• East Asian mathematical journal
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• v.34 no.5
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• pp.549-570
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• 2018

We establish existence and bifurcation of global positive solutions for parametrized nonhomogeneous elliptic equations involving critical Sobolev-Hardy exponents and Hardy terms. The main approach to the problem is the variational method.

• Journal of the Korean Mathematical Society
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• v.56 no.2
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• pp.475-484
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• 2019

We study the set of critical exponents of discrete groups acting on regular trees. We prove that for every real number ${\delta}$ between 0 and ${\frac{1}{2}}\;{\log}\;q$, there is a discrete subgroup ${\Gamma}$ acting without inversion on a (q+1)-regular tree whose critical exponent is equal to ${\delta}$. Explicit construction of edge-indexed graphs corresponding to a quotient graph of groups are given.

• Communications of the Korean Mathematical Society
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• v.14 no.1
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• pp.81-97
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• 1999

A sufficient condition for pseudo-Laplacian equations involving critical Sobolev exponents to have positive solutions is established.

• East Asian mathematical journal
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• v.30 no.3
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• pp.335-353
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• 2014

We establish multiple extence of positive solutions for parameterized nonhomogeneous elliptic equations involving critical Sobolev exponent. The approach to the problem is variational method.

• Journal of the Korean Mathematical Society
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• v.44 no.5
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• pp.1051-1063
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• 2007

The aim of this note is to advertise a method which turns out to be powerful enough to be used successfully in problems which are apparently unrelated. It is based on a modification of a construction that we first introduced in [2].

• Bulletin of the Korean Mathematical Society
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• v.46 no.6
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• pp.1159-1173
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• 2009

This paper deals with the critical blow-up and extinction exponents for the non-Newton polytropic filtration equation. We reveals a fact that the equation admits two critical exponents $q_1,\;q_2\;{\in}\;(0,+{\infty})$) with $q_1\;{<}\;q_2$. In other words, when q belongs to different intervals (0, $q_1),\;(q_1,\;q_2),\;(q_2,+{\infty}$), the solution possesses complete different properties. More precisely speaking, as far as the blow-up exponent is concerned, the global existence case consists of the interval (0, $q_2$]. However, when q ${\in}\;(q_2,+{\infty}$), there exist both global solutions and blow-up solutions. As for the extinction exponent, the extinction case happens to the interval ($q_1,+{\infty}$), while for q ${\in}\;(0,\;q_1$), there exists a non-extinction bounded solution for any nonnegative initial datum. Moreover, when the critical case q = $q_1$ is concerned, the other parameter ${\lambda}$ will play an important role. In other words, when $\lambda$ belongs to different interval (0, ${\lambda}_1$) or (${\lambda}_1$,+${\infty}$), where ${\lambda}_1$ is the first eigenvalue of p-Laplacian equation with zero boundary value condition, the solution has completely different properties.

• Journal of the Korean Mathematical Society
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• v.54 no.3
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• pp.821-833
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• 2017

We get a result that shows the existence of infinitely many solutions for a class of the elliptic systems involving subcritical Sobolev exponents nonlinear terms with even functionals on the bounded domain with smooth boundary. We get this result by variational method and critical point theory induced from invariant subspaces and invariant functional.

• Korean Journal of Mathematics
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• v.23 no.2
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• pp.259-267
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• 2015

This paper is devoted to investigate the existence of the solutions for a class of the noncooperative elliptic system involving critical Sobolev exponents. We show the existence of the negative solution for the problem. We show the existence of the unique negative solution for the system of the linear part of the problem under some conditions, which is also the negative solution of the nonlinear problem. We also consider the eigenvalue problem of the matrix.

• Korean Journal of Mathematics
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• v.23 no.4
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• pp.619-630
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• 2015

This paper is devoted to investigate the multiple solutions for a class of the cooperative elliptic system involving subcritical Sobolev exponents on the bounded domain with smooth boundary. We first show the uniqueness and the negativity of the solution for the linear system of the problem via the direct calculation. We next use the variational method and the mountain pass theorem in the critical point theory.

• Journal of applied mathematics & informatics
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• v.28 no.5_6
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• pp.1143-1156
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• 2010

In this paper, we study the following p&q-Laplacian problem with critical Sobolev and Hardy exponents {$-{\Delta}_pu-{\Delta}_qu={\mu}\frac{{\mid}u{\mid}^{p^*(s)-2}u}{{\mid}x{\mid}^s}+{\lambda}f(x,\;u)$, in $\Omega$, u=0, on $\Omega$, where ${\Omega}\;{\subset}\;\mathbb{R}^{\mathbb{N}}$ is a bounded domain and ${\Delta}_ru=div({\mid}{\nabla}u{\mid}^{r-2}{\nabla}u)$ is the r-Laplacian of u. By using the variational method and concentration-compactness principle, we obtain the existence of infinitely many small solutions for above problem which are the complement of previously known results.