## Annals of Functional Analysis

### Solutions of nonlinear elliptic problems with lower order terms

#### Abstract

We give an existence result for strongly nonlinear elliptic equations of the form $-{\rm div}(a(x,u,\nabla u))+g(x,u,\nabla u)+H(x,\nabla u) = \mu\ \text{in}\ \Omega,$ where the right hand side belongs to $L^1(\Omega)+W^{-1,p'}(\Omega)$ and $- {\rm div}(a(x,u,\nabla u))$ is a Leray--Lions type operator with growth $|\nabla u|^{p-1}$ in $\nabla u$. The critical growth condition on $g$ is with respect to $\nabla u$ and no growth condition with respect to $u$, while the function $H(x,\nabla u)$ grows as $|\nabla u|^{p-1}$.

#### Article information

Source
Ann. Funct. Anal., Volume 6, Number 1 (2015), 34-53.

Dates
First available in Project Euclid: 19 December 2014

https://projecteuclid.org/euclid.afa/1419001449

Digital Object Identifier
doi:10.15352/afa/06-1-4

Mathematical Reviews number (MathSciNet)
MR3297786

Zentralblatt MATH identifier
1319.35038

#### Citation

Akdim, Youssef; Benkirane, Abdelmoujib; El Moumni, Mostafa. Solutions of nonlinear elliptic problems with lower order terms. Ann. Funct. Anal. 6 (2015), no. 1, 34--53. doi:10.15352/afa/06-1-4. https://projecteuclid.org/euclid.afa/1419001449

#### References

• A. Alvino and G. Trombetti, Sulle migliori costanti di maggiorazione per una classe di equazioni ellittiche degeneri, (Italian) Ricerche Mat. 27 (1978), 413–428.
• E. Beckenbak and R. Beliman, Inequalities, Springer-Verlag, 1965.
• A. Benkirane, Approximations de type Hedberg dans les espaces $W\sp mL\log L(\Omega)$ et applications. (French) [Hedberg-type approximations in the spaces $W\sp mL\log L(\Omega)$ and applications], Ann. Fac. Sci. Toulouse Math. (5) 11 (1990), no. 2, 67–78.
• A. Benkirane and J.P. Gossez, An approximation theorem for higher order Orlicz-Sobolev spaces, Studia Math 92 (1989), 231–255.
• A. Benkirane and A. Elmahi, An existence theorem form a strongly nonlinear problems in Orlicz space, Nonlinear Anal. 3 (1999), no. 6, 11–24.
• A. Benkirane and A. Elmahi, Strongly nonlinear elliptic equations having natural growth terms and $L^1$ data, Nonlinear Anal. 39 (2000), no. 4, 403–411.
• A. Benkirane, A. Elmahi and D. Meskine, An existence theorem for a class of elliptic problems in $L^1$ , Appl. Math. (Warsaw) 29 (2002), no. 4, 439–457.
• A. Youssfi, A. Benkirane and M. El Moumni, Bounded solutions of unilateral problems for strongly nonlinear equations in Orlicz spaces, Electron. J. Qual. Theory Differ. Equ. 2013, No. 21, 25pp.
• A. Bensoussan, L. Boccardo and F. Murat, On a non linear partial differential equation having natural growth terms and unbounded solution, Ann. Inst. Poincaré 5 (1988), 347–364.
• L. Boccardo and T. Gallouët, Nonliner elliptic equations with right hand side measure, Comm. Partial Differential Equations 17 (1992), no. 3-4, 641–655.
• L. Boccardo and T. Gallouët, Strongly nonlinear elliptic equation having natural growth terms and $L^{1}$ data, Nonlinear Anal. 19 (1992), 573–578.
• L. Boccardo,T. Gallouët and L. Orsina, Existence and nonexistence of solutions for some nonlinear elliptic equations, J. Anal. Math. 73 (1997), 203–223.
• L. Boccardo, F. Murat and J.P. Puel, Existance of bounded solution for non linear elliptic unilateral problems, Annali Mat. Pura Appl. 152 (1988), 183–196.
• H. Brezis and W. Strauss, Semilinear second-order elliptic equations in $L^1$, J. Math. Soc. Japan 25, (1973), no. 4, 565–590.
• T. Del Vecchio, Nonlinear elliptic equations with measure data, Potential Anal. 4 1995), no. 2, 185–203.
• T. Gallouët and R. Herbin, Existence of a solution to a coupled elliptic system, Appl. Math. Lett. 7 (1994), 49–55.
• T. Goudon and M. Saad, On a Fokker-Planck equation arising in population dynamics, Rev. Mat. Complut. 11 (1998), 353–372.
• R. Lewandowski, The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy-viscosity, Nonlinear Anal. T.M.A 28 (1997), 393–417.
• J.-L. Lions, Mathematical topics in fluid mechanics, incompressible models, Oxford Lecture Series in Math and its Applications 3 Clarendon Press, 1996.
• G. H. Hardy, Littlewood and G. Polya, Inequalities , Cambrige University Press, Cambrige, 1964.
• J.-L. Lions, Quelques méthodes de résolution des problème aux limites non lineaires, Dundo, Paris, 1969.
• V.M. Monetti and L. Randazzo, Existence results for nonlinear elliptic equations with $p$-growth in the gradient, Ricerche Mat. 49 (2000), no. 1, 163–181.
• V. Radulescu and M. Willem, Elliptic systems involving finite Radon measures, Differential Integral Equations 16 (2003), no. 2, 221–229.
• A. Porretta, Existence for elliptic equations in $L^{1}$ having lower order terms with natural growth, Portugual. Math. 57 (2000), 179–190.
• G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann.Mat. Pura Appl. 120 (1979), 159–184.