• 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 $M^p_{c_1,c_2}$.

• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.125-144
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• 2017

Three nontrivial nonnegative solutions for some critical quasilinear elliptic systems with lower-order negative perturbations are obtained by using the Ekeland's variational principle and the mountain pass theorem.

• The Pure and Applied Mathematics
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• v.18 no.3
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• pp.245-253
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• 2011

Some new results on the intersection property of all nonzero solutions of a class of planar systems of Li$\acute{e}$nard type with vertical isoclines are obtained. The results of this paper generalize some previous results on this field.

• Journal of applied mathematics & informatics
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• v.38 no.5_6
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• pp.407-414
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• 2020

In this article, a second-order Sturm-Liouville problem with impulsive effects and involving the one-dimensional p-Laplacian is considered. The existence of at least three weak solutions via variational methods and critical point theory is obtained.

• Journal of the Korean Mathematical Society
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• v.31 no.3
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• pp.401-416
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• 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 $\partial\Omega$.

• Journal of the Korean Mathematical Society
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• v.37 no.3
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• pp.339-358
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• 2000

Let be a bonded domain in N with smooth boundary . In this paper, we consider the existence of solutions of the following problem: (1.1)-div{} - + = , , , , , , where q > 1, p$\geq$1, $\delta$>0, , the Laplacian in N and is a positive function like as .

• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1297-1314
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• 2019

In this work, we investigate the following fractional boundary value problems $$\{_tD^{\alpha}_T({\mid}_0D^{\alpha}_t(u(t)){\mid}^{p-2}_0D^{\alpha}_tu(t))\\={\nabla}W(t,u(t))+{\lambda}g(t){\mid}u(t){\mid}^{q-2}u(t),\;t{\in}(0,T),\\u(0)=u(T)=0,$$ where ${\nabla}W(t,u)$ is the gradient of W(t, u) at u and $W{\in}C([0,T]{\times}{\mathbb{R}}^n,{\mathbb{R}})$ is homogeneous of degree r, ${\lambda}$ is a positive parameter, $g{\in}C([0,T])$, 1 < r < p < q and ${\frac{1}{p}}<{\alpha}<1$. Using the Fibering map and Nehari manifold, for some positive constant ${\lambda}_0$ such that $0<{\lambda}<{\lambda}_0$, we prove the existence of at least two non-trivial solutions

• 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.

• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1435-1458
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• 2018

In this paper, we would like to give an answer to Problem 1 below issued firstly in [17]. In fact, by imposing some conditions on the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced mean curvature flow considered here, we can obtain that the first eigenvalues of the Laplace and the p-Laplace operators are monotonic under this flow. Surprisingly, during this process, we get an interesting byproduct, that is, without any complicate constraint, we can give lower bounds for the first nonzero closed eigenvalue of the Laplacian provided additionally the second fundamental form of the initial hypersurface satisfies a pinching condition.

• 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.