Browse > Article
http://dx.doi.org/10.11568/kjm.2018.26.4.683

ONE-DIMENSIONAL JUMPING PROBLEM INVOLVING p-LAPLACIAN  

Jung, Tacksun (Department of Mathematics, Kunsan National University)
Choi, Q-Heing (Department of Mathematics Education, Inha University)
Publication Information
Korean Journal of Mathematics / v.26, no.4, 2018 , pp. 683-700 More about this Journal
Abstract
We get one theorem which shows existence of solutions for one-dimensional jumping problem involving p-Laplacian and Dirichlet boundary condition. This theorem is that there exists at least one solution when nonlinearities crossing finite number of eigenvalues, exactly one solutions and no solution depending on the source term. We obtain these results by the eigenvalues and the corresponding normalized eigenfunctions of the p-Laplacian eigenvalue problem when 1 < p < ${\infty}$, variational reduction method and Leray-Schauder degree theory when $2{\leq}$ p < ${\infty}$.
Keywords
One-dimensional jumping problem; one-dimensional p-Laplacian eigenvalue problem; variational reduction method; jumping nonlinearity; variational reduction method; Leray-Schauder degree theory;
Citations & Related Records
연도 인용수 순위
  • Reference
1 R. Manasevich, J. Mawhin, Periodic solutions for nonlinear systems with p-Laplacian- like operators, J. Differential Equations 145 (1998), 367-393.   DOI
2 Q. H. Choi and T. Jung, A nonlinear suspension bridge equation with nonconstant load, Nonlinear Analysis TMA. 35 (1999), 649-668.   DOI
3 Q. H. Choi and T. Jung, An application of a variational reduction method to a nonlinear wave equation, J. Diff. Eq. 117 (1995), 390-410.   DOI
4 R. Manasevich, J. Mawhin, Boundary value problems for nonlinear perturbations of vector p-Laplacian-like operators, J. Korean Math. Society 37 (2000), 665-685.
5 P. J. McKenna, W. Walter, Nonlinear oscillations in a suspension bridge, Arch. Rat. Mech. Anal. 98 (1987), 167-177.   DOI
6 D. O'Regan, Some general existence principles and results for $({\phi}(y = qf(t, y, y'), 0 < t < 1, SIAM J. Math. Anal. 24 (1993), 648-668.   DOI
7 S. Solimini, Some remarks on the number of solutions of some nonlinear ellipltic problems, Ann. Inst. Henri Poincare 2 (2) (1985), 143-156.   DOI
8 M. del Pino, M. Elgueta and R. Manasevich, A homotopic deformation along p of a Leray Schauder degree result and existence for ($\left|uu')' + f(x, u) = 0, u(0) = u(T) = 0, p > 1, J. Differential Equations 80 (1898), 1-13.
9 Q. H. Choi and T. Jung, Multiplicity results for the nonlinear suspension bridge equation, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 9 (2002), 29-38.
10 Q. H. Choi, T. Jung and P. J. McKenna, The study of a nonlinear suspension bridge equation by a variational reduction method, Applicable Analysis 50 (1993), 73-92.   DOI
11 M. Ghergu and V. Radulescu, Singular elliptic problems, bifurcation and asymptotic analysis, Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press.
12 Y.-H. Kim, L. Wang and C. Zhang, Global bifurcation for a class of degenerate elliptic equations with variable exponents, J. Math. Anal Appl. 371 (2010), 624-637.   DOI
13 A. Le Eigenvalue Problems for the p-Laplacian, Nonlinear Analysis 64 (2006), 1057-1099.