Multiple Solutions for Problems Involving p(x)-Laplacian and p(x)-Biharmonic Operators
DOI:
https://doi.org/10.15407/mag20.02.235Анотація
У роботі розглянуто таку $p(x)$-бігармонічну задачу з нелінійністю Гарді:
\begin{equation*}
\left\{
\begin{aligned}
&\Delta_{p(x)}^{2}u-\Delta_{p(x)}u =\lambda \frac{|u|^{p(x)-2}u}{\delta(x)^{2p(x)}}
+f(x,u) && \mbox{in }\Omega,
\\
&u=0 && \mbox{on }\partial\Omega,
\\
&|\nabla u|^{p(x)-2}\frac{\partial u}{\partial n}=g(x,u) && \mbox{on }\partial\Omega,
\end{aligned}
\right.
\end{equation*}
де $ \Omega\subset R^{N}$ $( N\geq 3 )$, $\Delta_{p(x)}$ є $p(x)$-лапласіаном і $\Delta_{p(x)}^{2}$ є $p(x)$- бігармонічним оператором. Точніше, для доведення існування і множинності розв'язків варіаційні методи скомбіновано з теорією узагальнених просторів Лебега і Соболєва за відповідних умов на нелінійності $f$ і $g$.
Mathematical Subject Classification 2020: 35J20, 35J60, 35G30, 35J35
Ключові слова:
p(x)-бiгармонiчний оператор, p(x)-лапласiан, теорема про симетричний гiрський перевал, узагальнений простiр СоболєваПосилання
R. Alsaedi, A. Dhifli, and A. Ghanmi, Low perturbations of $p$-biharmonic equations with competing nonlinearities, Complex Var. Elliptic Equ. 66 (2020), No. 4, 642--657. https://doi.org/10.1080/17476933.2020.1747057
A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), No. 4, 349--381. https://doi.org/10.1016/0022-1236(73)90051-7
A. Bahrouni and V.D. Rǎdulescu, On a new fractional Sobolev space and application to nonlocal variational problems with variable exponent, Discrete Contin. Dyn. Syst. Ser. A, 11 (2018), No. 3, 379--389. https://doi.org/10.3934/dcdss.2018021
R. Chammem, A. Ghanmi, and A. Sahbani, Existence and multiplicity of solution for some Styklov problem involving $p(x)$-Laplacian operator, Appl. Anal. 101 (2021), No. 7, 2401--2417. https://doi.org/10.1080/00036811.2020.1807014
R. Chammem and A. Sahbani, Existence and multiplicity of solution for some Styklov problem involving $(p_{1}(x),p_{2}(x))$-Laplacian operator, Appl. Anal. 102 (2023), No. 3, 709--724. https://doi.org/10.1080/00036811.2021.1961758
R. Chammem, A. Ghanmi, and A. Sahbani, Nehari manifold for singular fractional $p(x,·)$-Laplacian problem, Complex Var. Elliptic Equ. 68 (2023), No. 9, 1603--1625. https://doi.org/10.1080/17476933.2022.2069757
Y. Chen, S. Levine, and M. Rao, Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math. 66 (2006), 1383--1406. https://doi.org/10.1137/050624522
A. Drissi, A. Ghanmi, and D.D. Repovš, Singular $p$-biharmonic problems involving the Hardy-Sobolev exponent, Electron. J. Differ. Equ., 2023 (2023), No. 61, 1--12. https://doi.org/10.58997/ejde.2023.61
A. El khalil, M. El Moumni, M.D. Morchid Alaoui, and A. Touzani, $p(x)$-Biharmonic operator involving p(x)-Hardy’s inequality, Georgian Math. J., 27 (2018), 233--247. https://doi.org/10.1515/gmj-2018-0013
X. Fan and D. Zhao, On the spaces $ L^{p}(Ω) $ and $ W^{m,p(x)}(Ω) $, J. Math. Anal. Appl. 263 (2001), 424--446. https://doi.org/10.1006/jmaa.2000.7617
A. Ghanmi and A. Sahbani, Existence results for $p(x)$-biharmonic problems involving a singular and a Hardy type nonlinearities, AIMS Math. 8 (2023), No. 12, 29892--29909. https://doi.org/10.3934/math.20231528
M. Jennane, Infinitely many weak solutions for problems involving both $p(x)$-Laplacian and $p(x)$-biharmonic operators, Eur. J. Math. Stat. 3 (2022), No. 4, 71--80. https://doi.org/10.24018/ejmath.2022.3.4.141
C. Ji and W.Wang, On the $p$-biharmonic equation involving concave-convex nonlinearities and sign-changing weight function. Electron. J. Qual. Theory Differ. Equ., 2 (2012), 1--17. https://doi.org/10.14232/ejqtde.2012.1.2
K. Kefi and V.D. Radulescu, On a p(x)-biharmonic problem with singular weights, Z. Angew. Math. Phys. 68 (2017), 68--80. https://doi.org/10.1007/s00033-017-0827-3
M. Laghzal, A. El Khalil, M. Alaoui, and A. Touzani, Eigencurves of the $p(x)$-biharmonic operator with a Hardy-type term, Moroccan J. Pure Appl. Anal. 6 (2020), No. 2, 198--209. https://doi.org/10.2478/mjpaa-2020-0015
M. Ruzicka, Electrortheological fluids: modeling and mathematical theory, Springer, Berlin, 2000. https://doi.org/10.1007/BFb0104030
J. Sun, J. Chu, and T. Wu, Existence and multiplicity of nontrivial solutions for some biharmonic equations with p-Laplacian, J. Differ. Equ, 262 (2017), 945--977. https://doi.org/10.1016/j.jde.2016.10.001
J. Sun and T. Wu, Existence of nontrivial solutions for a biharmonic equation with $p$-Laplacian and singular sign-changing potential, Appl. Math. Lett. 66 (2017), 61--67. https://doi.org/10.1016/j.aml.2016.11.001
S. Saiedinezhad, Existence of solutions to biharmonic equations with sing changing coefficients, Electron. J. Differ. Equ. 99 (2018), 1--9.
A. Sahbani, Infinitely many solutions for problems involving Laplacian and biharmonic operators, Complex Var. Elliptic Equ., 1--14. https://doi.org/10.1080/17476933.2023.2287007
L. Xu and H. Chen, Existence of positive ground solutions for biharmonic equations via Pohožaev-Nehari manifold, Topol. Meth. Nonl. Anal. 52 (2018), 541--560. https://doi.org/10.12775/TMNA.2018.015
V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 675--710 (Russian); Engl. transl.: Math. USSR-Izv. 29 (1987), 33--66. https://doi.org/10.1070/IM1987v029n01ABEH000958
A.B. Zang, Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces, Nonlinear Anal. 69 (2008), 3629--3636. https://doi.org/10.1016/j.na.2007.10.001
