EXISTENCE AND MULTIPLICITY OF NONTRIVIAL SOLUTIONS FOR KLEIN-GORDON-MAXWELL SYSTEM WITH A PARAMETER
1
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
This paper is concerned with the following Klein-Gordon-Maxwell system: $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u+\lambda V(x)u-(2\omega+\phi)\phi u=f(x,u), \mbox{ \ \ }x\in\mathbb{R}^{3},\\ \Delta\phi=(\omega+\phi)u^{2},\mbox{ \ \ }x\in\mathbb{R}^{3}, \end{array} \right. $$ where $\omega>0$ is a constant and $\lambda$ is the parameter. Under some suitable assumptions on $V(x)$ and $f(x, u)$, we establish the existence and multiplicity of nontrivial solutions of the above system via variational methods. Our conditions weaken the Ambrosetti Rabinowitz type condition.
Abstract
This paper is concerned with the following Klein-Gordon-Maxwell system: $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u+\lambda V(x)u-(2\omega+\phi)\phi u=f(x,u), \mbox{ \ \ }x\in\mathbb{R}^{3},\\ \Delta\phi=(\omega+\phi)u^{2},\mbox{ \ \ }x\in\mathbb{R}^{3}, \end{array} \right. $$ where $\omega>0$ is a constant and $\lambda$ is the parameter. Under some suitable assumptions on $V(x)$ and $f(x, u)$, we establish the existence and multiplicity of nontrivial solutions of the above system via variational methods. Our conditions weaken the Ambrosetti Rabinowitz type condition.
참고문헌 (19)
- A. Azzollini, "Impoved estimates and a limit case for the electrostatic Klein-Gordon-Maxwell system", Proc. R. Soc. Edinb. Sect. A, 141(3), pp.449~463 (2011)
- A. Azzollini, "Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations", Topol. Methods Nonlinear Anal, 35(1), pp.33~42 (2010)
- V. Benci, "Solitary waves of the nonlinear Klein-Gordon-Maxwell equation coupled with Maxwell equations", Rev. Math. Phys, 14(4), pp.409~420 (2002)
- D. Cassani, "Existence and non-existence of solitary waves for the critical Klein-Gordon-Maxwell equation coupled with Maxwell’s equations", Nonlinear Anal, 58(7-8), pp.733~747 (2004)
- S. J. Chen, "Multiple solutions for the nonhomogeneous Klein-Gordon equation coupled with Born-Infield theory on R3", J. Math. Anal. App, 400(2), pp.517~524 (2013)
- S. J. Chen, "Multiple solutions for nonhomogeneous Schrodinger-Maxwell equations and Klein-Gordon-Maxwell equations on R3", Nodea-Nonlinear Differential Equations Appl, 17(5), pp.559~574 (2010)
- P. Chen, "Infinitely many solutions for Schrodinger-Maxwell equations with indefinite sign subquadratic potentials", Appl. Math. Comput, 226, pp.492~502 (2014)
- T. D’Aprile, "Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrodinger-Maxwell equations", Proc. R. Soc. Edinburgh Sect. A, 134(5), pp.893~906 (2004)
- Y. H. Ding, "Bound states for semilinear Schrodinger equations with sign-changing potential", Calc. Var. Partial Differential Equations, 29(3), pp.397~419 (2007)
- L. Ding, "Infinitely many standing wave solutions for the nonlinear Klein-Gordon-Maxwell system with sign-changing potential", Comput. Math. Appl, 68(5), pp.589~595 (2014)
- I. Ekeland, "Convexity Methods in Hamiltonian Mechanics" (1990), Springer
- Y. X. Guo, "Existence and multiplicity of nontrivial solutions for p-Laplacian schrodinger-Kirchhoff-type equations", J. Math. Anal. App, 428(2), pp.1054~1069 (2015)
- L. Lin, "Infinitely many solutions for a nonlinear Klein-Gordon-Maxwell system", Nonlinear Anal, 110, pp.157~169 (2014)
- P. H. Rabinowitz, "Minimax methods in critical point theory with applications to differential equations" (1986), American Mathematical Society
- F. Z.Wang, "Solitary waves for the Klein-Gordon-Maxwell system with critical exponent", Nonlinear Anal, 74, pp.827~835 (2011)
- L. P. Xu, "Multiplicity of small negative-energy solutions for a class of nonlinear Schrodinger-Poisson systems", Appl. Math. Comput, 243(3), pp.817~824 (2014)
- L. P. Xu, "Multiplicity results for fourth order elliptic equations of Kirchhoff-type", Acta. Math. Sci. Ser. B Engl. Ed, 35(5), pp.1067~1076 (2015)
- L. P. Xu, "Existence and multiplicity of solutions for nonhomogeneous Klein-Gordon-Maxwell equations", Elentron. J. Differential Equation, 2015(102), pp.1~12 (2015)
- W. Zou, "Critical Point Theory and its Applications" (2006), Springer