2020 Volume 10 Issue 5
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Zhi-Yun Tang, Zeng-Qi Ou. INFINITELY MANY SOLUTIONS FOR A NONLOCAL PROBLEM[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 1912-1917. doi: 10.11948/20190286
Citation: Zhi-Yun Tang, Zeng-Qi Ou. INFINITELY MANY SOLUTIONS FOR A NONLOCAL PROBLEM[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 1912-1917. doi: 10.11948/20190286

INFINITELY MANY SOLUTIONS FOR A NONLOCAL PROBLEM

  • Consider a class of nonlocal problems $ \left\{\begin{array}{lr} -\left(a-b \int_{\Omega}|\nabla u|^{2} d x\right) \Delta u=f(x, u), & x \in \Omega, \\ u=0, & x \in \partial \Omega, \end{array}\right.$ where $ a>0, b>0 $, $ \Omega\subset \mathbb{R}^N $ is a bounded open domain, $ f:\overline{\Omega} \times \mathbb R \longrightarrow \mathbb R $ is a Carath$ \acute{\mbox{e}} $odory function. Under suitable conditions, the equivariant link theorem without the $ (P.S.) $ condition due to Willem is applied to prove that the above problem has infinitely many solutions, whose energy increasingly tends to $ a^2/(4b) $, and they are neither large nor small.
    MSC: 35G20, 35J60, 35J75
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  • [1] Y. Duan, X. Sun and H. Li, Existence and multiplicity of positive solutions for a nonlocal problem, J. Nonlinear Sci. Appl., 2017, 10, 6056-6061. doi: 10.22436/jnsa.010.11.40

    CrossRef Google Scholar

    [2] X. He and W. Zou, Multiplicity of solutions for a class of Kirchhoff type problems, Acta Math. Appl. Sin. Engl. Ser., 2010, 26, 387-394. doi: 10.1007/s10255-010-0005-2

    CrossRef Google Scholar

    [3] C. Lei, J. Liao and H. Suo, Multiple positive solutions for nonlocal problems involving a sign-changing potential, Electron. J. Differential Equations, 2017, 9, 1-8.

    Google Scholar

    [4] C. Lei, C. Chu and H. Suo, Positive solutions for a nonlocal problem with singularity, Electron. J. Differential Equations, 2017, 85, 1-9.

    Google Scholar

    [5] H. Pan and C. Tang, Existence of infinitely many solutions for semilinear elliptic equations, Electron. J. Differential Equations, 2016, 167, 1-11.

    Google Scholar

    [6] J. Sun and C. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal., 2011, 74, 1212-1222. doi: 10.1016/j.na.2010.09.061

    CrossRef Google Scholar

    [7] M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhauser Boston, Inc., Boston, MA, 1996.

    Google Scholar

    [8] Y. Wu and T. An, Infinitely many solutions for a class of semilinear elliptic equations, J. Math. Anal. Appl., 2014, 414, 285-295. doi: 10.1016/j.jmaa.2014.01.003

    CrossRef Google Scholar

    [9] Y. Ye and C. Tang, Multiplicity of solutions for elliptic boundary value problems, Electron. J. Differential Equations, 2014, 140, 1-13.

    Google Scholar

    [10] G. Yin and J. Liu, Existence and multiplicity of nontrivial solutions for a nonlocal problem, Boundary Value Problem, 2015, 26, 1-7.

    Google Scholar

    [11] K. Yosida, Functional analysis. Reprint of the sixth edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.

    Google Scholar

    [12] X. Zhang, Existence and multiplicity of solutions for a class of elliptic boundary value problems, J. Math. Anal. Appl., 2014, 410, 213-226. doi: 10.1016/j.jmaa.2013.08.001

    CrossRef Google Scholar

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