• East Asian mathematical journal
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• 제34권5호
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• pp.703-713
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• 2018

Let $N{\geq}3$, $2^*=2N/(N-2)$ and $p{\in}(2,2^*)$. Our purpose in this paper is to consider multiple existence of solutions of problem $$-{\Delta}u-{\frac{\mu}{{\mid}x{\mid}^2}}+{\alpha}u={\mid}u{\mid}^{p-2}u+{\lambda}f\;u{\in}H^1({\mathbb{R}}^n)$$, where a, ${\lambda}$ > 0, ${\mu}{\in}(0,(N-2)^2/4)f{\in}H^{-1}({\mathbb{R}}^N)$, $f{\geq}0$ and $f{\neq}0$.

• 대한수학회보
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• 제52권2호
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• pp.421-438
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• 2015

In this paper, we study the existence of classical and generalized solutions for nonlinear differential inclusions $x^{\prime}(t){\in}F(t,x(t))$ in Hilbert spaces in which the multifunction F on the right-hand side is hemicontinuous and satisfies the semimonotone condition or is condensing. Our existence results are obtained via the selection and fixed point methods by reducing the problem to an ordinary differential equation. We first prove the existence theorem in finite dimensional spaces and then we generalize the results to the infinite dimensional separable Hilbert spaces. Then we apply the results to prove the existence of the mild solution for semilinear evolution inclusions. At last, we give an example to illustrate the results obtained in the paper.

• Korean Journal of Mathematics
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• 제17권4호
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• pp.473-480
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• 2009

We prove the existence and multiplicity of nontrivial solutions for a fourth order problem ${\Delta}^2u+c{\Delta}u={\alpha}u-{\beta}(u+1)^-$ in ${\Omega}$, ${\Delta}u=0$ and $u=0$ on ${\partial}{\Omega}$, where ${\lambda}_1{\leq}c{\leq}{\lambda}_2$ (where $({\lambda}_i)_{i{\geq}1}$ is the sequence of the eigenvalues of $-{\Delta}$ in$H_0^1({\Omega})$) and ${\Omega}$ is a bounded open set in $R^N$ with smooth boundary ${\partial}{\Omega}$. The results are proved by applying minimax arguments and linking theory.

• 대한수학회보
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• 제57권2호
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• pp.331-343
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• 2020

This paper is concerned with the global behavior of components of radial nodal solutions of semilinear elliptic problems -Δv = λh(x, v) in Ω, v = 0 on ∂Ω, where Ω = {x ∈ R^N : r₁ < |x| < r₂} with 0 < r₁ < r₂, N ≥ 2. The nonlinear term is continuous and satisfies h(x, 0) = h(x, s₁(x)) = h(x, s₂(x)) = 0 for suitable positive, concave function s₁ and negative, convex function s₂, as well as sh(x, s) > 0 for s ∈ ℝ \ {0, s₁(x), s₂(x)}. Moreover, we give the intervals for the parameter λ which ensure the existence and multiplicity of radial nodal solutions for the above problem. For this, we use global bifurcation techniques to prove our main results.

• 대한수학회보
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• 제44권4호
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• pp.731-742
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• 2007

Let X be a Banach space, $A:D(A){\subset}X{\rightarrow}X$ the generator of a compact $C_0-semigroup\;S(t):X{\rightarrow}X,\;t{\geq}0$, D a locally closed subset in X, and $f:(a,b){\times}C([-q,0];X){\rightarrow}X$ a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order that D be a viable domain of the semi linear differential equation of retarded type $$u#(t)=Au(t)+f(t,u_t),\;t{\in}[t_0,\;t_0+T],{u_t}_0={\phi}{\in}C([-q,0];X)$$ is the tangency condition $$\limits_{h{\downarrow}0}^{lim\;inf\;h^{-1}d(S(h)v(0)+hf(t,v);D)=0}$$ for almost every $t{\in}(a,b)$ and every $v{\in}C([-q,0];X)\;with\;v(0){\in}D$.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제24권2호
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• pp.229-242
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• 2020

Functional fractional differential inclusions with impulse effects in general Banach spaces are studied. We discuss the situation when the semigroup generated by the linear part is equicontinuous and the multifunction is Caratheodory. First, we define the PC-mild solutions for functional fractional semilinear impulsive differential inclusions. We then prove the existence of PC-mild solutions for such inclusions by using the fixed point theorem, multivalued properties and applications of NCHM (noncompactness Hausdorff measure). Eventually, we enhance the acquired results by giving an example.

• 대한수학회논문집
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• 제38권1호
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• pp.137-151
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• 2023

In this paper we prove the existence of nontrivial weak solutions to the boundary value problem -G₁u = u³ + f(x, y, u) in Ω, u ≥ 0 in Ω, u = 0 on ∂Ω, where Ω is a bounded domain with smooth boundary in ℝ³, G₁ is a Grushin type operator, and f(x, y, u) is a lower order perturbation of u³ with f(x, y, 0) = 0. The nonlinearity involved is of critical exponent, which differs from the existing results in [11, 12].

• 대한기계학회:학술대회논문집
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• 대한기계학회 2000년도 춘계학술대회논문집A
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• pp.591-595
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• 2000

Sound absorbing characteristics of poroelastic materials is known to be greatly affected by high intensity acoustic waves. However, this effect has not been considered yet. In this study, the extended semilinear model based on Biot's theory for the porous materials and the characteristics of nonlinear waves in poroelastic sound absorbing materials were introduced. The expressions for the finite-amplitude acoustic plane waves were presented. By combining each nonlinear wave with appropriate matching conditions, we could investigate the effects of finite-amplitude acoustic waves on absorption characteristics of poroelastic materials. In the most ideal case considered in this paper, the absorption coefficient was found to become larger than that of linear incident waves. It was shown that the absorption coefficient became greater especially as frequency goes higher and as distance from the source goes larger. These phenomena may be inferred to result from 'dissipation effects due to nonlinearity'. This finding may have important implications for high intensity noise control.