2019 Volume 9 Issue 1
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Xiumei He, Xian Wu. MULTIPLE SIGN-CHANGING SOLUTIONS FOR A CLASS OF SEMILINEAR ELLIPTIC EQUATIONS IN $\mathbb{R}$N[J]. Journal of Applied Analysis & Computation, 2019, 9(1): 12-30. doi: 10.11948/2019.12
Citation: Xiumei He, Xian Wu. MULTIPLE SIGN-CHANGING SOLUTIONS FOR A CLASS OF SEMILINEAR ELLIPTIC EQUATIONS IN $\mathbb{R}$N[J]. Journal of Applied Analysis & Computation, 2019, 9(1): 12-30. doi: 10.11948/2019.12

MULTIPLE SIGN-CHANGING SOLUTIONS FOR A CLASS OF SEMILINEAR ELLIPTIC EQUATIONS IN $\mathbb{R}$N

  • In this paper, we study the following semilinear elliptic equations $ -\triangle u+V(x)u=f(x, u), \ \ x\in \mathbb{R}^{N}, $ where $V\in C(\mathbb{R}^{N}, \mathbb{R})$ and $f\in C(\mathbb{R}^{N}\times\mathbb{R}, \mathbb{R})$. Under some suitable conditions, we prove that the equation has three solutions of mountain pass type: one positive, one negative, and sign-changing. Furthermore, if $f$ is odd with respect to its second variable, this problem has infinitely many sign-changing solutions.
    MSC: 35J20, 35J70, 35P05, 35P30, 34B15, 58E05, 47H04
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  • [1] C. O. Alves and M. A. S. Souto, Existence of solutions for a class of elliptic equations in $\mathbb{R}$N with vanishing potentials, J. Differential Equations, 2012, 252(10), 5555-5568. doi: 10.1016/j.jde.2012.01.025

    CrossRef Google Scholar

    [2] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 1973, 14, 349-381. doi: 10.1016/0022-1236(73)90051-7

    CrossRef Google Scholar

    [3] T. Bartsch, Z. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Comm. Partial Differential Equations, 2004, 29(1-2), 25-42.

    Google Scholar

    [4] T. Bartsch and Z.-Q. Wang, Sign changing solutions of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 1999, 13(2), 191-198. doi: 10.12775/TMNA.1999.010

    CrossRef Google Scholar

    [5] T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2005, 22(3), 259-281. doi: 10.1016/j.anihpc.2004.07.005

    CrossRef Google Scholar

    [6] C. J. Batkam, An elliptic equation under the effect of two nonlocal terms, Math. Methods Appl. Sci., 2016, 39(6), 1535-1547. doi: 10.1002/mma.v39.6

    CrossRef Google Scholar

    [7] C. J. Batkam, Multiple sign-changing solutons to a class of kirchhoff type problems, Election J. Differential Equations, 2016. ArXiv: 1501. 05733.

    Google Scholar

    [8] M.-F. Bidaut-Véron, G. Hoang, Q.-H. Nguyen and L. Véron, An elliptic semilinear equation with source term and boundary measure data: the supercritical case, J. Funct. Anal., 2015, 269(7), 1995-2017. doi: 10.1016/j.jfa.2015.06.020

    CrossRef Google Scholar

    [9] J. a. M. do Ó, E. Medeiros and U. Severo, On the existence of signed and signchanging solutions for a class of superlinear Schrödinger equations, J. Math. Anal. Appl., 2008, 342(1), 432-445. doi: 10.1016/j.jmaa.2007.11.058

    CrossRef Google Scholar

    [10] X. He and W. Zou, Multiplicity of solutions for a class of elliptic boundary value problems, Nonlinear Anal., 2009, 71(7-8), 2606-2613. doi: 10.1016/j.na.2009.01.111

    CrossRef Google Scholar

    [11] M. W. J. Mawhin, Critical Point Theory and Hamiltonian System, SpringerVerleg, New York, 1989.

    Google Scholar

    [12] Q. Jiang and C.-L. Tang, Existence of a nontrivial solution for a class of superquadratic elliptic problems, Nonlinear Anal., 2008, 69(2), 523-529. doi: 10.1016/j.na.2007.05.038

    CrossRef Google Scholar

    [13] S. J. Li and M. Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl., 1995, 189(1), 6-32. doi: 10.1006/jmaa.1995.1002

    CrossRef Google Scholar

    [14] J. Liu and J. Chen, Sign changing solutions and multiple solutions of an elliptic eigenvalue problem with constraint in H1(RN), Comput. Math. Appl., 2010, 59(8), 3005-3013. doi: 10.1016/j.camwa.2010.02.019

    CrossRef Google Scholar

    [15] Z. Liu, Z.-Q. Wang and J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl. (4), 2016, 195(3), 775-794. doi: 10.1007/s10231-015-0489-8

    CrossRef Google Scholar

    [16] A. Qian, Infinitely many sign-changing solutions for a Schrödinger equation, Adv. Difference Equ., 2011, 2011:39, 6. doi: 10.1186/1687-1847-2011-39

    CrossRef Google Scholar

    [17] X. H. Tang, Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity, J. Math. Anal. Appl., 2013, 401(1), 407-415. doi: 10.1016/j.jmaa.2012.12.035

    CrossRef Google Scholar

    [18] J. Wang, Energy bound for sign changing solutions of an asymptotically linear elliptic equation in $\mathbb{R}$N, Nonlinear Anal., 2011, 74(13), 4474-4480. doi: 10.1016/j.na.2011.04.011

    CrossRef Google Scholar

    [19] T. Weth, Energy bounds for entire nodal solutions of autonomous superlinear equations, Calc. Var. Partial Differential Equations, 2006, 27(4), 421-437. doi: 10.1007/s00526-006-0015-3

    CrossRef Google Scholar

    [20] M. Willem, Minimax Theorems, Birkhauser, Boston, 1996.

    Google Scholar

    [21] X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in RN, Nonlinear Anal. Real World Appl., 2011, 12(2), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023

    CrossRef Google Scholar

    [22] X. Wu and K. Wu, Existence of positive solutions, negative solutions and high energy solutions for quasi-linear elliptic equations on $\mathbb{R}$N, Nonlinear Anal. Real World Appl., 2014, 16, 48-64. doi: 10.1016/j.nonrwa.2013.09.005

    CrossRef Google Scholar

    [23] X. Wu and K. Wu, Geometrically distinct solutions for quasilinear elliptic equations, Nonlinearity, 2014, 27(5), 987-1001. doi: 10.1088/0951-7715/27/5/987

    CrossRef Google Scholar

    [24] Q. Zhang and C. Liu, Multiple solutions for a class of semilinear elliptic equations with general potentials, Nonlinear Anal., 2012, 75(14), 5473-5481. doi: 10.1016/j.na.2012.04.052

    CrossRef Google Scholar

    [25] W. Zhang and X. Liu, Infinitely many sign-changing solutions for a quasilinear elliptic equation in $\mathbb{R}$N, J. Math. Anal. Appl., 2015, 427(2), 722-740. doi: 10.1016/j.jmaa.2015.02.070

    CrossRef Google Scholar

    [26] F. Zhao, L. Zhao and Y. Ding, Existence and multiplicity of solutions for a non-periodic Schrödinger equation, Nonlinear Anal., 2008, 69(11), 3671-3678. doi: 10.1016/j.na.2007.10.024

    CrossRef Google Scholar

    [27] W. M. Zou, Sign-changing Critical Point TheorySign-changing Critical Point Theory, New York, New, 2008.

    Google Scholar

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