$L^\infty$-bounds of solutions for a class of strongly nonlinear elliptic equations in Musielak spaces

Mohamed Bourahma, Abdelmoujib Benkirane, Jaouad Bennouna, Mostafa El Moumni

DOI: http://dx.doi.org/10.12775/TMNA.2020.011


In this paper we establish the existence of bounded solutions to a strongly nonlinear elliptic problem of the form $$ -\mathop{\rm div}\mathcal{A}(x,u,{\nabla}u)+g(x,u,\nabla u)= f \quad\text{in }{\Omega}, $$ with $u\in W^1_0L_\varphi({\Omega})\cap L^{\infty}(\Omega)$, where $$ \mathcal{A}(x,s,\xi)\cdot\xi\geq \overline{\varphi}_{x}^{-1} (\varphi(x,h(|s|)))\varphi(x,|\xi|), $$% $h\colon {\mathbb{R}^+} \to \mathopen ]0,1] $ is a continuous decreasing function with unbounded primitive and $g$ is a non-linearity satisfying $|g(x,s,\xi)|\leq\beta(s)\varphi(x,|\xi|)$. We assume the $\Delta_{2}$-condition on the Musielak function $\varphi$.


Elliptic problems; Musielak-Orlicz-Sobolev spaces; $L^\infty$-estimates; bounded solution

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R. Adams, Sobolev spaces, Academic Press Inc, New York, 1975.

Y. Ahmida, I. Chlebicka, P. Gwiazda and A. Youssfi, Gossez’s approximation theorems in Musielak–Orlicz–Sobolev spaces, J. Funct. Anal. (2018), DOI: 10.1016/j.jfa.2018.05.015.

M. Ait Khellou and A. Benkirane, Correction to: Elliptic inequalities with L1 data in Musielak–Orlicz spaces, Monatsh. Math. 187 (2018), 181–187, DOI: 10.1007/s00605-0181210-y.

A. Alvino, L. Boccardo, V. Ferone, L. Orsina and G. Trombetti, Existence results for nonlinear elliptic equations with degenerate coercivity, Ann. Mat. Pura Appl. Ser. (4) 182, (2003), no. 1, 53-79.

A. Alvino, V. Ferone and G. Trombetti, A priori estimates for a class of non uniformly elliptic equations, Atti Semin. Mat. Fis. Univ. Modena 46 (1998), 381–391.

A. Benkirane and M. Sidi El Vally (Ould Mohameden Val), An existence result for nonlinear elliptic equations in Musielak–Orlicz–Sobolev spaces, Bull. Belg. Math. Soc. Simon Stevin 20 (2013), no. 1, 57–75.

A. Benkirane and M. Sidi El Vally (Ould Mohameden Val), Variational inequalities in Musielak–Orlicz–Sobolev spaces, Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 5, 787–811.

A. Benkirane and A. Youssfi, existence of bounded solutions for a class of strongly nonlinear elliptic equations in Orlicz-Sobolev spaces, Aust. J. Math. Anal. 5 (2008), no. 1, 1–26.

C. Bennett and R. Sharpley, Interpolation of operators, Academic Press, Boston, 1988.

L. Boccardo, A. Dall’Aglio and L. Orsina, Existence and regularity results for some elliptic equations with degenerate coercivity, Atti Semin. Mat. Fis. Univ. Modena 46 (1998), 51–81.

L. Boccardo, F. Murat and J.P. Puel, L∞ estimate for some nonlinear elliptic partial differential equations and application to an existence result, SIAM J. Math. Anal. 2 (1992), 326–333.

L. Boccardo, S. Segura de león and C. Trombetti, Bounded and unbounded solution for a class of quasi-linear elliptic problems with aquadratic gradient term, J. Math. Pures Appl. 80 (2001), 919–940.

M. Bourahma, A. Benkirane and J. Bennouna, Existence of renormalized solutions for some nonlinear elliptic equations in Orlicz spaces, J. Rend. Circ. Mat. Palermo Ser. II (2019), DOI: 10.1007/s12215-019-00399-z.

I. Chlebicka, P. Gwiazda and A. Zatorska-Goldstein, Well-posedness of parabolic equations in the non-reflexive and anisotropic Musielak–Orlicz spaces in the class of renormalized solutions, arXiv:1707.06097v4 [math.AP] 13 Apr 2018.

A. Elmahi and D. Meskine, Nonlinear elliptic problems having natural growth and L1 data in Orlicz spaces, Ann. Mat. Pura Appl. 184 (2005), no. 2, 161–184.

V. Ferone, M.R. Posteraro and J.M. Rakotosone, L∞ estimates for nonlinear elliptic problem with p-growth in the gradient, J. Inequal. Appl. 3 (1999), 109–125.

J.P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. soc. 190, (1974), 163–205.

J.P. Gossez, Surjectivity results for pseudo-monotone mappings in complementary systems, J. Math. Anal. Appl. 53, (1976), 484–494.

J.P. Gossez, A strongly nonlinear elliptic problem in Orlicz–Sobolev spaces, Proc. Sympos. Pure Math. 45 (1986), 455–462.

P. Gwiazda, I. Skrzypczak and A. Zatorska-Goldstein, Existence of renormalized solutions to elliptic equation in Musielak–Orlicz space, J. Differential Equations 264 (2018), 341–377.

P. Gwiazda, A. Świerczewska-Gwiazda and A. Wróblewska, Monotonicity methods in generalized Orlicz spaces for a class of non-Newtonian fluids, Math. Methods Appl. Sci. 33 (2010), 125–137.

P. Gwiazda, P. Wittbold, A. Wróblewska, and A. Zimmermann, Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces, J. Differential Equations 253 (2012), 635–666.

P. Harjulehto and P. Hästö, Orlicz Spaces and Generalized Orlicz Spaces, Lecture Notes in Mathematics, vol. 2236, Springer, Cham, 2019.

M. Krasnosel’skiı̆ and Ya. Rutikiı̆, Convex functions and Orlicz spaces, Groningen, Nordhooff (1969).

J. Musielak, Orlicz spaces and modular spaces, Lecture Note in Mahtematics, 1034, Springer, Berlin, 1983.

G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa (4) 3 (1976), 697–718.

G. Talenti, Nonlinear elliptic equations, Rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl. (4) 120 (1979), 159–184.

G. Talenti, Linear elliptic PDE’s: Level sets, rearrangements and a priori estimates of solutions, Boll. Un. Mat. Ital. 4-B(6), (1985), 917–949.

C. Trombetti, Non-uniformly elliptic equations with natural growth in the gradient, Potential Anal. 18 (2003), 391–404.

A. Youssfi, Existence of bounded solution for nonlinear degenerate elliptic equations in Orlicz spaces, Electron. J. Differential Equations 2007 (2007), no. 54, 1–13.


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