## Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum.(English. Abridged French version)Zbl 1016.35026

This paper is devoted to noncoercive nonlinear problems whose prototype is $\begin{cases}-\Delta_p u+ b(x)|\nabla u|^\lambda= \mu\quad &\text{in }\Omega,\\ u= 0\quad &\text{on }\partial\Omega,\end{cases}\tag{1}$ where $$\Omega$$ is a bounded domain in $$\mathbb{R}^N$$ $$(W\geq 2)$$, $$\Delta_p$$ is the $$p$$-Laplacian $$(1< p< N)$$, $$\mu$$ is a Radon measure with bounded variation on $$\Omega$$, $$\lambda\geq 0$$, $$b\in L^\infty(\Omega)$$. Under some natural assumptions on the data of (1) the authors prove uniqueness and existence of a solution of (1).

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35J60 Nonlinear elliptic equations
Full Text:

### References:

 [1] Bénilan, P.; Boccardo, L.; Gallouët, T.; Gariepy, R.; Pierre, M.; Vazquez, J.L., An L^{1}-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. scuola norm. sup. Pisa cl. sci., 22, 241-273, (1995) · Zbl 0866.35037 [2] M.F. Betta, A. Mercaldo, F. Murat, M.M. Porzio, Existence of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side a measure, J. Math. Pures Appl., to appear · Zbl 1165.35365 [3] M.F. Betta, A. Mercaldo, F. Murat, M.M. Porzio, Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side $$L\^{}\{1\}(Ω)$$, ESAIM Control Optim. Calc. Var., Special issue dedicated to the memory of Jacques-Louis Lions (2002), to appear · Zbl 1092.35032 [4] M.F. Betta, A. Mercaldo, F. Murat, M.M. Porzio, Uniqueness results for nonlinear elliptic equations with a lower order term, in preparation · Zbl 1125.35343 [5] Boccardo, L.; Gallouët, T., Nonlinear elliptic and parabolic equations involving measure data, J. funct. anal., 87, 149-169, (1989) · Zbl 0707.35060 [6] Boccardo, L.; Gallouët, T.; Orsina, L., Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. inst. H. Poincaré anal. non linéaire, 13, 539-551, (1996) · Zbl 0857.35126 [7] Bottaro, G.; Marina, M.E., Problema di Dirichlet per equazioni ellittiche di tipo variazionale su insiemi non limitati, Boll. un. mat. ital., 8, 46-56, (1973) · Zbl 0291.35021 [8] Dal Maso, G.; Murat, F.; Orsina, L.; Prignet, A., Renormalized solutions for elliptic equations with general measure data, Ann. scuola norm. sup. Pisa cl. sci., 28, 741-808, (1999) · Zbl 0958.35045 [9] Del Vecchio, T., Nonlinear elliptic equations with measure data, Pot. analysis, 4, 185-203, (1995) · Zbl 0815.35023 [10] P.-L. Lions, F. Murat, Solutions renormalisées d’équations elliptiques non linéaires, to appear [11] F. Murat, Soluciones renormalizadas de EDP elipticas no lineales, Preprint 93023, Laboratoire d’Analyse Numérique de l’Université Paris VI, 1993 [12] Murat, F., Équations elliptiques non linéaires avec second membre L^{1} ou mesure, (), A12-A24 [13] Serrin, J., Pathological solutions of elliptic differential equations, Ann. scuola norm. sup. Pisa cl. sci., 18, 385-387, (1964) · Zbl 0142.37601 [14] Stampacchia, G., Le problème de Dirichlet pour LES équations elliptiques du second ordre à coefficients discontinus, Ann. inst. Fourier (Grenoble), 15, 189-258, (1965) · Zbl 0151.15401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.