• 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/.

• Communications of the Korean Mathematical Society
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• v.10 no.1
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• pp.49-55
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• 1995

This paper is the third of a series in which smoothness properties of function in several variables are discussed. The germ of the whole theory was laid in the works by Folland and Stein [4]. On nilpotent Lie groups, they difined analogues of the classical $L^p$ Sobolev or potential spaces in terms of fractional powers of sub-Laplacian, L and extended several basic theorems from the Euclidean theory of differentaiability to these spaces: interpolation properties, boundedness of singular integrals,..., and imbeding theorems. In this paper we study the analogue to the extension theorem for the Folland-Stein spaces. The analogue to Stein's restriction theorem were studied by M. Mekias [5] and Y.M. Kim [6]. First, we have the space of Bessel potentials on the Heisenberg group introduced by Folland [4].

• Journal of the Korean Mathematical Society
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• v.54 no.5
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• pp.1573-1594
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• 2017

This paper deals with the existence and energy estimates of solutions for a class of degenerate nonlocal problems involving sub-linear nonlinearities, while the nonlinear part of the problem admits some hypotheses on the behavior at origin or perturbation property. In particular, for a precise localization of the parameter, the existence of a non-zero solution is established requiring the sublinearity of nonlinear part at origin and infinity. We also consider the existence of solutions for our problem under algebraic conditions with the classical Ambrosetti-Rabinowitz. In what follows, by combining two algebraic conditions on the nonlinear term which guarantees the existence of two solutions as well as applying the mountain pass theorem given by Pucci and Serrin, we establish the existence of the third solution for our problem. Moreover, concrete examples of applications are provided.

• Honam Mathematical Journal
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• v.43 no.2
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• pp.289-304
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• 2021

In this paper, we establish several geometric characterizations focusing on the relationship between the squared norm of the second fundamental form and the warping function of SCR-lightlike warped product submanifolds in an indefinite Kaehler manifold. In particular, we find an estimate for the squared norm of the second fundamental form h in terms of the Hessian of the warping function λ for SCR-lightlike warped product submanifolds of an indefinite complex space form. Consequently, we derive an optimal inequality, namely $${\parallel}h{\parallel}^2{\geq}2q\{{\Delta}(ln{\lambda})+{\parallel}{\nabla}(ln{\lambda}){\parallel}^2+\frac{c}{2}p\}$$, for SCR-lightlike warped product submanifolds in an indefinite complex space form. We also provide one non-trivial example for this class of warped products in an indefinite Kaehler manifold.

• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1255-1268
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• 2022

We consider the Schrödinger type operator 𝓛 = (-𝚫_ℍⁿ)² + V² on the Heisenberg group ℍⁿ, where 𝚫_ℍⁿ is the sub-Laplacian and the non-negative potential V belongs to the reverse Hölder class RH_s for s ≥ Q/2 and Q ≥ 6. We shall establish the (L^p, L^q) estimates for the Riesz transforms T_𝛼,𝛽,j = V^2𝛼𝛁^j_ℍⁿ𝓛^-𝛽, j = 0, 1, 2, 3, where 𝛁_ℍⁿ is the gradient operator on ℍⁿ, 0 < α ≤ 1-j/4, j/4 < 𝛽 ≤ 1, and 𝛽 - 𝛼 ≥ j/4.

• Nonlinear Functional Analysis and Applications
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• v.28 no.3
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• pp.601-612
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• 2023

The existence of solution of the fractional order differential equations is very important mathematical field. Thus, in this work, we discuss, under some hypothesis, the existence of a positive solution for the nonlinear fourth order fractional boundary value problem which includes the p-Laplacian transform. The proposed method in the article is based on the fixed point theorem. More precisely, Krasnosilsky's theorem on a fixed point and some properties of the Green's function were used to study the existence of a solution for fourth order fractional boundary value problem. The main theoretical result of the paper is explained by example.

• 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 applied mathematics & informatics
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• v.28 no.5_6
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• pp.1171-1184
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• 2010

In this paper, by using fixed point theorems in cones, the existence of positive solutions is considered for nonlinear m-point boundary value problem for the following second-order dynamic equations on time scales $({\phi}(u^{\Delta}))^{\nabla}+a(t)f(t,\;u(t))=0$, t $\in$ (0, T), $u(0)=\sum\limits^{m-2}_{i=1}a_iu(\xi_i)$, $\phi(u^{\Delta}(T))=\sum\limits^{m-2}_{i=1}b_i{\phi}(u^{\Delta}(\xi_i))$, where $\phi$ : R $\rightarrow$ R is an increasing homeomorphism and positive homomorphism and ${\phi}(0)=0$. In [27], we obtained the existence results of the above problem for an increasing homeomorphism and positive homomorphism with sign changing nonlinearity. The purpose of this paper is to supplement with a result in the case when the nonlinear term f is nonnegative, and the most point we must point out for readers is that there is only the p-Laplacian case for increasing homeomorphism and positive homomorphism due to the sign restriction. As an application, one example to demonstrate our results are given.