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Existence of bounded solutions for non linear elliptic unilateral problems. (English) Zbl 0687.35042
Let ${\Omega }$ be a bounded domain in ${ℝ}^{n}$, $\psi$ a measurable function on ${\Omega }$, $p>1$, and $A\left(u\right)=diva\left(x,u,Du\right)+{a}_{0}\left(x,u,Du\right)$ an elliptic quasilinear differential operator whose coefficients a, ${a}_{0}$ satisfy natural regularity and growth conditions which in particular guarantee that A is a continuous, pseudomonotone operator from the Sobolev space ${W}_{0}^{1,p}\left({\Omega }\right)$ into its dual. The authors prove the existence of a solution $u\in {W}_{0}^{1,p}\left({\Omega }\right)\cap {L}^{\infty }\left({\Omega }\right)$ of the variational inequality $u\ge \psi$, $+\int H\left(x,u,Du\right)\left(v-u\right)dx\ge 0$ for all $v\in {W}_{0}^{1,p}\left({\Omega }\right)\cap {L}^{\infty }\left({\Omega }\right)$ such that $v\ge \psi$. Here it is important to notice that the inhomogeneous term H is allowed to grow like ${|Du|}^{p}$. The proof is carried out by an approximation of H by bounded functions ${H}_{ϵ}$ for which a solution ${u}_{ϵ}$ of the corresponding problem is known to exist. Then it is shown that the family ${u}_{ϵ}$ is compact in ${W}_{0}^{1,p}\left({\Omega }\right)$.
Reviewer: F.Tomi

##### MSC:
 35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000) 47H05 Monotone operators (with respect to duality) and generalizations 35J65 Nonlinear boundary value problems for linear elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 49J40 Variational methods including variational inequalities
##### References:
 [1] H. Amann -M. G. Crandall,On some existence theorems for semilinear elliptic equa- tions, Indiana Univ. Math. J.,27 (1978), pp. 779–790. · Zbl 0391.35030 · doi:10.1512/iumj.1978.27.27050 [2] L. Boccardo,An L s-estimate for the gradient of solutions of some nonlinear unilateral problems, Ann. Mat. Pura Appl.,141 (1985), pp. 277–287. · Zbl 0599.49009 · doi:10.1007/BF01763177 [3] L. Boccardo -F. Murat -J. P. Puel,Existence de solutions faibles pour des équations elliptiques quasilinéaires à croissance quadratique, in Konlinear partial differential equations and their applications, Collège de France Seminar, Vol. IV, ed. byH. Brezis andJ. L. Lions, Research Notes in Mathematics,84 Pitman, London, (1983), pp. 19–73. [4] L. Boccardo -F. Murat -J. P. Puel,Résultats d’existence pour certains problèmes elliptiques quasilinéaires, Ann. Scuola Norm. Sup. Pisa,11 (1984), pp. 213–235. [5] A, Bensoussan -J. Frehse,Nonlinear elliptic systems in stochastic game theory, J. reine ang. Math.,350 (1984), pp. 23–67. [6] A. Bensoussan -J. Frehse -U. Mosco,A stochastic impulse control problem with quadratic growth Mamiltonian and the corresponding quasi-variational inequality, J. reine ang. Math.,331 (1982), pp. 124–145. [7] H. Brezis,Equations et inéquations non linéaires dans les espaces en dualité, Ann. Inst. Fourier,18 (1968), pp. 115–175. [8] H. Brezis,Problèmes unilatéraux, J. Math. Pures et Appl.,51 (1972), pp. 1–168. [9] F. E. Browder,Existence theorems for nonlinear partial differential equations, in Proceedings of Symposia in Pure Mathematics, Vol. XVI, ed. byS. S. Chern andS. Smale, American Mathematical Society, Providence, (1970), pp. 1–60. [10] P. Donato -D. Giachetti,Quasilinear elliptic equations with quadratic growth in unbounded domains, Nonlinear Anal. T.M.A.,10 (8) (1986), pp. 791–804. · Zbl 0602.35036 · doi:10.1016/0362-546X(86)90038-6 [11] P. Donato -D. Giachetti,Unilateral problems with quadratic growth in unbounded domains, Boll. Un. Mat. Ital., (6)5 A (1986), pp. 361–369. [12] D. Kinderlehrer -G. Stampacchia,An introduction to variational inequalities and their applications, Academic Press, New York, 1980. [13] P. Hartman -G. Stampacchia,On some nonlinear elliptic differential functional equations, Acta Math.,115 (1966), pp. 153–188. · Zbl 0142.38102 · doi:10.1007/BF02392210 [14] J. Leray -J. L. Lions,Quelques résultats de Visik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France,93 (1965), pp. 97–107. [15] J. L. Lions,Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod et Gauthier Villars, Paris, 1969. [16] P. L. Lions,Résolution de problèmes elliptiques quasilinéaires, Arch. Rat. Mech. Anal.,74 (1980), pp. 336–353. · Zbl 0449.35036 · doi:10.1007/BF00249679 [17] P. L. Lions,Quelques remarques sur les problèmes elliptiques quasilinéaires du second ordre, J. Anal. Math.,45 (1985), pp. 234–254. · Zbl 0614.35034 · doi:10.1007/BF02792551 [18] J. M.Rakotoson - R.Temam,Relative rearrangement in quasilinear variational inequalities, to appear.