Quasilinear elliptic problems on non-reflexive Orlicz-Sobolev spaces

Edcarlos D. Silva, Marcos Leandro Carvalho, K. Silva, José Valdo Gonçalves

DOI: http://dx.doi.org/10.12775/TMNA.2019.078

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.

Keywords


Variational methods; quasilinear elliptic problems; positive solutions; reflexive and non-reflexive Banach spaces

Full Text:

PREVIEW FULL TEXT

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 (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. Garcı́a-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 logLgrowth, 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 RN , 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. Garcı́a-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 (or slowly) 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 Uralt́seva 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: 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. Fucı́k, 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.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism