## Topological Methods in Nonlinear Analysis

### Quasilinear elliptic problems on non-reflexive Orlicz-Sobolev spaces

#### Abstract

In the paper the existence, uniqueness and the multiplicity of solutions for a quasilinear elliptic problems driven by the $\Phi$-Laplacian operator is established. Here we consider the non-reflexive case taking into account the Orlicz and Orlicz-Sobolev framework. The non-reflexive case occurs when the $N$-function $\widetilde{\Phi}$ does not verify the $\Delta_{2}$-condition. In order to prove our main results we employ variational methods, regularity results and truncation arguments.

#### Article information

Source
Topol. Methods Nonlinear Anal., Advance publication (2019), 26 pp.

Dates
First available in Project Euclid: 7 October 2019

https://projecteuclid.org/euclid.tmna/1570413619

Digital Object Identifier
doi:10.12775/TMNA.2019.078

#### Citation

Silva, Edcarlos D.; Carvalho, Marcos L. M.; Silva, Kaye; Gonçalves, José V. Quasilinear elliptic problems on non-reflexive Orlicz-Sobolev spaces. Topol. Methods Nonlinear Anal., advance publication, 7 October 2019. doi:10.12775/TMNA.2019.078. https://projecteuclid.org/euclid.tmna/1570413619

#### References

• R.A. Adams and J.F. Fournier, Sobolev Spaces, Academic Press, New York, (2003).
• L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. 19 (1992), no. 6, 581–597.
• L. Boccardo and L. Orsina, Leray–Lions operators with logarithmic growth, J. Mat. Anal. Appl. 423 (2015), 608–622.
• G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istituto Lombardo, Accademia di Scienze e Lettere, Rendiconti, Scienze Matematiche, Fisiche, Chimiche e Geologiche. A 112 (1978), no. 2, 332–336.
• M.L.M. Carvalho, J.V. Goncalves and E.D. Silva, On quasilinear elliptic problems without the Ambrosetti–Rabinowitz condition, J. Anal. Mat. Appl 426 (2015), 466–483.
• N.T. Chung and H.Q. Toan, On a nonlinear and non-homogeneous problem without \rom(AR) type condition in Orlicz–Sobolev spaces, Appl. Math. Comput. 219 (2013), 7820–7829.
• P. Clément, B. de Pagter, G. Sweers and F. de Thélin, Existence of solutions to a semilinear elliptic system through Orlicz–Sobolev spaces, Mediterr. J. Math. 1 (2004), no. 3, 241–267.
• P. Clément, M. García-Huidobro, R. Manásevich and K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. Partial Differential Equations 11 (2000), 33–62.
• F.J.S.A. Corrêa, M.L.M. Carvalho, J.V. Gonçalves and K. O. Silva, On the existence of infinite sequences of ordered positive solutions of nonlinear elliptic eigenvalue problems, Adv. Nonlinear Stud. 16 (2016), no. 3, 439–458.
• A.P. di Napoli, Existence and regularity results for a class of equations with logarithmic growth, Nonlinear Anal. 125 (2015), 290–309.
• T.K. Donaldson and N.S. Trudinger, Orlicz–Sobolev spaces and imbedding theorems, J. Funct. Anal. 8 (1971), 52–75.
• L. Esposito and G. Mingione, Partial regularity for minimizers of convex integrals with $L$ logL-growth, NoDEA Nonlinear Differential Equations Appl.7 (2000), 107–125.
• M. Fuchs and G. Seregin, A regularity theory for variational integrals with L logL-growth, Calc. Var. Partial Differential Equations 6 (1998), 171–187.
• M. Fuchs and G. Seregin, Variational methods for fluids of Prandtl–Eyring type and plastic materials with logarithmic hardening, Math. Methods Appl. Sci. 22 (1999), 317–351.
• N. Fukagai, M. Ito, K. Narukawa, Positive solutions of quasilinearelliptic equations with critical Orlicz–Sobolev nonlinearity on $\mathbb{R}^N$, Funkc. Ekv. 49 (2006), 235–267.
• N. Fukagai and K. Narukawa, On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems, Ann. Mat. 186 (2007), 539–564.
• M. García-Huidobro, V.K. Le, R. Manásevich and K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz–Sobolev space setting, NoDEA Nonlinear Differential Equations Appl. 6 (1999), 207–225.
• J.P. Gossez, Orlicz–Sobolev spaces and nonlinear elliptic boundary value problems, Nonlinear Analysis, Function Spaces and Applications, (Proc. Spring School, Horni Bradlo, 1978), Teubner, Leipzig, 1979, pp. 59–94.
• J.P. Gossez, Nonlinear elliptic boundary value problems for equations with raplidy \rom(or slowly\rom) increasing coefficients, Trans. Amer. Math. Soc. 190 (1974), 163–205.
• V.K. Le, A global bifurcation result for quasilinear elliptic equations in Orlicz–Sobolev spaces, Topol. Methods Nonlinear Anal. 15 (2000), 301–327.
• G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203–1219.
• G.M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural\'tseva for elliptic equation, Comm. Partial Differential Equations 16 (1991), 311–361.
• M. Mihailescu and V. Radulescu, Existence and multiplicity of solutions for quasilinear nonhomogeneous problems\rom: An Orlicz–Sobolev space setting, J. Math. Anal. Appl. 330 (2007), 416–432.
• D. Mugnai and N. S. Papageorgiou, Wang's multiplicity result for superlinear $(p,q)$-equations without the Ambrosetti–Rabinowitz condition, Trans. Amer. Math. Soc. 366 (2014), 4919–4937.
• L. Pick, A. Kufner, O. John and S. Fucík, Function Spaces, Vol. 1, second evised and extended edition, De Gruyter Series in Nonlinear Analysis and Applications, Berlin, 2013.
• P. Pucci and J. Serrin, The strong maximum principle revisited, J. Differential Equations 196 (2004), 1–66.
• M.N. Rao and Z.D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1985.
• Z. Tan and F. Fang, Orlicz–Sobolev versus Hölder local minimizer and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl. 402 (2013), 348–370.
• C. Zhang and S. Zhou, On a class of non-uniformly elliptic equations, NoDEA Nonlinear Differential Equations Appl. 19 (2012), 345–363.