## Rocky Mountain Journal of Mathematics

### Eigenvalues of some $p(x)$-biharmonic problems under Neumann boundary conditions

#### Abstract

In this paper, we study the following $p(x)$-biharmonic problem in Sobolev spaces with variable exponents $$\begin{cases} \triangle ^{2}_{p(x)}u=\lambda ({\partial F } (x,u)/{\partial u}) & x\in \Omega , \\ {\partial u}/{\partial n}=0 & x\in \partial \Omega ,\\ {\partial }(|\triangle u|^{p(x)-2}\triangle u)/{\partial n} =a(x)|u|^{p(x)-2}u & x\in \partial \Omega . \end{cases}$$ By means of the variational approach and Ekeland's principle, we establish that the above problem admits a nontrivial weak solution under appropriate conditions.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 8 (2018), 2543-2558.

Dates
First available in Project Euclid: 30 December 2018

https://projecteuclid.org/euclid.rmjm/1546138820

Digital Object Identifier
doi:10.1216/RMJ-2018-48-8-2543

Mathematical Reviews number (MathSciNet)
MR3894992

Zentralblatt MATH identifier
06999273

#### Citation

Hsini, Mounir; Irzi, Nawal; Kefi, Khaled. Eigenvalues of some $p(x)$-biharmonic problems under Neumann boundary conditions. Rocky Mountain J. Math. 48 (2018), no. 8, 2543--2558. doi:10.1216/RMJ-2018-48-8-2543. https://projecteuclid.org/euclid.rmjm/1546138820

#### References

• S.N. Antontsev and S.I Shmarev, A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness and localization properties of solutions, Nonlin. Anal. Th. Meth. Appl. 60 (2005), 515–545.
• A. Ayoujil and A.R. El Amrouss, On the spectrum of a fourth order elliptic equation with variable exponent, Nonlin. Anal. 71 (2009), 4916–4926.
• K. Ben Haddouch, Z. El Allali, A. Ayoujil and N. Tsouli, Continuous spectrum of a fourth order eigenvalue problem with variable exponent under Neumann boundary conditions, Ann. Univ. Craiova, Math. Comp. Sci. 42 (2015), 42–55.
• M. Cencelj, D. Repovš and Z. Virk, Multiple perturbations of a singular eigenvalue problem, Nonlin. Anal. 119 (2015), 37–45.
• Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math. 66 (2006), 1383–1406.
• D. Edmunds and J. Rakosnik, Sobolev embeddings with variable exponent, Stud. Math. 143 (2000), 267–293.
• A.R. El Amrouss, F. Moradi and M. Moussaoui, Existence and multiplicity of solutions for a $p(x)$-biharmonic problem with Neumann boundary condition, preprint.
• X. Fan, Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl. 339 (2008), 1395–1412.
• X. Fan and X. Han, Existence and multiplicity of solutions for $p(x)$-Laplacian equations in ${\mathbb R}^N$, Nonlin. Anal. 59 (2004), 173–188.
• X. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$, J. Math. Anal. Appl. 262 (2001), 749–760.
• X.L. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl. 263 (2001), 424–446.
• Y. Fu and Y. Shan, On the removability of isolated singular points for elliptic equations involving variable exponent, Adv. Nonlin. Anal. 5 (2016), 121–132.
• B. Ge, Q.-M. Zhou and Y.-H. Wu, Eigenvalues of the $p(x)$-biharmonic operator with indefinite weight, Springer, Berlin, 2014.
• P. Harjulehto, P. Hästö, U.V. Le and M. Nuortio, Overview of differential equations with non-standard growth, Nonlin. Anal. 72 (2010), 4551–4574.
• K. Kefi, On the Robin problem with indefinite weight in Sobolev spaces with variable exponents, Z. Anal. Anwend. 37 (2018), 25–38.
• ––––, $p(x)$-Laplacian with indefinite weight, Proc. Amer. Math. Soc. 139 (2011), 4351–4360.
• K. Kefi and V. Rãdulescu, On a $p(x)$-biharmonic problem with singular weights, Z. angew. Math. Phys. (2017), 68–80.
• K. Kefi and K. Saoudi, On the existence of a weak solution for some singular $p(x)$-biharmonic equation with Navier boundary conditions. Adv. Nonlin. Anal., DOI:.
• R.A. Mashiyev, S. Ogras, Z. Yucedag and M. Avci, Existence and multiplicity of weak solutions for nonuniformly elliptic equations with non-standard growth condition, Compl. Var. Ellip. Eqs. 57 (2012), 579–595.
• M. Mihcailescu, P. Pucci and V. Radulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl. 340 (2008), 687–698.
• ––––, Nonhomogeneous boundary value problems in anisotropic Sobolev spaces, C.R. Acad. Sci. Paris 345 (2007), 561–566.
• G. Molica Bisci and D. Repovš, Multiple solutions for elliptic equations involving a general operator in divergence form, Ann. Acad. Sci. Fenn. Math. 39 (2014), 259–273.
• W. Orlicz, Über konjugierte Exponentenfolgen, Stud. Math. 3 (1931), 200–212.
• V.D. Rădulescu, Nonlinear elliptic equations with variable exponent: Old and new, Nonlin. Anal. 121 (2015), 336–369.
• V.D. Rădulescu and D.D. Repovš, Partial differential equations with variable exponents: Variational methods and qualitative analysis, Mono. Res. Notes Math., Taylor & Francis, London, 2015.
• M. Ruzicka, Electrorheological fluids: Modeling and mathematical theory, Springer, Berlin, 2000.
• S. Taarabti, Z. El Allali and K. Ben Hadddouch, Eigenvalues of the $p(x)$-biharmonic operator with indefinite weight under Neumann boundary conditions, Bol. Soc. Paran. Mat. 36 (2018), 195–213.
• A. Zang and Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue Sobolev spaces, Nonlin. Anal. Th. Meth. Appl. 69 (2008), 3629–3636.
• V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk 50 (1986), 675–710; Math. USSR-Izv. 29 (1987), 33–66 (in English).