• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.109-123
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• 2003

A tree is called starlike if it has exactly one vertex of degree greate. than two. In [4] it was proved that two starlike trees G and H are cospectral if and only if they are isomorphic. We prove here that there exist no two non-isomorphic Laplacian cospectral starlike trees. Further, let G be a simple graph of order n with vertex set V(G) : {1,2, …, n} and let H = {$H_1$, $H_2$, …, $H_{n}$} be a family of rooted graphs. According to [2], the rooted product G(H) is the graph obtained by identifying the root of $H_{i}$ with the i-th vertex of G. In particular, if H is the family of the paths $P_k_1,P_k_2,...P_k_2$ with the rooted vertices of degree one, in this paper the corresponding graph G(H) is called the sunlike graph and is denoted by G($k_1,k_2,...k_n$). For any $(x_1,x_2,...,x_n)\;\in\;{I_*}^n$, where $I_{*}$ = : {0,1}, let G$(x_1,x_2,...,x_n)$ be the subgraph of G which is obtained by deleting the vertices $i_1,i_2,...i_j\;\in\;V(G)\;(O\leq j\leq n)$, provided that $x_i_1=x_i_2=...=x_i_j=o.\;Let \;G[x_1,x_2,...x_n]$ be characteristic polynomial of G$(x_1,x_2,...,x_n)$, understanding that G[0,0,...,0] $\equiv$1. We prove that $G[k_1,k_2,...,k_n]-\sum_{x\in In}[{\prod_{\imath=1}}^n\;P_k_i+x_i-2(\lambda)](-1)...G[x_1,x_2,...,X_n]$ where x=($x_1,x_2,...,x_n$);G[$k_1,k_2,...,k_n$] and $P_n(\lambda)$ denote the characteristic polynomial of G($k_1,k_2,...,k_n$) and $P_n$, respectively. Besides, if G is a graph with $\lambda_1(G)\;\geq1$ we show that $\lambda_1(G)\;\leq\;\lambda_1(G(k_1,k_2,...,k_n))<\lambda_1(G)_{\lambda_1}^{-1}(G}$ for all positive integers $k_1,k_2,...,k_n$, where $\lambda_1$ denotes the largest eigenvalue.

• Journal of the Korean Mathematical Society
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• v.56 no.4
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• pp.857-867
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• 2019

Suppose that G = (V, E) is a connected locally finite graph with the vertex set V and the edge set E. Let ${\Omega}{\subset}V$ be a bounded domain. Consider the following quasilinear elliptic equation on graph G $$\{-{\Delta}_{pu}={\lambda}K(x){\mid}u{\mid}^{p-2}u+f(x,u),\;x{\in}{\Omega}^{\circ},\\u=0,\;x{\in}{\partial}{\Omega},$$ where ${\Omega}^{\circ}$ and ${\partial}{\Omega}$ denote the interior and the boundary of ${\Omega}$, respectively, ${\Delta}_p$ is the discrete p-Laplacian, K(x) is a given function which may change sign, ${\lambda}$ is the eigenvalue parameter and f(x, u) has exponential growth. We prove the existence and monotonicity of the principal eigenvalue of the corresponding eigenvalue problem. Furthermore, we also obtain the existence of a positive solution by using variational methods.

• Communications of the Korean Mathematical Society
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• v.29 no.1
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• pp.155-161
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• 2014

Let M be a complete Riemannian manifold and let N be a Riemannian manifold of non-positive sectional curvature. Assume that $Ric^M{\geq}-\frac{4(p-1)}{p^2}{\mu}_0$ at all $x{\in}M$ and Vol(M) is infinite, where ${\mu}_0$ > 0 is the infimum of the spectrum of the Laplacian acting on $L^2$-functions on M. Then any p-harmonic map ${\phi}:M{\rightarrow}N$ of finite p-energy is constant Also, we study Liouville type theorem for p-harmonic morphism.

• Journal of the Korean Mathematical Society
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• v.35 no.2
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• pp.465-490
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• 1998

In this paper, we consider the existence and asymptotic behavior of solutions of the following problem: $u_{tt}$ -(t, x) - (∥∇u(t, x)∥(equation omitted) ＋ ∥∇v(t, x) (equation omitted)$^{\gamma}$ $\Delta$u(t, x)＋$\delta$│ $u_{t}$ (t, x)│sup p-1/ $u_{t}$ (t, x) = $\mu$│u(t, x) $^{q-1}$u(t, x), x$\in$$\Omega$, t$\in$[0, T], $v_{tt}$ (t, x) - (∥∇uu(t, x) (equation omitted) ＋ ∥∇v(t, x) (equation omitted)sup ${\gamma}$/ $\Delta$v(t, x)＋$\delta$ │ $v_{t}$ (t, x)│sup p-1/ $u_{t}$ (t, x) = $\mu$ u(t, x) $^{q-1}$u(t, x), x$\in$$\Omega$, t$\in$[0, T], u(0, x) = $u_{0}$ (x), $u_{t}$ (0, x) = $u_1$(x), x$\in$$\Omega$, u(0, x) = $v_{0}$ (x), $v_{t}$ (0, x) = $v_1$(x), x$\in$$\Omega$, u│∂$\Omega$=v│∂$\Omega$=0 T > 0, q > 1, p $\geq$1, $\delta$ > 0, $\mu$ $\in$ R, ${\gamma}$ $\geq$ 1 and $\Delta$ is the Laplacian in $R^{N}$.X> N/.

• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1337-1358
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• 2018

In this paper, we study the solvability of the nonlinear Dirichlet problem with sum of the operators of independent non standard growths $$-div\({\mid}{\nabla}u{\mid}^{p_1(x)-2}{\nabla}u\)-\sum\limits^n_{i=1}D_i\({\mid}u{\mid}^{p_0(x)-2}D_iu\)+c(x,u)=h(x),\;{\in}{\Omega}$$ in a bounded domain ${\Omega}{\subset}{\mathbb{R}}^n$. Here, one of the operators in the sum is monotone and the other is weakly compact. We obtain sufficient conditions and show the existence of weak solutions of the considered problem by using monotonicity and compactness methods together.

• Nonlinear Functional Analysis and Applications
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• v.29 no.2
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• pp.333-360
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• 2024

This paper is concerned with the radial solutions of a nonlinear elliptic equation ∆_pu + |x|^𝑙₁ |u|^q₁-1 u + |x|^𝑙₂ |u|^q₂-1 u = 0, x ∈ ℝ^N, where p > 2, N ≥ 1, q₂ > q₁ ≥ 1, -p < 𝑙₂ < 𝑙₁ ≤ 0 and -N < 𝑙₂ < 𝑙₁ ≤ 0. We prove the existence of global solutions, we give their classification and we present the explicit behavior of positive solutions near the origin and infinity.

• Korean Journal of Mathematics
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• v.27 no.2
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• pp.437-444
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• 2019

For c > -1, let ${\nu}_c$ denote a weighted radial measure on ${\mathbb{C}}$ normalized so that ${\nu}_c(D)=1$. For $c_1,c_2>-1$ and $f{\in}L^1(D^2,\;{\nu}_{c_1}{\times}{\nu}_{c_2})$, we define the weighted Berezin transform $B_{c_1,c_2}f$ on $D^2$ by $$(B_{c_1,c_2})f(z,w)={\displaystyle{\smashmargin2{\int\nolimits_D}{\int\nolimits_D}}}f({\varphi}_z(x),\;{\varphi}_w(y))\;d{\nu}_{c_1}(x)d{\upsilon}_{c_2}(y)$$. This paper is about the space $M^p_{c_1,c_2}$ of function $f{\in}L^p(D^2,\;{\nu}_{c_1}{\times}{\nu}_{c_2})$ ) satisfying $B_{c_1,c_2}f=f$ for $1{\leq}p<{\infty}$. We find the identity operator on $M^p_{c_1,c_2}$ by using invariant Laplacians and we characterize some special type of functions in $M^p_{c_1,c_2}$.

• Journal of the Korean Mathematical Society
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• v.57 no.3
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• pp.747-775
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• 2020

In this work we investigate the nonlocal elliptic equation with critical Hardy-Sobolev exponents as follows, $$(P)\;\{(-{\Delta}_p)^su={\lambda}{\mid}u{\mid}^{q-2}u+{\frac{{\mid}u{\mid}^{p{^*_s}(t)-2}u}{{\mid}x{\mid}^t}}{\hspace{10}}in\;{\Omega},\\u=0{\hspace{217}}in\;{\mathbb{R}}^N{\backslash}{\Omega},$$ where Ω ⊂ ℝ^N is an open bounded domain with Lipschitz boundary, 0 < s < 1, λ > 0 is a parameter, 0 < t < sp < N, 1 < q < p < p^∗_s where $p^*_s={\frac{N_p}{N-sp}}$, $p^*_s(t)={\frac{p(N-t)}{N-sp}}$, are the fractional critical Sobolev and Hardy-Sobolev exponents respectively. The fractional p-laplacian (-∆_p)^su with s ∈ (0, 1) is the nonlinear nonlocal operator defined on smooth functions by $\displaystyle(-{\Delta}_p)^su(x)=2{\lim_{{\epsilon}{\searrow}0}}\int{_{{\mathbb{R}}^N{\backslash}{B_{\epsilon}}}}\;\frac{{\mid}u(x)-u(y){\mid}^{p-2}(u(x)-u(y))}{{\mid}x-y{\mid}^{N+ps}}dy$, x ∈ ℝ^N. The main goal of this work is to show how the usual variational methods and some analysis techniques can be extended to deal with nonlocal problems involving Sobolev and Hardy nonlinearities. We also prove that for some α ∈ (0, 1), the weak solution to the problem (P) is in C^1,α(${\bar{\Omega}}$).

• Kyungpook Mathematical Journal
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• v.64 no.1
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• pp.113-131
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• 2024

In this paper we give conditions for the existence of, and describe the asymtotic behavior of, radial positive solutions of the nonlinear elliptic equation of Emden-Fowler type with convection term ∆_p u + 𝛼|u|^q-1u + 𝛽x.∇(|u|^q-1u) = 0 for x ∈ ℝ^N, where p > 2, q > 1, N ≥ 1, 𝛼 > 0, 𝛽 > 0 and ∆_p is the p-Laplacian operator. In particular, we determine ${\lim}_{r{\rightarrow}}{\infty}\,r^{\frac{p}{q+1-p}}\,u(r)$ when $\frac{{\alpha}}{{\beta}}$ > N > p and $q\,{\geq}\,{\frac{N(p-1)+p}{N-p}}$.

• Journal of the Korean Mathematical Society
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• v.58 no.3
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• pp.703-722
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• 2021

Let G = (V, E) be a connected locally finite and weighted graph, ∆_p be the p-th graph Laplacian. Consider the p-th nonlinear equation -∆_pu + h|u|^p-2u = f(x, u) on G, where p > 2, h, f satisfy certain assumptions. Grigor'yan-Lin-Yang [24] proved the existence of the solution to the above nonlinear equation in a bounded domain Ω ⊂ V. In this paper, we show that there exists a strictly positive solution on the infinite set V to the above nonlinear equation by modifying some conditions in [24]. To the m-order differential operator 𝓛_m,p, we also prove the existence of the nontrivial solution to the analogous nonlinear equation.