• Title, Summary, Keyword: Nehari manifold

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THE NEHARI MANIFOLD APPROACH FOR DIRICHLET PROBLEM INVOLVING THE p(x)-LAPLACIAN EQUATION

  • Mashiyev, Rabil A.;Ogras, Sezai;Yucedag, Zehra;Avci, Mustafa
    • Journal of the Korean Mathematical Society
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    • v.47 no.4
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    • pp.845-860
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    • 2010
  • In this paper, using the Nehari manifold approach and some variational techniques, we discuss the multiplicity of positive solutions for the p(x)-Laplacian problems with non-negative weight functions and prove that an elliptic equation has at least two positive solutions.

MULTIPLE SOLUTIONS FOR A p-LAPLACIAN SYSTEM WITH NONLINEAR BOUNDARY CONDITIONS

  • Zhou, Jun;Kim, Chan-Gyun
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.1
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    • pp.99-113
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    • 2014
  • A nonlinear elliptic problem involving p-Laplacian and nonlinear boundary condition is considered in this paper. By using the method of Nehari manifold, it is proved that the system possesses two nontrivial nonnegative solutions if the parameter is small enough.

NEHARI MANIFOLD AND MULTIPLICITY RESULTS FOR A CLASS OF FRACTIONAL BOUNDARY VALUE PROBLEMS WITH p-LAPLACIAN

  • Ghanmi, Abdeljabbar;Zhang, Ziheng
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.5
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    • pp.1297-1314
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    • 2019
  • In this work, we investigate the following fractional boundary value problems $$\{_tD^{\alpha}_T({\mid}_0D^{\alpha}_t(u(t)){\mid}^{p-2}_0D^{\alpha}_tu(t))\\={\nabla}W(t,u(t))+{\lambda}g(t){\mid}u(t){\mid}^{q-2}u(t),\;t{\in}(0,T),\\u(0)=u(T)=0,$$ where ${\nabla}W(t,u)$ is the gradient of W(t, u) at u and $W{\in}C([0,T]{\times}{\mathbb{R}}^n,{\mathbb{R}})$ is homogeneous of degree r, ${\lambda}$ is a positive parameter, $g{\in}C([0,T])$, 1 < r < p < q and ${\frac{1}{p}}<{\alpha}<1$. Using the Fibering map and Nehari manifold, for some positive constant ${\lambda}_0$ such that $0<{\lambda}<{\lambda}_0$, we prove the existence of at least two non-trivial solutions

EXISTENCE AND CONCENTRATION RESULTS FOR KIRCHHOFF-TYPE SCHRÖ DINGER SYSTEMS WITH STEEP POTENTIAL WELL

  • Lu, Dengfeng
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.2
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    • pp.661-677
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    • 2015
  • In this paper, we consider the following Kirchhoff-type Schr$\ddot{o}$dinger system $$\{-\(a_1+b_1{\int}_{\mathbb{R^3}}{\mid}{\nabla}u{\mid}^2dx\){\Delta}u+{\gamma}V(x)u=\frac{2{\alpha}}{{\alpha}+{\beta}}{\mid}u{\mid}^{\alpha-2}u{\mid}v{\mid}^{\beta}\;in\;\mathbb{R}^3,\\-\(a_2+b_2{\int}_{\mathbb{R^3}}{\mid}{\nabla}v{\mid}^2dx\){\Delta}v+{\gamma}W(x)v=\frac{2{\beta}}{{\alpha}+{\beta}}{\mid}u{\mid}^{\alpha}{\mid}v{\mid}^{\beta-2}v\;in\;\mathbb{R}^3,\\u,v{\in}H^1(\mathbb{R}^3),$$ where $a_i$ and $b_i$ are positive constants for i = 1, 2, ${\gamma}$ > 0 is a parameter, V (x) and W(x) are nonnegative continuous potential functions. By applying the Nehari manifold method and the concentration-compactness principle, we obtain the existence and concentration of ground state solutions when the parameter ${\gamma}$ is sufficiently large.