• Title/Summary/Keyword: nonlinear elliptic equations

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NEW EXACT TRAVELLING WAVE SOLUTIONS FOR SOME NONLINEAR EVOLUTION EQUATIONS

  • Lee, Youho;An, Jaeyoung;Lee, Mihye
    • Journal of the Chungcheong Mathematical Society
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    • v.24 no.2
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    • pp.359-370
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    • 2011
  • In this work, we obtain new solitary wave solutions for some nonlinear partial differential equations. The Jacobi elliptic function rational expansion method is used to establish new solitary wave solutions for the combined KdV-mKdV and Klein-Gordon equations. The results reveal that Jacobi elliptic function rational expansion method is very effective and powerful tool for solving nonlinear evolution equations arising in mathematical physics.

BOUNDARY VALUE PROBLEM FOR A CLASS OF THE SYSTEMS OF THE NONLINEAR ELLIPTIC EQUATIONS

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.17 no.1
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    • pp.67-76
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    • 2009
  • We show the existence of at least two nontrivial solutions for a class of the systems of the nonlinear elliptic equations with Dirichlet boundary condition under some conditions for the nonlinear term. We obtain this result by using the variational linking theory in the critical point theory.

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SOLVABILITY FOR A CLASS OF THE SYSTEMS OF THE NONLINEAR ELLIPTIC EQUATIONS

  • Jung, Tack-Sun;Choi, Q-Heung
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.1
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    • pp.1-10
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    • 2012
  • Let ${\Omega}$ be a bounded subset of $\mathbb{R}^n$ with smooth boundary. We investigate the solvability for a class of the system of the nonlinear elliptic equations with Dirichlet boundary condition. Using the mountain pass theorem we prove that the system has at least one nontrivial solution.

GLOBAL GRADIENT ESTIMATES FOR NONLINEAR ELLIPTIC EQUATIONS

  • Ryu, Seungjin
    • Journal of the Korean Mathematical Society
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    • v.51 no.6
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    • pp.1209-1220
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    • 2014
  • We prove global gradient estimates in weighted Orlicz spaces for weak solutions of nonlinear elliptic equations in divergence form over a bounded non-smooth domain as a generalization of Calder$\acute{o}$n-Zygmund theory. For each point and each small scale, the main assumptions are that nonlinearity is assumed to have a uniformly small mean oscillation and that the boundary of the domain is sufficiently flat.

Oscillation of Second Order Nonlinear Elliptic Differential Equations

  • Xu, Zhiting
    • Kyungpook Mathematical Journal
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    • v.46 no.1
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    • pp.65-77
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    • 2006
  • By using general means, some oscillation criteria for second order nonlinear elliptic differential equation with damping $$\sum_{i,j=1}^{N}D_i[a_{ij}(x)D_iy]+\sum_{i=1}^{N}b_i(x)D_iy+p(x)f(y)=0$$ are obtained. These criteria are of a high degree of generality and extend the oscillation theorems for second order linear ordinary differential equations due to Kamenev, Philos and Wong.

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EXTENDED JACOBIN ELLIPTIC FUNCTION METHOD AND ITS APPLICATIONS

  • Chen, Huaitang;Zhang, Hongqing
    • Journal of applied mathematics & informatics
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    • v.10 no.1_2
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    • pp.119-130
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    • 2002
  • An extended Jacobin elliptic function method is presented for constructing exact travelling wave solutions of nonlinear partial differential equations(PDEs) in a unified way. The main idea of this method is to take full advantage of the elliptic equation that Jacobin elliptic functions satisfy and use its solutions to replace Jacobin elliptic functions in Jacobin elliptic function method. It is interesting that many other methods are special cases of our method. Some illustrative equations are investigated by this means.

HYBRIDIZABLE DISCONTINUOUS GALERKIN METHOD FOR ELLIPTIC EQUATIONS WITH NONLINEAR COEFFICIENTS

  • MINAM, MOON
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.26 no.4
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    • pp.244-262
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    • 2022
  • In this paper, we analyze the hybridizable discontinuous Galerkin (HDG) method for second-order elliptic equations with nonlinear coefficients, which are used in many fields. We present the HDG method that uses a mixed formulation based on numerical trace and flux. Under assumptions on the nonlinear coefficient and H2-regularity for a dual problem, we prove that the discrete systems are well-posed and the numerical solutions have the optimal order of convergence as a mesh parameter. Also, we provide a matrix formulation that can be calculated using an iterative technique for numerical experiments. Finally, we present representative numerical examples in 2D to verify the validity of the proof of Theorem 3.10.

UNIQUENESS AND MULTIPLICITY OF SOLUTIONS FOR THE NONLINEAR ELLIPTIC SYSTEM

  • Jung, Tacksun;Choi, Q-Heung
    • Journal of the Chungcheong Mathematical Society
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    • v.21 no.1
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    • pp.139-146
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    • 2008
  • We investigate the uniqueness and multiplicity of solutions for the nonlinear elliptic system with Dirichlet boundary condition $$\{-{\Delta}u+g_1(u,v)=f_1(x){\text{ in }}{\Omega},\\-{\Delta}v+g_2(u,v)=f_2(x){\text{ in }}{\Omega},$$ where ${\Omega}$ is a bounded set in $R^n$ with smooth boundary ${\partial}{\Omega}$. Here $g_1$, $g_2$ are nonlinear functions of u, v and $f_1$, $f_2$ are source terms.

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SYMMETRY AND MONOTONICITY OF SOLUTIONS TO FRACTIONAL ELLIPTIC AND PARABOLIC EQUATIONS

  • Zeng, Fanqi
    • Journal of the Korean Mathematical Society
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    • v.58 no.4
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    • pp.1001-1017
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    • 2021
  • In this paper, we first apply parabolic inequalities and a maximum principle to give a new proof for symmetry and monotonicity of solutions to fractional elliptic equations with gradient term by the method of moving planes. Under the condition of suitable initial value, by maximum principles for the fractional parabolic equations, we obtain symmetry and monotonicity of positive solutions for each finite time to nonlinear fractional parabolic equations in a bounded domain and the whole space. More generally, if bounded domain is a ball, then we show that the solution is radially symmetric and monotone decreasing about the origin for each finite time. We firmly believe that parabolic inequalities and a maximum principle introduced here can be conveniently applied to study a variety of nonlocal elliptic and parabolic problems with more general operators and more general nonlinearities.