## Taiwanese Journal of Mathematics

### Eigenvalue Problems for Fractional $p(x,y)$-Laplacian Equations with Indefinite Weight

Nguyen Thanh Chung

#### Abstract

In this paper, we consider a class of eigenvalue problems for fractional $p(x,y)$-Laplacian equations with indefinite weight in fractional Sobolev space with variable exponent. Under some suitable conditions on the growth rates involved in the problem, we establish some results on the existence of a continuous family of eigenvalues using variational techniques and Ekeland's variational principle.

#### Article information

Source
Taiwanese J. Math., Advance publication (2019), 21 pages.

Dates
First available in Project Euclid: 23 April 2019

https://projecteuclid.org/euclid.twjm/1555984822

Digital Object Identifier
doi:10.11650/tjm/190404

#### Citation

Chung, Nguyen Thanh. Eigenvalue Problems for Fractional $p(x,y)$-Laplacian Equations with Indefinite Weight. Taiwanese J. Math., advance publication, 23 April 2019. doi:10.11650/tjm/190404. https://projecteuclid.org/euclid.twjm/1555984822

#### References

• A. Abdou and A. Marcos, Existence and multiplicity of solutions for a Dirichlet problem involving perturbed $p(x)$-Laplacian operator, Electron. J. Differential Equations 2016 (2016), no. 197, 19 pp.
• E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal. 156 (2001), no. 2, 121–140.
• C. O. Alves and M. C. Ferreira, Existence of solutions for a class of $p(x)$-Laplacian equations involving a concave-convex nonlinearity with critical growth in $\mathbb{R}^N$, Topol. Methods Nonlinear Anal. 45 (2015), no. 2, 399–422.
• S. Antontsev, F. Mirandac and L. Santos, Blow-up and finite time extinction for $p(x,t)$-curl systems arising in electromagnetism, J. Math. Anal. Appl. 440 (2016), no. 1, 300–322.
• G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in $\mathbb{R}^N$, J. Differential Equations 255 (2013), no. 8, 2340–2362.
• A. Baalal and M. Berghout, Traces and fractional Sobolev extension domains with variable exponent, Int. J. Math. Anal. (Ruse) 12 (2018), no. 2, 85–98.
• A. Bahrouni, Comparison and sub-supersolution principles for the fractional $p(x)$-Laplacian, J. Math. Anal. Appl. 458 (2018), no. 2, 1363–1372.
• A. Bahrouni and V. D. Rădulescu, On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent, Discrete Contin. Dyn. Syst. Ser. S 11 (2018), no. 3, 379–389.
• A. Bahrouni and D. Repovš, Existence and nonexistence of solutions for $p(x)$-curl systems arising in electromagnetism, Complex Var. Elliptic Equ. 63 (2018), no. 2, 292–301.
• G. Barletta, A. Chinn\`\i and D. O'Regan, Existence results for a Neumann problem involving the $p(x)$-Laplacian with discontinuous nonlinearities, Nonlinear Anal. Real World Appl. 27 (2016), 312–325.
• R. Bartolo and G. M. Bisci, Asymptotically linear fractional $p$-Laplacian equations, Ann. Mat. Pura Appl. (4) 196 (2017), no. 2, 427–442.
• Z. Binlin, G. M. Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity 28 (2015), no. 7, 2247–2264.
• G. M. Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications 162, Cambridge University Press, Cambridge, 2016.
• M. Bouslimi and K. Kefi, Existence of solution for an indefinite weight quasilinear problem with variable exponent, Complex Var. Elliptic Equ. 58 (2013), no. 12, 1655–1666.
• L. Diening, P. Harjulehto, P. Hästö and M. R\ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics 2017, Springer, Heidelberg, 2011.
• D. E. Edmunds and J. Rákosník, Sobolev embeddings with variable exponent, Studia Math. 143 (2000), no. 3, 267–293.
• I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353.
• A. Fiscella, G. M. Bisci and R. Servadei, Multiplicity results for fractional Laplace problems with critical growth, Manuscripta Math. 155 (2018), no. 3-4, 369–388.
• U. Kaufmann, J. D. Rossi and R. Vidal, Fractional Sobolev spaces with variable exponents and fractional $p(x)$-Laplacians, Electronic J. Qual. Theory Differ. Equ. 2017 (2017), no. 76, 10 pp.
• K. Kefi, $p(x)$-Laplacian with indefinite weight, Proc. Amer. Math. Soc. 139 (2011), no. 12, 4351–4360.
• M. Mihăilescu and V. Rădulescu, Eigenvalue problems with weight and variable exponent for the Laplace operator, Anal. Appl. (Singap.) 8 (2010), no. 3, 235–246.
• ––––, Concentration phenomena in nonlinear eigenvalue problems with variable exponents and sign-changing potential, J. Anal. Math. 111 (2010), 267–287.
• J. Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics 1034, Springer-Verlag, Berlin, 1983.
• L. M. Del Pezzo and J. D. Rossi, Traces for fractional Sobolev spaces with variable exponents, Adv. Oper. Theory 2 (2017), no. 4, 435–446.
• P. Pucci, M. Xiang and B. Zhang, Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal. 5 (2016), no. 1, 27–55.
• V. D. Rădulescu, Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal. 121 (2015), 336–369.
• M. R\ružička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics 1748, Springer-Verlag, Berlin 2000.
• R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. 33 (2013), no. 5, 2105–2137.
• M. Xiang, B. Zhang and D. Yang, Multiplicity results for variable-order fractional Laplacian equations with variable growth, Nonlinear Anal. 178 (2019), 190–204.