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MULTIPLE SOLUTIONS FOR $p(x)$-LAPLACIAN EQUATIONS WITH NONLINEARITY SUBLINEAR AT ZERO

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  29 January 2024

SHIBO LIU Open the ORCID record for SHIBO LIU [Opens in a new window]
SHIBO LIU*
Affiliation:
Department of Mathematics and Systems Engineering, Florida Institute of Technology, Melbourne 32901, FL, USA
*
e-mail: [email protected]
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Abstract

We consider the Dirichlet problem for $p(x)$-Laplacian equations of the form

$$ \begin{align*} -\Delta_{p(x)}u+b(x)\vert u\vert ^{p(x)-2}u=f(x,u),\quad u\in W_{0}^{1,p(x)}(\Omega). \end{align*} $$

The odd nonlinearity $f(x,u)$ is $p(x)$-sublinear at $u=0$ but the related limit need not be uniform for $x\in \Omega $. Except being subcritical, no additional assumption is imposed on $f(x,u)$ for $|u|$ large. By applying Clark’s theorem and a truncation method, we obtain a sequence of solutions with negative energy and approaching the zero function $u=0$.

Keywords

p(x)-LaplacianClark’s theoremtruncation(PS)c condition

MSC classification

Primary: 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 110 , Issue 2 , October 2024 , pp. 346 - 354
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

Chinnì, A., Sciammetta, A. and Tornatore, E., ‘Existence of non-zero solutions for a Dirichlet problem driven by $\left(p(x),q(x)\right)$ -Laplacian’, Appl. Anal. 101(15) (2022), 5323–5333.10.1080/00036811.2021.1889523CrossRefGoogle Scholar
Diening, L., Harjulehto, P., Hästö, P. and Růžička, M., Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017 (Springer, Heidelberg, 2011).10.1007/978-3-642-18363-8CrossRefGoogle Scholar
Fan, X. and Zhang, Q., ‘Existence of solutions for $p(x)$ -Laplacian Dirichlet problem’, Nonlinear Anal. 52(8) (2003), 1843–1852.10.1016/S0362-546X(02)00150-5CrossRefGoogle Scholar
Fan, X. and Zhao, D., ‘On the spaces ${L}^{p(x)}(\Omega)$ and ${W}^{m,p(x)}(\Omega )$ ’, J. Math. Anal. Appl. 263(2) (2001), 424–446.10.1006/jmaa.2000.7617CrossRefGoogle Scholar
He, W. and Wu, Q., ‘Multiplicity results for sublinear elliptic equations with sign-changing potential and general nonlinearity’, Bound. Value Probl. 2020 (2020), Article no. 159, 9 pages.10.1186/s13661-020-01456-8CrossRefGoogle Scholar
Jikov, V. V., Kozlov, S. M. and Oleĭnik, O. A., Homogenization of Differential Operators and Integral Functionals (Springer-Verlag, Berlin, 1994). Translated from the Russian by G. A. Yosifian.10.1007/978-3-642-84659-5CrossRefGoogle Scholar
Liang, S. and Zhang, J., ‘Infinitely many small solutions for the $p(x)$ -Laplacian operator with nonlinear boundary conditions’, Ann. Mat. Pura Appl. (4) 192(1) (2013), 1–16.10.1007/s10231-011-0209-yCrossRefGoogle Scholar
Liu, Z. and Wang, Z.-Q., ‘On Clark’s theorem and its applications to partially sublinear problems’, Ann. Inst. H. Poincaré Anal. Non Linéaire 32(5) (2015), 1015–1037.10.1016/j.anihpc.2014.05.002CrossRefGoogle Scholar
Růžička, M., Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, 1748 (Springer-Verlag, Berlin, 2000).10.1007/BFb0104029CrossRefGoogle Scholar
Taarabti, S., ‘Positive solutions for the $p(x)$ -Laplacian: application of the Nehari method’, Discrete Contin. Dyn. Syst. Ser. S 15(1) (2022), 229–243.10.3934/dcdss.2021029CrossRefGoogle Scholar
Tavares, L. and Sousa, J. V. C., ‘Multiplicity results for a system involving the $p(x)$ -Laplacian operator’, Appl. Anal. 102(5) (2023), 1271–1280.10.1080/00036811.2021.1979226CrossRefGoogle Scholar