• Title, Summary, Keyword: mountain pass theorem

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MULTIPLICITY OF SOLUTIONS FOR A CLASS OF NON-LOCAL ELLIPTIC OPERATORS SYSTEMS

  • Bai, Chuanzhi
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.3
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    • pp.715-729
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    • 2017
  • In this paper, we investigate the existence and multiplicity of solutions for systems driven by two non-local integrodifferential operators with homogeneous Dirichlet boundary conditions. The main tools are the Saddle point theorem, Ekeland's variational principle and the Mountain pass theorem.

SINGULAR POTENTIAL BIHARMONIC PROBLEM

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.21 no.4
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    • pp.483-493
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    • 2013
  • We investigate the multiplicity of the solutions for a class of the system of the biharmonic equations with some singular potential nonlinearity. We obtain a theorem which shows the existence of the nontrivial weak solution for a class of the system of the biharmonic equations with singular potential nonlinearity and Dirichlet boundary condition. We obtain this result by using variational method and the generalized mountain pass theorem.

TWO JUMPING NONLINEAR TERMS AND A NONLINEAR WAVE EQUATION

  • Jung, Tacksun;Choi, Q-Heung
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.4
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    • pp.675-687
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    • 2009
  • We find the multiple nontrivial solutions of the equation of the form $u_{tt}-u_{xx}=b_1[(u+1)^{+}-1]+b_2[(u+2)^{+}-2]$ with Dirichlet boundary condition. Here we reduce this problem into a two-dimensional problem by using variational reduction method and apply the Mountain Pass theorem to find the nontrivial solutions.

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THREE NONTRIVIAL NONNEGATIVE SOLUTIONS FOR SOME CRITICAL p-LAPLACIAN SYSTEMS WITH LOWER-ORDER NEGATIVE PERTURBATIONS

  • Chu, Chang-Mu;Lei, Chun-Yu;Sun, Jiao-Jiao;Suo, Hong-Min
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.1
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    • pp.125-144
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    • 2017
  • Three nontrivial nonnegative solutions for some critical quasilinear elliptic systems with lower-order negative perturbations are obtained by using the Ekeland's variational principle and the mountain pass theorem.

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.

MULTIPLICITY RESULT OF THE SOLUTIONS FOR A CLASS OF THE ELLIPTIC SYSTEMS WITH SUBCRITICAL SOBOLEV EXPONENTS

  • JUNG, TACKSUN;CHOI, Q-HEUNG
    • Korean Journal of Mathematics
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    • v.23 no.4
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    • pp.619-630
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    • 2015
  • This paper is devoted to investigate the multiple solutions for a class of the cooperative elliptic system involving subcritical Sobolev exponents on the bounded domain with smooth boundary. We first show the uniqueness and the negativity of the solution for the linear system of the problem via the direct calculation. We next use the variational method and the mountain pass theorem in the critical point theory.

CRITICAL POINT THEORY AND AN ASYMMETRIC BEAM EQUATION WITH TWO JUMPING NONLINEAR TERMS

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.17 no.3
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    • pp.299-314
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    • 2009
  • We investigate the multiple nontrivial solutions of the asymmetric beam equation $u_{tt}+u_{xxxx}=b_1[{(u + 2)}^+-2]+b_2[{(u + 3)}^+-3]$ with Dirichlet boundary condition and periodic condition on t. We reduce this problem into a two-dimensional problem by using variational reduction method and apply the Mountain Pass theorem to find the nontrivial solutions of the equation.

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MULTIPLE SOLUTIONS RESULT FOR THE MIXED TYPE NONLINEAR ELLIPTIC PROBLEM

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.19 no.4
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    • pp.423-436
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    • 2011
  • We obtain a theorem that shows the existence of multiple solutions for the mixed type nonlinear elliptic equation with Dirichlet boundary condition. Here the nonlinear part contain the jumping nonlinearity and the subcritical growth nonlinearity. We first show the existence of a positive solution and next find the second nontrivial solution by applying the variational method and the mountain pass method in the critical point theory. By investigating that the functional I satisfies the mountain pass geometry we show the existence of at least two nontrivial solutions for the equation.

MOUNTAIN PASS GEOMETRY APPLIED TO THE NONLINEAR MIXED TYPE ELLIPTIC PROBLEM

  • Jung Tacksun;Choi Q-Heung
    • Korean Journal of Mathematics
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    • v.17 no.4
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    • pp.419-428
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    • 2009
  • We show the existence of at least one nontrivial solution of the homogeneous mixed type nonlinear elliptic problem. Here mixed type nonlinearity means that the nonlinear part contain the jumping nonlinearity and the critical growth nonlinearity. We first investigate the sub-level sets of the corresponding functional in the Soboles space and the linking inequalities of the functional on the sub-level sets. We next investigate that the functional I satisfies the mountain pass geometry in the critical point theory. We obtain the result by the mountain pass method, the critical point theory and variational method.

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NONLINEAR BIHARMONIC PROBLEM WITH VARIABLE COEFFICIENT EXPONENTIAL GROWTH TERM

  • Choi, Q-Heung;Jung, Tacksun
    • Korean Journal of Mathematics
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    • v.18 no.3
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    • pp.277-288
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    • 2010
  • We consider the nonlinear biharmonic equation with coefficient exponential growth term and Dirichlet boundary condition. We show that the nonlinear equation has at least one bounded solution under the suitable conditions. We obtain this result by the variational method, generalized mountain pass theorem and the critical point theory of the associated functional.