}{q}$ and ${\alpha}_i,{\beta}_i,c_i$ are positive parameters. Here $M_i,A_i,B_i,f,g,h,k$ are continuous functions and we discuss the existence of positive solution when they satisfy certain additional conditions. Our approach is based on the sub and super solutions method.

• Communications of the Korean Mathematical Society
• /
• v.27 no.3
• /
• pp.557-564
• /
• 2012

This study concerns the existence of positive solution for the following nonlinear system $$\{-div(|x|^{-ap}|{\nabla}u|^{p-2}{\nabla}u)=|x|^{-(a+1)p+c_1}({\alpha}_1f(v)+{\beta}_1h(u)),x{\in}{\Omega},\\-div(|x|^{-bq}|{\nabla}v|q^{-2}{\nabla}v)=|x|^{-(b+1)q+c_2}({\alpha}_2g(u)+{\beta}_2k(v)),x{\in}{\Omega},\\u=v=0,x{\in}{\partial}{\Omega}$$, where ${\Omega}$ is a bounded smooth domain of $\mathbb{R}^N$ with $0{\in}{\Omega}$, 1 < $p,q$ < N, $0{{\leq}}a<\frac{N-p}{p}$, $0{{\leq}}b<\frac{N-q}{q}$ and $c_1$, $c_2$, ${\alpha}_1$, ${\alpha}_2$, ${\beta}_1$, ${\beta}_2$ are positive parameters. Here $f,g,h,k$ : $[0,{\infty}){\rightarrow}[0,{\infty})$ are nondecresing continuous functions and $$\lim_{s{\rightarrow}{\infty}}\frac{f(Ag(s)^{\frac{1}{q-1}})}{s^{p-1}}=0$$ for every A > 0. We discuss the existence of positive solution when $f,g,h$ and $k$ satisfy certain additional conditions. We use the method of sub-super solutions to establish our results.

• Communications of the Korean Mathematical Society
• /
• v.24 no.2
• /
• pp.281-290
• /
• 2009

The famous Guo-Krasnosel'skii fixed point theorems concerning cone expansion and compression of norm type and order type are extended, respectively. As an application, the existence of multiple positive solutions for systems of Hammerstein type integral equations is considered.