본문 바로가기
KERD
논문KCI 등재

Existence and multiplicity of solutions for Kirchhoff-Schr\"{o}dinger-Poisson system with concave and convex nonlinearities

Haibo Chen|Guofeng Che
대한수학회 (The Korean Mathematical Society)|2020.11|대한수학회지 (Journal of the KMS)|제57권 제6호|pp. 1551-1571|DOI 10.4134/JKMS.j190833

1

조회

0

다운로드

0

인용

0

참고문헌

키워드

Ekeland's variational principleKirchhoff-Schr\"{o}dinger-Poisson systemconcave and convex nonlinearitiesMountain Pass Theorem
주제분류수학

초록

This paper is concerned with the following Kirchhoff-Schr\"{o}d\-inger-Poisson system $$ \scalebox{0.84}{$\displaystyle\left\{\!\! \begin{array}{ll} \displaystyle -\big(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\mathrm{d}x\big)\Delta u + V(x)u+\mu\phi u=\lambda f(x)|u|^{p-2}u+g(x)|u|^{q-2}u, &\mbox{ in }\mathbb{R}^{3},\\ -\Delta \phi= \mu |u|^{2}, &\mbox{ in }\mathbb{R}^{3},\\ \end{array} \right.$} $$ where $a>0,~b,~\mu\geq0$, $p\in(1,2)$, $q\in[4,6)$ and $\lambda>0$ is a parameter. Under some suitable assumptions on $V(x)$, $f(x)$ and $g(x)$, we prove that the above system has at least two different nontrivial solutions via the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory. Some recent results from the literature are improved and extended.

Abstract

This paper is concerned with the following Kirchhoff-Schr\"{o}d\-inger-Poisson system $$ \scalebox{0.84}{$\displaystyle\left\{\!\! \begin{array}{ll} \displaystyle -\big(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\mathrm{d}x\big)\Delta u + V(x)u+\mu\phi u=\lambda f(x)|u|^{p-2}u+g(x)|u|^{q-2}u, &\mbox{ in }\mathbb{R}^{3},\\ -\Delta \phi= \mu |u|^{2}, &\mbox{ in }\mathbb{R}^{3},\\ \end{array} \right.$} $$ where $a>0,~b,~\mu\geq0$, $p\in(1,2)$, $q\in[4,6)$ and $\lambda>0$ is a parameter. Under some suitable assumptions on $V(x)$, $f(x)$ and $g(x)$, we prove that the above system has at least two different nontrivial solutions via the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory. Some recent results from the literature are improved and extended.

참고문헌 (38)

  1. C. O. Alves, "Positive solutions for a quasilinear elliptic equation of Kirchhotype", Comput. Math. Appl., 49(1), pp.85~93 (2005)[DOI]
  2. A. Ambrosetti, "Dual variational methods in critical point theory and applications", J. Funct. Anal, 14, pp.349~381 (1973)[DOI]
  3. A. Arosio, "On the well-posedness of the Kirchhostring", Trans. Amer. Math. Soc, 348(1), pp.305~330 (1996)[DOI]
  4. A. Azzollini, "Ground state solutions for the nonlinear Schrodinger-Maxwell equations", J. Math. Anal. Appl, 345(1), pp.90~108 (2008)[DOI]
  5. T. Bartsch, "Existence and multiplicity results for some superlinear elliptic problems on RN", Comm. Partial Dierential Equations, 20(9-10), pp.1725~1741 (1995)[DOI]
  6. V. Benci, "An eigenvalue problem for the Schrodinger-Maxwell equations", Topol. Methods Nonlinear Anal, 11(2), pp.283~293 (1998)
  7. M. M. Cavalcanti, "Global existence and uniform decay rates for the Kirchho-Carrier equation with nonlinear dissipation", Adv. Dierential Equations, 6(6), pp.701~730 (2001)
  8. G. Che, "Existence and multiplicity of systems of Kirchho-type equations with general potentials", Math. Methods Appl. Sci, 40(3), pp.775~785 (2017)[DOI]
  9. G. Che, "Innitely many solutions for the Klein-Gordon equation with sublinear nonlinearity coupled with Born-Infeld theory", Bull. Iran. Math. Soc (2019)[DOI]
  10. G. Che, "Existence and multiplicity of positive solutions for Kirchho-Schrodinger-Poisson system with critical growth", Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Mat. RACSAM, 114(2) (2020)
  11. G. Che, "Existence and multiplicity of positive solutions for fractional Laplacian systems with nonlinear coupling", J. Math. Phys, 60, pp.28~ (2019)[DOI]
  12. G. Che, "Bound state positive solutions for a class of elliptic system with Hartree nonlinearity", Commun. Pure Appl. Anal, 19(7), pp.3697~3722 (2020)[DOI]
  13. G. Che, "Existence and concentration of positive ground states for a 1-Laplacian problem in RN", Appl. Math. Lett, 100, pp.7~ (2020)[DOI]
  14. P. D'Ancona, "Global solvability for the degenerate Kirchhoequation with real analytic data", Invent. Math, 108(2), pp.247~262 (1992)[DOI]
  15. I. Ekeland, "On the variational principle", J. Math. Anal. Appl, 47, pp.324~353 (1974)[DOI]
  16. L. Jeanjean, "A positive solution for a nonlinear Schrodinger equation on RN", Indiana Univ. Math. J, 54(2), pp.443~464 (2005)
  17. R. Kajikiya, "A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations", J. Funct. Anal., 225(2), pp.352~370 (2005)[DOI]
  18. G. Kirchho, "Mechanik" (1883), Teubner
  19. J.-L. Lions, "Contemporary developments in continuum mechanics and partial dierential equations", pp.284~346 (1977)
  20. Z. Liu, "On ground state solutions for the Schrodinger-Poisson equations with critical growth", J. Math. Anal. Appl, 412(1), pp.435~448 (2014)
  21. Z. Liu, "Existence of positive ground state solutions for Kirchhotype problems", Nonlinear Anal, 120, pp.1~13 (2015)[DOI]
  22. Z. Liu, "Existence of ground state solutions for the Schrodinger-Poisson systems", Appl. Math. Comput, 244, pp.312~323 (2014)
  23. A. Mao, "Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems", J. Math. Anal. Appl., 383(1), pp.239~243 (2011)[DOI]
  24. A. Mao, "Sign-changing and multiple solutions of Kirchhotype problems without the P.S. condition", Nonlinear Anal, 70(3), pp.1275~1287 (2009)[DOI]
  25. D. Mugnai, "Addendum to: Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem", NoDEA Nonlinear Dierential Equations Appl, 11(3), pp.379~391 (2012)[DOI]
  26. D. Naimen, "The critical problem of Kirchhotype elliptic equations in dimension four", J. Dierential Equations, 257(4), pp.1168~1193 (2014)[DOI]
  27. S. I. Pohozaev, "A certain class of quasilinear hyperbolic equation", Mat. Sb. (NS), 96, pp.152~168 (1975)
  28. D. Ruiz, "The Schrodinger-Poisson equation under the eect of a nonlinear local term", J. Funct. Anal, 237(2), pp.655~674 (2006)
  29. J. Sun, "On ground state solutions for some non-autonomous Schrodinger-Poisson systems", J. Dierential Equations, 252(5), pp.3365~3380 (2012)
  30. J. Sun, "Positive solutions of asymptotically linear Schrodinger-Poisson systems with a radial potential vanishing at innity", Nonlinear Anal, 74(2), pp.413~423 (2011)
  31. J. Sun, "Ground state solutions for an indenite Kirchhotype problem with steep potential well", J. Dierential Equations, 256(4), pp.1771~1792 (2014)[DOI]
  32. J. Sun, "Existence and multiplicity of solutions for an indenite Kirchho-type equation in bounded domains", Proc. Roy. Soc. Edinburgh Sect. A, 146(2), pp.435~448 (2016)[DOI]
  33. X. H. Tang, "Ground state sign-changing solutions for Kirchhotype problems in bounded domains", J. Dierential Equations, 261(4), pp.2384~2402 (2016)[DOI]
  34. M. Willem, "Minimax theorems", Progress in Nonlinear Dierential Equations and their Applications, 24 (1996), Birkhauser Boston, Inc[DOI]
  35. L. Xu, "Multiplicity of small negative-energy solutions for a class of nonlinear Schrodinger-Poisson systems", Appl. Math. Comput, 243, pp.817~824 (2014)
  36. G. Zhao, "Existence of innitely many solutions to a class of Kirchho-Schrodinger-Poisson system", Appl. Math. Comput., 256, pp.572~581 (2015)[DOI]
  37. W. Zou, "Variant fountain theorems and their applications", Manuscripta Math, 104(3), pp.343~358 (2001)[DOI]
  38. W. Zou, "Critical Point Theory and Its Applications" (2006), Springer

관련 논문AI추천

Ground state sign-changing solutions for a class of Schr\"{o}dinger-Poisson-Kirchhoff type problems with a critical nonlinearity in $\mathbb{R}^{3}$Aixia Qian, Mingming Zhang대한수학회 · 대한수학회지2021 · 제58권 제5호Multiplicity results for nonlinear Schr\"{o}dinger-Poisson systems with subcritical or critical growthShangjiang Guo, Zhisu Liu대한수학회 · 대한수학회지2016 · 제53권 제2호Multiple solutions for a class of quasilinear Schr\"{o}dinger system in $\mathbb{R}^N$Caisheng Chen, Qiang Chen대한수학회 · 대한수학회보2016 · 제53권 제6호Existence and multiplicity of solutions for nonlinear Schr\"{o}dinger-Kirchhoff-type equationsHaibo Chen, Hongliang Liu, Liping Xu대한수학회 · 대한수학회지2016 · 제53권 제1호MULTIPLICITY OF SOLUTIONS FOR QUASILINEAR SCHRÖDINGER TYPE EQUATIONS WITH THE CONCAVE-CONVEX NONLINEARITIES김윤호, 김인현, 박기섭, Chenshuo Li대한수학회 · 대한수학회지2021 · 제58권 제6호Existence and multiplicity results for a new Kirchhoff type problem with critical exponent and indefinite potentialXiaotao Qian, Zhigao Shi대한수학회 · 대한수학회보2025 · 제62권 제2호

이 간행물의 다른 논문 (13건)

Erd\H os-Ko-Rado type theorems for simplicial complexes via algebraic shifting김연진pp. 1323-1333A certain K\"ahler potential of the Poincar\'e metric and its characterizationSungmin Yoo, 최영준, 이강혁pp. 1335-1345Bifurcation problem for a class of quasilinear fractional Schr\"{o}dinger equationsImed Abidpp. 1347-1372Cyclotomic quiver Hecke algebras corresponding to minuscule representations박의용pp. 1373-1388On the extension dimension of module categoriesTiwei Zhao, Yeyang Pengpp. 1389-1406
원문 보기