• 충청수학회지
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• 제37권4호
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• pp.181-188
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• 2024

Many phenomena occurring in the natural environment have been modeled and studied using mathematical methods. In particular, investigating the existence and multiplicity of positive solutions, which represent the coexistence of equations, is always an intriguing research topic. To study the multiplicity of these positive solutions, it is necessary to analyze the behavior of positive solutions concerning a given parameter in the equation. In this research, we present a semi-linear partial differential equation to explain a series of natural phenomena through the study of positive solution behavior. We aim to investigate the existence and multiplicity of positive solutions that are not constant under homogeneous Neumann boundary conditions. Specifically, we apply the Mountain Pass theorem to demonstrate the existence of positive solutions for this equation, and further, we use the Leray-Schauder degree theory to explore sufficient conditions for the existence of two or more positive solutions.

• Journal of applied mathematics & informatics
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• 제20권1_2호
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• pp.491-511
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• 2006

The existence of the admissible solution for conservation laws of trilinear type occurring material sciences was proved by Abeyaratne and Knowles. LeFloch proved the existence of admissible solutions of conservation laws of this type via Glimm's method. In this paper we introduce a front tracking solution and prove the existence of the front tracking solution. We also investigate the admissibility of solutions via the Front Tracking Method.

• 충청수학회지
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• 제33권1호
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• pp.19-32
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• 2020

We discuss the coexistence of positive solutions to certain strongly-coupled predator-prey elliptic systems under the homogeneous Dirichlet boundary conditions. The sufficient condition for the existence of positive solutions is expressed in terms of the spectral property of differential operators of nonlinear Schrödinger type which reflects the influence of the domain and nonlinearity in the system. Furthermore, applying the obtained results, we investigate the sufficient conditions for the existence of positive solutions of a predator-prey system with degenerate diffusion rates.

• Korean Journal of Mathematics
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• 제17권1호
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• pp.103-112
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• 2009

We show the existence of the solutions for the following nonlinear elliptic problem under the Dirichlet boundary condition. To show the existence of the solutions we use the variational formulation.

• Journal of applied mathematics & informatics
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• 제33권5_6호
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• pp.749-757
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• 2015

In this paper, we prove existence of radial positive solutions for the following boundary value problem

• 충청수학회지
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• 제12권1호
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• pp.73-93
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• 1999

In this paper we prove the existence of solutions for the nonlinear hyperbolic system of conservation laws in several space variables provided that the initial data are very close to each other. We also give an example and discuss the existence of solutions for large initial data.

• 충청수학회지
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• 제34권1호
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• pp.27-35
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• 2021

We study the existence of positive T-periodic solutions of ratio-dependent predator-prey systems with time periodic and spatially dependent coefficients. The fixed point theorem by H. Amann is used to obtain necessary and sufficient conditions for the existence of positive T-periodic solutions.

• Journal of applied mathematics & informatics
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• 제28권3_4호
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• pp.625-637
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• 2010

In this paper, we are concerned with the quasilinear elliptic systems with boundary blow-up conditions in a smooth bounded domain. Using the method of lower and upper solutions, we prove the sufficient conditions for the existence of the positive solution. Our main results are new and extend the results in [Mingxin Wang, Lei Wei, Existence and boundary blow-up rates of solutions for boundary blow-up elliptic systems, Nonlinear Analysis(In Press)].

• 대한수학회지
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• 제54권5호
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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.

• East Asian mathematical journal
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• 제34권5호
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• pp.651-660
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• 2018

This paper concerned the existence of positive solutions to the second order differential systems with strongly coupled integral boundary value conditions. By using Krasnoselskii fixed point theorem, we prove the existence of positive solutions according to the parameters under the proper nonlinear growth conditions.