Multiplicity of positive solutions for a Kirchhoff type problem without asymptotic conditions
DOI:
https://doi.org/10.12775/TMNA.2022.031Keywords
Kirchhoff type problem, multiple positive solutions, variational methodsAbstract
In this paper, we are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem \[ \begin{cases} -\bigg({\varepsilon}^2a+{\varepsilon}b\int_{\mathbb{R}^3} |\n u|^2dx\bigg)\Delta u+u=Q(x)|u|^{p-2}u, & x\in\mathbb{R}^3,\\ u\in H^1\big(\mathbb{R}^3\big), \quad u> 0, & x\in\mathbb{R}^3, \end{cases} \] where $\varepsilon> 0$ is a small parameter, $a,b> 0$ are constants, $4< p< 6$, $Q$ is a nonnegative continuous potential and does not satisfy any asymptotic condition. Combining Nehari manifold and concentration compactness principle, we study how the shape of the graph of $Q(x)$ affects the number of positive solutions.References
A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal. 140 (1997), 285–300.
A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal. 159 (2001), 253–271.
A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996), 305–330.
H. Brézis and E.H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.
D. Cao and E.S. Noussair, Multiple positive and nodal solutions for semilinear elliptic problems with critical exponents, Indiana Univ. Math. J. 44 (1995), 1249–1271.
D. Cao and E.S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in RN , Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 567–588.
D. Cassani, Z. Liu, C. Tarsi and J. Zhang, Multiplicity of sign-changing solutions for Kirchhoff-type equations, Nonlinear Anal. 186 (2019), 145–161.
J. Chen, Multiple positive solutions to a class of Kirchhoff equation on R3 with indefinite nonlinearity, Nonlinear Anal. 96 (2014), 134–145.
S. Cingolani and N. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal. 10 (1997), 1–13.
M. Del Pino and P.L. Felmer, Semi-classical states for nonlinear Schrödinger equations: a variational reduction method, Math. Ann. 324 (2002), 1–32.
H. Fan, Positive solutions for a Kirchhoff-type problem involving multiple competitive potentials and critical Sobolev exponent, Nonlinear Anal. 198 (2020), Article 111869.
A. Floer and A. Weinstein, Nospreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (1986), 397–408.
X. He and W. Zou, Multiplicity of solutions for a class of Kirchhoff type problems, Acta Math. Appl. Sin. 26 (2010), 387–394.
X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3 , J. Differential Equations 252 (2012), 1813–1834.
G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
C. Lei, G. Liu and L. Guo, Multiple positive solutions for a Kirchhoff problem with a critical nonlinearity, Nonlinear Anal. Real World Appl. 31 (2016), 343–355.
J.L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud. 30 (1978), 284–346.
J.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part II, Ann. Inst. H. Poincaré Anal. Nonlinéaire 1 (1984), 223–283.
G. Li and H. Ye, Existence of positive solutions for nonlinear Kirchhoff type problems in R3 with critical Sobolev exponent and sign-changing nonlinearities, Math. Method. Appl. Sci. 7 (2013), 97–114.
Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations 253 (2012), 2285–2294.
Y. Liu and S. Guo, Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent, Z. Angew. Math. Phys. 66 (2015), 747–769.
T.F. Ma and J.E. Munoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett. 16 (2003), 243–248.
J.Y. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys. 131 (1990), 223–253.
K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (2006), 246–255.
P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270–291.
X. Qian and J. Chen, Multiple positive and sign-changing solutions of an elliptic equation with fast increasing weight and critical growth, J. Math. Anal. Appl. 465 (2018), 1186–1208.
X. Qian and W. Chao, Positive solutions for a Kirchhoff type problem with fast increasing weight and critical nonlinearity, Electron. J. Qual. Theory Differ. Equ. 27 (2019), 1–17.
Y. Sun and X. Liu, Existence of positive solutions for Kirchhoff type problems with critical exponent, J. Partial Differ. Equ. 25 (2012), 187–198.
D. Sun and Z. Zhang, Uniqueness, existence and concentration of positive ground state solutions for Kirchhoff type problems in R3 , J. Math. Anal. Appl. 461 (2018), 128–149.
J. Wang, L. Tian, J. Xu, and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations 253 (2012), 2314–2351.
J. Wang and L. Xiao, Existence and concentration of solutions for a Kirchhoff type problem with potential, Discrete Contin. Dyn. Syst. 36 (2016), 7137–7168.
X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys. 53 (1993), 224–229.
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Xiaotao Qian
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
