본문 바로가기
KERD
논문KCI 등재

Positive solution and ground state solution for a Kirchhoff type equation with critical growth

Caixia Chen|Aixia Qian
대한수학회 (The Korean Mathematical Society)|2022.07|대한수학회보 (Bulletin of the KMS)|제59권 제4호|pp. 961-977|DOI 10.4134/BKMS.b210567

2

조회

0

다운로드

0

인용

0

참고문헌

키워드

Kirchhoff type equationcritical nonlinearityconcentration-compactness principleground state solution
주제분류수학

초록

In this paper, we consider the following Kirchhoff type equation on the whole space $$\left\{\aligned &-(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx)\triangle u = u^{5} + \lambda k(x)g(u), \ x\in \mathbb{R}^{3},\\ & u\in\mathcal{D}^{1,2}(\mathbb{R}^{3}),\endaligned\right.$$ where $\lambda>0$ is a real number and $k, g$ satisfy some conditions. We mainly investigate the existence of ground state solution via variational method and concentration-compactness principle.

Abstract

In this paper, we consider the following Kirchhoff type equation on the whole space $$\left\{\aligned &-(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx)\triangle u = u^{5} + \lambda k(x)g(u), \ x\in \mathbb{R}^{3},\\ & u\in\mathcal{D}^{1,2}(\mathbb{R}^{3}),\endaligned\right.$$ where $\lambda>0$ is a real number and $k, g$ satisfy some conditions. We mainly investigate the existence of ground state solution via variational method and concentration-compactness principle.

참고문헌 (20)

  1. G. Cerami, "Positive solutions for some non-autonomous Schrodinger Poisson systems", J. Differential Equations, 248(3), pp.521~543 (2010)[DOI]
  2. G. M. Figueiredo, "Existence of a positive solution for a Kirchhoffproblem type with critical growth via truncation argument", J. Math. Anal. Appl, 401(2), pp.706~713 (2013)[DOI]
  3. X. He, "Existence and concentration behavior of positive solutions for a Kirchhoffequation in R3", J. Differential Equations, 252(2), pp.1813~1834 (2012)[DOI]
  4. T. Hu, "Multiplicity of positive solutions for Kirchhofftype problems in R3", Topol. Methods Nonlinear Anal, 50(1), pp.231~252 (2017)[DOI]
  5. Y. Huang, "On finding solutions of a Kirchhofftype problem", Proc. Amer. Math. Soc, 144(7), pp.3019~3033 (2016)[DOI]
  6. L. Huang, "On the Schr¨odinger-Poisson system with a general indefinite nonlinearity", Nonlinear Anal. Real World Appl, 28, pp.1~19 (2016)[DOI]
  7. C. -Y. Lei, "Multiple positive solutions for Kirchhofftype of problems with singularity and critical exponents", J. Math. Anal. Appl, 421(1), pp.521~538 (2015)[DOI]
  8. C. -Y. Lei, "Multiple positive solutions for a Kirchhofftype problem with a critical nonlinearity", Nonlinear Anal. Real World Appl, 31, pp.343~355 (2016)[DOI]
  9. C. -Y. Lei, "On ground state solutions for a Kirchhofftype equation with critical growth", Comput. Math. Appl, 72(3), pp.729~740 (2016)[DOI]
  10. G. Li, "Existence of positive ground state solutions for the nonlinear Kirchhofftype equations in R3", J. Differential Equations, 257(2), pp.566~600 (2014)[DOI]
  11. S. Liang, "Existence of solutions for Kirchhofftype problems with critical nonlinearity in R3", Nonlinear Anal. Real World Appl, 17, pp.126~136 (2014)[DOI]
  12. P. -L. Lions, "The concentration-compactness principle in the calculus of variations. The limit case. I", Rev. Mat. Iberoamericana, 1(1), pp.145~201 (1985)[DOI]
  13. Pierre-Louis Lions, "The Concentration-Compactness Principle in the Calculus of Variations. The Limit Case, Part 2", Revista Matemática Iberoamericana, pp.45~121 (1985), European Mathematical Society - EMS - Publishing House GmbH[DOI]
  14. Z. Liu, "Existence of positive ground state solutions for Kirchhofftype equa"tion with general critical growth", Topol. Methods Nonlinear Anal, 49(1), pp.165~182 (2017)[DOI]
  15. Zhaoli Liu, "Infinitely many sign-changing solutions for the nonlinear Schrödinger–Poisson system", Annali di Matematica Pura ed Applicata, 195(3), pp.775~794 (2015), Springer Science and Business Media LLC[DOI]
  16. Anass Ourraoui, "On a p-Kirchhoffproblem involving a critical nonlinearity", Comptes Rendus Mathematique, 352(4), pp.295~298 (2014), Elsevier BV[DOI]
  17. A. Qian, "Ground state and nodal solutions for a Schrodinger Poisson equation with critical growth", J. Math. Phys, 59(12), pp.20~ (2018)[DOI]
  18. M. Shao, "Signed and sign-changing solutions of Kirchhofftype problems", J. Fixed Point Theory Appl, 20(1), pp.20~ (2018)[DOI]
  19. J. Wang, "Multiplicity and concentration of positive solutions for a Kirchhofftype problem with critical growth", J. Differential Equations, 253(7), pp.2314~2351 (2012)[DOI]
  20. M. Willem, "Progress in Nonlinear Differential Equations and their Applications" (1996), Birkhauser Boston, Inc

관련 논문AI추천

Existence and multiplicity results for a new Kirchhoff type problem with critical exponent and indefinite potentialXiaotao Qian, Zhigao Shi대한수학회 · 대한수학회보2025 · 제62권 제2호Positive solutions to p-Kirchhoff-type elliptic equation with general subcritical growthHuixing Zhang, Ran Zhang대한수학회 · 대한수학회보2017 · 제54권 제3호Ground 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 제5호Multiple solutions for a critical fractional Kirchhoff type equation with supercritical exponent대한수학회 · 대한수학회보2026 · 제63권 제1호Existence and multiplicity of solutions for Kirchhoff-Schr\"{o}dinger-Poisson system with concave and convex nonlinearitiesGuofeng Che, Haibo Chen대한수학회 · 대한수학회지2020 · 제57권 제6호EXISTENCE AND CONCENTRATION RESULTS FOR KIRCHHOFF-TYPE SCHR¨O DINGER SYSTEMS WITH STEEP POTENTIAL WELLDengfeng Lu대한수학회 · 대한수학회보2015 · 제52권 제2호

이 간행물의 다른 논문 (17건)

Integrability of an almost complex structure on $S^4\times_f V^2$어윤희, 전선향, 조종택pp. 801-810$P$-extremal functions and Bernstein-Markov properties associated to compact sets in $\mathbb R^d$Tang Van Long, Hoang Thieu Anh, Kieu Phuong Chi, Nguyen Quang Dieupp. 811-825Uniqueness of meromorphic solutions of a certain type of difference equationsJun-Fan Chen, Shu-Qing Linpp. 827-841B\'ezout rings and weakly B\'ezout ringsHaitham El Alaouipp. 843-852On right regularity of commutators정다운, 류성주, 성효진, 이양, 박상원, 이창익pp. 853-868
원문 보기