INFINITELY MANY LOW- AND HIGH-ENERGY SOLUTIONS FOR A CLASS OF ELLIPTIC EQUATIONS WITH VARIABLE EXPONENT

Chang-Mu Chu; Haidong Liu

doi:10.11948/20190319

2020 Volume 10 Issue 5

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2020 > 10(5): 2024-2035

Next Article Previous Article

Chang-Mu Chu, Haidong Liu. INFINITELY MANY LOW- AND HIGH-ENERGY SOLUTIONS FOR A CLASS OF ELLIPTIC EQUATIONS WITH VARIABLE EXPONENT[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 2024-2035. doi: 10.11948/20190319

Citation:

Chang-Mu Chu, Haidong Liu. INFINITELY MANY LOW- AND HIGH-ENERGY SOLUTIONS FOR A CLASS OF ELLIPTIC EQUATIONS WITH VARIABLE EXPONENT[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 2024-2035. doi: 10.11948/20190319

INFINITELY MANY LOW- AND HIGH-ENERGY SOLUTIONS FOR A CLASS OF ELLIPTIC EQUATIONS WITH VARIABLE EXPONENT

Chang-Mu Chu^1, ,,
Haidong Liu²

1.
School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, 550025, China
2.
College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, China

Corresponding author: Email: gzmychuchangmu@sina.com.(C. Chu)
Fund Project: C. Chu is supported by National Natural Science Foundation of China (No. 11861021). H. Liu is supported National Natural Science Foundation of China (Nos. 11701220, 11926334, 11926335)

Abstract

Abstract

This paper is concerned with the $ p(x) $-Laplacian equation of the form $\left\{ \begin{array}{l} - {\Delta _{p(x)}}u = Q(x)|u{|^{r(x) - 2}}u,{\rm{ in}}\;\Omega ,\\ {\rm{u = 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; on}}\;\partial \Omega , \end{array} \right.$ where $ \Omega\subset{\mathbb{R}}^N $ is a smooth bounded domain, $ 1<p^- = \min_{x\in\overline{\Omega}}p(x)\leq p(x)\leq\max_{x\in\overline{\Omega}}p(x) = p^+<N $, $ 1\leq r(x)<p^{*}(x) = \frac{Np(x)}{N-p(x)} $, $ r^- = \min_{x\in \overline{\Omega}}r(x)<p^- $, $ r^+ = \max_{x\in\overline{\Omega}}r(x)>p^+ $ and $ Q: \overline{\Omega}\to{\mathbb{R}} $ is a nonnegative continuous function. We prove that (0.1) has infinitely many small solutions and infinitely many large solutions by using the Clark's theorem and the symmetric mountain pass lemma.
- $p(x)$-Laplacian /
- variable exponent /
- infinitely many solutions /
- Clark's theorem, symmetric mountain pass lemma

FullText(HTML)

References

[1]	S. Aouaoui, Multiple solutions to some degenerate quasilinear equation with variable exponents via perturbation method, J. Math. Anal. Appl., 2018, 458(2018), 1568-1596. Google Scholar
[2]	S. Aouaoui, Eigenvalue problem with nonstandard concave and convex nonlinearities, Mediterr. J. Math., 2014, 11, 1149-1169. doi: 10.1007/s00009-013-0333-6 CrossRef Google Scholar
[3]	S. Aouaoui, Existence and multiplicity results for some eigenvalue problems involving variable exponents, Nonlinear Anal., 2013, 80, 76-87. doi: 10.1016/j.na.2012.12.007 CrossRef Google Scholar
[4]	S. Aouaoui, Existence of solutions for eigenvalue problems with nonstandard growth conditions, Electron. J. Differ. Eq., 2013, 176, 1-14. doi: 10.1186/1687-2770-2013-177 CrossRef Google Scholar
[5]	G. M. Bisci, V.D. Radulescu and R. Servadei, Low- and high-energy solutions of nonlinear elliptic oscillatory problems, C. R. Math. Acad. Sci. Paris, 2014, 352(2), 117-122. doi: 10.1016/j.crma.2013.11.015 CrossRef Google Scholar
[6]	J. Chabrowski and Y. Fu, Existence of solutions for $p(x)$-Laplacian problems on a bounded domain, J. Math. Anal. Appl., 2005, 306, 604-618. doi: 10.1016/j.jmaa.2004.10.028 CrossRef Google Scholar
[7]	Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 2006, 66, 1383-1406. doi: 10.1137/050624522 CrossRef Google Scholar
[8]	X. Fan, Remarks on eigenvalue problems involving the $p(x)$-Laplacian, J. Math. Anal. Appl., 2009, 352, 85-98. doi: 10.1016/j.jmaa.2008.05.086 CrossRef Google Scholar
[9]	X. Fan, Global $C.{1, \alpha}$ regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 2007, 235, 397-417. doi: 10.1016/j.jde.2007.01.008 CrossRef Google Scholar
[10]	X. Fan and Q. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problem, Nonlinear Anal., 2003, 52, 1843-1852. doi: 10.1016/S0362-546X(02)00150-5 CrossRef Google Scholar
[11]	X. Fan and D. Zhao, On the generalized Orlicz-Sobolev space $W.{m, p(x)}(\Omega)$, J. Gansu Educ. College, 1998, 12, 1-6. Google Scholar
[12]	J. Gao, P. Zhao and Y. Zhang, Compact Sobolev embedding theorems involving symmetry and its application, Nonlinear Differential Equations Appl., 2010, 17, 161-180. doi: 10.1007/s00030-009-0046-5 CrossRef Google Scholar
[13]	J. Garcia-Mellian, J.D. Rossi and J.C.S De Lis, A variable exponent diffusion problem of concave-convex nature, Topological Methods in Nonlinear Analysis, 2016, 47, 613-639. Google Scholar
[14]	C. Ji and F. Fang, Infinitely many solutions for the $p(x)$-Laplacian equations without $(AR)$-type growth condition, Ann. Polon. Math., 2012, 105, 87-99. doi: 10.4064/ap105-1-8 CrossRef Google Scholar
[15]	R. Kajikiya, Superlinear elliptic equations with singular coefficients on the boundary, Nonlinear Anal., 2010, 73, 2117-2131. doi: 10.1016/j.na.2010.05.039 CrossRef Google Scholar
[16]	Y. Komiya and R. Kajikiya, Existence of infinitely many solutions for the $(p, q)$-Laplace equation, Nonlinear Differ. Equ. Appl., 2016, 49, 1-23. Google Scholar
[17]	Z. Liu and Z. Wang, On Clark's theorem and its applications to partially sublinear problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2015, 32, 1015-1037. doi: 10.1016/j.anihpc.2014.05.002 CrossRef Google Scholar
[18]	R. A. Mashiyev, S. Ogras, Z. Yucedag and M. Avci, The Nehari manifold approach for Dirichlet problem involving the $p(x)$-Laplacian equation, J. Korean Math. Soc., 2010, 47, 845-860. doi: 10.4134/JKMS.2010.47.4.845 CrossRef Google Scholar
[19]	M. Mih$\check{a}$ilescu and V. R$\check{a}$dulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc., 2017, 135, 2929-2937. Google Scholar
[20]	T. C. Nguyen, Multiple solutions for a class of $p(x)$-Laplacian problems involving concave-convex nonlinearities, Electron. J. Qual. Theory Differential Equations, 2013, 26, 1-17. Google Scholar
[21]	P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, 65, American Mathematical Society, Providence, RI, 1986. Google Scholar
[22]	V. R$\check{a}$dulescu, Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal., 2015, 121, 336-369. doi: 10.1016/j.na.2014.11.007 CrossRef Google Scholar
[23]	M. R$\mathring{\mbox{u}}$žička, Electrorheological fluids: modeling and mathematical theory, Volume 1748 of Lecture Notes in Mathematics, Springer, Berlin, 2000. Google Scholar
[24]	E. A. B. Silva and M. S. Xavier, Multiplicity of solutions for quasilinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2003, 20, 341-358. doi: 10.1016/S0294-1449(02)00013-6 CrossRef Google Scholar
[25]	Z. Tan and F. Fang, On superlinear $p(x)$-Laplacian problems without Ambrosetti and Rabinowitz condition, Nonlinear Anal., 2012, 75, 3902-3915. doi: 10.1016/j.na.2012.02.010 CrossRef Google Scholar
[26]	J. Yao and X. Wang, On an open problem involving the $p(x)$-Laplacian--a further study on the multiplicity of weak solutions to $p(x)$-Laplacian equations, Nonlinear Anal., 2008, 69, 1445-1453. doi: 10.1016/j.na.2007.06.044 CrossRef Google Scholar
[27]	Z. Yucedag, Existence of solutions for $p(x)$ Laplacian equations without Ambrosetti-Rabinowitz type condition, Bull. Malays. Math. Sci. Soc., 2015, 38, 1023-1033. doi: 10.1007/s40840-014-0057-1 CrossRef Google Scholar
[28]	A. Zang, $p(x)$-Laplacian equations satisfying Cerami condition, J. Math. Anal. Appl., 2008, 337, 547-555. doi: 10.1016/j.jmaa.2007.04.007 CrossRef Google Scholar
[29]	Q. Zhang and C. Zhao, Existence of strong solutions of a $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition, Comput. Math. Appl., 2015, 69, 1-12. doi: 10.1016/j.camwa.2014.10.022 CrossRef Google Scholar