Infinitely many sign-changing solutions for a semilinear elliptic equation with variable exponent

Changmu Chu; Yuxia Xiao; Yanling Xie; Changmu Chu; Yuxia Xiao; Yanling Xie

doi:10.3934/math.2021337

AIMS Mathematics

2021, Volume 6, Issue 6: 5720-5736. doi: 10.3934/math.2021337

Previous Article Next Article

Research article

Infinitely many sign-changing solutions for a semilinear elliptic equation with variable exponent

School of Data Science and Information Engineering, Guizhou Minzu University, Guizhou 550025, China

Received: 03 January 2021 Accepted: 18 March 2021 Published: 26 March 2021
MSC : 35J20, 35J62, 35Q55

This paper is devoted to study a class of semilinear elliptic equations with variable exponent. By means of perturbation technique, variational methods and a priori estimation, the existence of infinitely many sign-changing solutions to this class of problem is obtained.
- semilinear elliptic equation,
- variable exponent,
- variational methods,
- a priori estimate,
- sign-changing solution
Citation: Changmu Chu, Yuxia Xiao, Yanling Xie. Infinitely many sign-changing solutions for a semilinear elliptic equation with variable exponent[J]. AIMS Mathematics, 2021, 6(6): 5720-5736. doi: 10.3934/math.2021337

Related Papers:

Abstract

This paper is devoted to study a class of semilinear elliptic equations with variable exponent. By means of perturbation technique, variational methods and a priori estimation, the existence of infinitely many sign-changing solutions to this class of problem is obtained.

References

[1]	C. O. Alves, G. Ercole, M. D. Huam$\acute{a}$n Bola$\tilde{n}$os, Ground state solutions for a semilinear elliptic problem with critical-subcritical growth, Adv. Nonlinear Anal., 9 (2020), 108–123.
[2]	A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 347–381.
[3]	T. Bartsch, Z. Q. Wang, On the existence of sign changing solutions for semilinear dirichlet problems, Topol. Methods Nonlinear Anal., 7 (1996), 115–131. doi: 10.12775/TMNA.1996.005
[4]	T. Bartsch, T. Weth, M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96 (2005), 1–18. doi: 10.1007/BF02787822
[5]	D. M. Cao, S. L. Li, Z. Y. Liu, Nodal solutions for a supercritical semilinear problem with variable exponent, Calc. Var. Partial Differential Equations, 57 (2018), 19–38. doi: 10.1007/s00526-017-1293-7
[6]	A. Castro, J. Cossio, J. M. Neuberger, A sign-changing solution for a superlinear Dirichlet problem, Rocky Mt. J. Math., 27 (1997), 1041–1053.
[7]	D. G. Costa, C. A. Magalh$\tilde{a}$es, Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal., 23 (1994) 1401–1412.
[8]	M. F. Furtado, E. D. Silva, Superlinear elliptic problems under the non-quadraticity condition at infinity, P. Roy. Soc. Edinb. A, 145 (2015), 779–790. doi: 10.1017/S0308210515000141
[9]	M. Hashizume, M. Sano, Strauss's radial compactness and nonlinear elliptic equation involving a variable critical exponent, J. Funct. Spaces, 2018 (2018), 1–13.
[10]	K. Kurata, N. Shioji, Compact embedding from $W^{1, 2}_0(\Omega)$ to $L^{q(x)}(\Omega)$ and its application to nonlinear elliptic boundary value problem with variable critical exponent, J. Math. Anal. Appl., 339 (2008), 1386–1394. doi: 10.1016/j.jmaa.2007.07.083
[11]	S. J. Li, Z. Q. Wang, Ljusternik-Schnirelman theory in partially ordered Hilbert spaces, T. Am. Math. Soc., 354 (2002), 3207–3227. doi: 10.1090/S0002-9947-02-03031-3
[12]	Z. Liu, Z. Q. Wang, On the Ambrosetti-Rabinowitz super-linear condition, Adv. Nonlinear Stud., 4 (2004), 563–574.
[13]	J. Marcos do, B. Ruf, P. Ubilla, On supercritical Sobolev type inequalities and related elliptic equations, Calc. Var. Partial Dif., 55 (2016), 55–83. doi: 10.1007/s00526-016-0991-x
[14]	O. H. Miyagaki, M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differ. Equ., 245 (2008), 3628–3638. doi: 10.1016/j.jde.2008.02.035
[15]	A. X. Qian, S. J. Li, Multiple nodal solutions for elliptic equations, Nonlinear Anal., 57 (2004), 615–632. doi: 10.1016/j.na.2004.03.010
[16]	M. Schechter, W. Zou, Superlinear problems, Pacific J. Math., 214 (2004), 145–160. doi: 10.2140/pjm.2004.214.145
[17]	Z. Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 8 (1991), 43–57.
[18]	J. F. Zhao, X. Q. Liu, J. Q. Liu, $p$-Laplacian equations in ${\mathbb{R}}^N$ with finite potential via truncation method, the critical case, J. Math. Anal. Appl., 455 (2017), 58–88. doi: 10.1016/j.jmaa.2017.03.085
[19]	W. M. Zou, Sign-Changing Critical Point Theory, Springer, 2008.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)