On $\mathcal{p}(x)$-Kirchhoff-type equation involving $\mathcal{p}(x)$-biharmonic operator via genus theory

S. Taarabti; Z.  El Allali; K.  Ben Haddouch

doi:10.37863/umzh.v72i6.6019

S. Taarabti Nat. School Appl. Sci. Agadir Ibn Zohr Univ., Morocco https://orcid.org/0000-0002-3134-9091
Z. El Allali Multidisciplinary Faculty of Nador, Mohammed First Univ., Oujda, Morocco
K. Ben Haddouch Nat. School Appl. Sci. Fes Sidi Mohammed Ben Abdellah Univ., Morocco

DOI: https://doi.org/10.37863/umzh.v72i6.6019

Abstract

UDC 517.9

The paper deals with the existence and multiplicity of nontrivial weak solutions for the $p(x)$-Kirchhoff-type problem

$$ {-M}\!\left(\displaystyle\int\limits_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}\,dx\right)\!\Delta_{p(x)}^{2} u = f(x,u)\quad \mbox{in}\quad \Omega, $$

$$ u = \Delta u = 0\quad \mbox{on}\quad \partial\Omega.$$

By using variational approach and Krasnoselskii's genus theory, we prove the existence and multiplicity of solutions for the $p(x)$-Kirchhoff-type equation.

References

Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math., 66, No 4, 1383 – 1406 (2006) https://doi.org/10.1137/050624522 DOI: https://doi.org/10.1137/050624522

B. Cheng, X. Wu, J. Liu, Multiplicity of nontrivial solutions for Kirchhoff type problems , Boundary Value Problems, 2010, Article ID 268946 (2010), 13 p. https://doi.org/10.1155/2010/268946 DOI: https://doi.org/10.1155/2010/268946

M. Avci, B. Cekic, R. A. Mashiyev, Existence and multiplicity of the solutions of the $p(x)$-Kirchhoff type equation via genus theory , Math. Meth. Appl. Sci., 34, 1751 – 1759 (2011) https://doi.org/10.1002/mma.1485 DOI: https://doi.org/10.1002/mma.1485

J. J. Sun, C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations , Nonlinear Anal., 74, 1212 – 1222 (2011) https://doi.org/10.1016/j.na.2010.09.061 DOI: https://doi.org/10.1016/j.na.2010.09.061

K. C. Chang, Critical point theory and applications, Shanghai Sci. and Technol. Press, Shanghai (1986).

M. A. Krasnoselskii, Topological methods in the theory of nonlinear integral equations, MacMillan, New York (1964).

A. Zang, Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue – Sobolev spaces, Nonlinear Anal., 69, 3629 – 3636 (2008) https://doi.org/10.1016/j.na.2007.10.001 DOI: https://doi.org/10.1016/j.na.2007.10.001

F. J. S. A. Correˆa, G. M. Figueiredo, On a elliptic equation of p-Kirchhoff type via variational methods, Bull. Austr. Math. Soc., 74, 263 – 277 (2006) https://doi.org/10.1017/S000497270003570X DOI: https://doi.org/10.1017/S000497270003570X

X. L. Fan, D. Zhao, On the spaces $L^{p(x)}(Omega)$ and $W^{m,p(x)}(Omega)$, J. Math. Anal. and Appl., 263, 424–446 (2001) https://doi.org/10.1006/jmaa.2000.7617 DOI: https://doi.org/10.1006/jmaa.2000.7617

M. Milhailescu, Existence and multiplicity of solutions for a Neumann problem involving the $p(x)$-Laplacian operator, Nonlinear Anal., 67, 1419 – 1425 (2007) https://doi.org/10.1016/j.na.2006.07.027 DOI: https://doi.org/10.1016/j.na.2006.07.027

D. C. Clark, D. Gilbarg, A variant of the Ljusternik – Schnirelman theory, Indiana Univ. Math. J., 22, No 1, 65 – 74 (1972) https://doi.org/10.1512/iumj.1972.22.22008 DOI: https://doi.org/10.1512/iumj.1972.22.22008

A. R. El Amrouss, F. Moradi, M. Moussaoui, Existence of solutions for fourth-order PDEs with variable exponents, Electron. J. Different. Equat., 2009, No 153, 1 – 13 (2009).

X. L. Fan, X. Fan, A Knobloch-type result for $p(x)$-Laplacian systems, J. Math. and Appl., 282, 453 – 464 (2003) https://doi.org/10.1016/S0022-247X(02)00376-1 DOI: https://doi.org/10.1016/S0022-247X(02)00376-1

A. R. El Amrouss, A. Ourraoui, Existence of solutions for a boundary problem involving $p(x)$-biharmonic operator, Bol. Soc. Parana. Mat., (3)31, No 1, 179 – 192 (2013) https://doi.org/10.5269/bspm.v31i1.15148} DOI: https://doi.org/10.5269/bspm.v31i1.15148

J. H. Yao, Solution for Neumann boundary problems involving the $p(x)$-Laplacian operators, Nonlinear Anal., 68, 1271 – 1283 (2008) https://doi.org/10.1016/j.na.2006.12.020 DOI: https://doi.org/10.1016/j.na.2006.12.020

G. Kirchhoff, Mechanik, Teubner, Leipzig (1883).

N. T. Chung, Multiplicity results for a class of p(x)-Kirchhoff type equations with combined nonlinearities, Electron. J. Qual. Theory Different. Equat., 42, 1 – 13 (2012). DOI: https://doi.org/10.14232/ejqtde.2012.1.42

G. A. Afrouzi, M. Mirzapour, Eigenvalue problems for p(x)-Kirchhoff type equations, Electron. J. Different. Equal., 2013, No 253 (2013).

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory>, Izv. Akad. Nauk SSSR, Ser. Mat., 50, No 4, 675 – 710 (1986).

E. Acerbi, G. Mingione, Gradient estimate for the $p(x)$-Laplacian system, J. reine und angew. Math., 584, 117 – 148 (2005) https://doi.org/10.1515/crll.2005.2005.584.117 DOI: https://doi.org/10.1515/crll.2005.2005.584.117

O. Kova ̃c ̃ik, J. Ra ̃kosnik, On spaces $ L^{p(x)}$ and $W^{k,p(x)}$ , Czechoslovak Math. J., 41(116), 592 – 618 (1991). DOI: https://doi.org/10.21136/CMJ.1991.102493

X. L. Fan, J. S. Shen, D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)}$, J. Math. Anal. and Appl., 262,749 – 760 (2001) https://doi.org/10.1006/jmaa.2001.7618 DOI: https://doi.org/10.1006/jmaa.2001.7618

A. Ambrosetti, A. Malchiodi, Nonlinear analysis and semilinear elliptic problems, Cambridge Stud. Adv. Math., 14 xii+316 pp. ISBN: 978-0-521-86320-9; 0-521-86320-1 (2007) https://doi.org/10.1017/CBO9780511618260 DOI: https://doi.org/10.1017/CBO9780511618260