• Korean Journal of Mathematics
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• v.20 no.1
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• pp.107-116
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• 2012

We investigate the multiple solutions for a class of the elliptic system with the singular potential nonlinearity. We obtain a theorem which shows the existence of the solution for a class of the elliptic system with singular potential nonlinearity and Dirichlet boundary condition. We obtain this result by using variational method and critical point theory.

• East Asian mathematical journal
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• v.38 no.5
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• pp.603-610
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• 2022

In [6, 7] they introduced a new finite element method for accurate numerical solutions of Poisson equations with corner singularities. They consider the Poisson equations with homogeneous boundary conditions, compute the finite element solutions using standard FEM and use the extraction formula to compute the stress intensity factor(s), then they posed new PDE with a regular solution by imposing the nonhomogeneous boundary condition using the computed stress intensity factor(s), which converges with optimal speed. From the solution they could get an accurate solution just by adding the singular part. They considered a partial differential equation with the input function f ∈ L²(Ω). In this paper we consider a PDE with the input function f ∈ H¹(Ω) and find the corresponding singular and dual singular functions. We also induce the corresponding extraction formula which are the basic element for the approach.

• Journal of applied mathematics & informatics
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• v.27 no.3_4
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• pp.501-515
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• 2009

A second order nonlinear ordinary differential equation is obtained by solving the initial-boundary value problem of a class of elas-todynamics equations, which models the radially symmetric motion of a incompressible hyper-elastic solid sphere under a suddenly applied surface tensile load. Some new conclusions are presented. All existence conditions of nonzero solutions of the ordinary differential equation, which describes cavity formation and motion in the interior of the sphere, are presented. It is proved that the differential equation has singular periodic solutions only when the surface tensile load exceeds a critical value, in this case, a cavity would form in the interior of the sphere and the motion of the cavity with time would present a class of singular periodic oscillations, otherwise, the sphere remains a solid one. To better understand the results obtained in this paper, the modified Varga material is considered simultaneously as an example, and numerical simulations are given.

• The Pure and Applied Mathematics
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• v.21 no.3
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• pp.147-163
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• 2014

In this paper, we study the existence of positive solutions for singular impulsive differential equations with integral boundary conditions $$\{u^{{\prime}{\prime}}(t)+q(t)f(t,u(t),u^{\prime}(t))=0,\;t{\in}\mathbb{J}^{\prime},\\{\Delta}u(t_k)=I_k(u(t_k),u^{\prime}(t_k)),\;k=1,2,{\cdots},p,\\{\Delta}u^{\prime}(t_k)=-L_k(u(t_k),u^{\prime}(t_k)),\;k=1,2,{\cdots},p,\\u=(0)={\int}_{0}^{1}g(t)u(t)dt,\;u^{\prime}=0,$$) where the nonlinearity f(t, u, v) may be singular at v = 0. The proof is based on the theory of Leray-Schauder degree, together with a truncation technique. Some recent results in the literature are generalized and improved.

• Journal of the Korean Mathematical Society
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• v.39 no.3
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• pp.439-460
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• 2002

We prove the multiplicity of ordered positive radial solutions for a semilinear elliptic problem defined on an exterior domain. The key argument is to prove the existence of the third solution in presence of two known solutions. For this, we obtain some partial results related to three solutions theorem for certain singular boundary value problems. Proof are mainly based on the upper and lower solutions method and degree theory.

• Journal of the Korean Mathematical Society
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• v.54 no.2
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• pp.359-380
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• 2017

This paper is concerned with a kind of boundary value problem for singular second order differential systems with Laplacian operators. Using a multiple fixed point theorem, sufficient conditions to guarantee the existence of at least three positive solutions of this kind of boundary value problem are established. An example is presented to illustrate the main results.

• East Asian mathematical journal
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• v.33 no.3
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• pp.323-331
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• 2017

In this paper, we consider the existence of positive solutions for eigenvalue problems of nonlinear fractional differential equations with singular weights. We give various conditions on f and apply Krasnoselskii's Cone Fixed Point Theorem. As a result, we obtain several existence and nonexistence results corresponding to ${\lambda}$ in certain intervals.

• Journal of the Korean Mathematical Society
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• v.34 no.1
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• pp.247-255
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• 1997

In this paper, we are concerned with the existence of positive solutions for the boundary value problems of the form; $$(1) u"(t) + q(t)g(u(t)) = 0, 0 < t < 1 $$ $$ u(0) = 0 = u(1), $$ where q is singular at 0 and /or 1.or 1.

• Nonlinear Functional Analysis and Applications
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• v.29 no.2
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• pp.419-434
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• 2024

This paper is devoted to the existence of solutions for Kirchhoff type equations with singular nonlinearities, sub-critical and critical exponent. By using the Nehari manifold and Maximum principle theorem, the existence of at least two distinct positive solutions is obtained.

• East Asian mathematical journal
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• v.38 no.1
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• pp.33-41
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• 2022

In this paper, we study singular Dirichlet boundary value problems involving ϕ-Laplacian. Using fixed point index theory, the existence of positive solutions is established under the assumption that the nonlinearity f = f(u) has a positive falling zero and is either superlinear or sublinear at u = 0.