Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in $L^1\left(\Omega \right)$

Betta, M. F.; Mercaldo, A.; Murat, F.; Porzio, M. M.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 239-272.

In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is

$$\left\{\begin{array}{cc}-div\left(a\left(x\right)\right(1+{\left|\nabla u\right|}^2{)}^{\frac{p-2}{2}}{\nabla u)+b\left(x\right)(1+\left|\nabla u\right|}^2{)}^{\frac{\lambda }{2}}=f\hfill & \text{in}\Omega ,\hfill \\ u=0\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$$

where $\Omega$ is a bounded open subset of ${ℝ}^N$, $N\ge 2$, $2-1/N<p<N$, $a$ belongs to $L^{\infty }\left(\Omega \right)$, $a\left(x\right)\ge {\alpha }_0>0$, $f$ is a function in $L^1\left(\Omega \right)$, $b$ is a function in $L^r\left(\Omega \right)$ and $0\le \lambda <{\lambda }^{*}(N,p,r),$ for some $r$ and ${\lambda }^{*}(N,p,r)$.

[1] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J.L. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 (1995) 241-273. | Numdam | MR | Zbl

[2] M.F. Betta, A. Mercaldo, F. Murat and M.M. Porzio, Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum. C. R. Acad. Sci. Paris Sér. I Math. 332 (to appear). | MR | Zbl

[3] M.F. Betta, A. Mercaldo, F. Murat and M.M. Porzio, Existence of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side measure. J. Math. Pures Appl. (to appear). | MR

[4] M.F. Betta, A. Mercaldo, F. Murat and M.M. Porzio, Uniqueness results for nonlinear elliptic equations with a lower order term (to appear). | MR | Zbl

[5] L. Boccardo, T. Gallouët and L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996) 539-551. | Numdam | MR | Zbl

[6] G. Bottaro and M.E. Marina, Problema di Dirichlet per equazioni ellittiche di tipo variazionale su insiemi non limitati. Boll. Un. Mat. Ital. 8 (1973) 46-56. | MR | Zbl

[7] A. Dall'Aglio, Approximated solutions of equations with $L^1$ data. Application to the $H$-convergence of quasi-linear parabolic equations. Ann. Mat. Pura Appl. 170 (1996) 207-240. | Zbl

[8] G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions for elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28 (1999) 741-808. | Numdam | MR | Zbl

[9] G. Dolzmann, N. Hungerbühler and S. Müller, Uniqueness and maximal regularity for nonlinear elliptic systems of $n$-Laplace type with measure valued right-hand side. J. Reine Angew. Math. 520 (2000) 1-35. | MR | Zbl

[10] A. Fiorenza and C. Sbordone, Existence and uniqueness results for solutions of nonlinear equations with right-hand side in $L^1\left(\Omega \right)$. Studia Math. 127 (1998) 223-231. | MR | Zbl

[11] L. Greco, T. Iwaniec and C. Sbordone, Inverting the p-harmonic operator. Manuscripta Math. 92 (1997) 249-258. | MR | Zbl

[12] O. Guibé, Remarks on the uniqueness of comparable renormalized solutions of elliptic equations with measure data. Ann. Mat. Pura Appl. Ser. IV 180 (2002) 441-449. | MR | Zbl

[13] P.-L. Lions and F. Murat, Solutions renormalisées d'équations elliptiques non linéaires (to appear).

[14] F. Murat, Soluciones renormalizadas de EDP elipticas no lineales, Preprint 93023. Laboratoire d'Analyse Numérique de l'Université Paris VI (1993).

[15] F. Murat, Équations elliptiques non linéaires avec second membre $L^1$ ou mesure, in Actes du 26${}^e$ Congrès National d'Analyse Numérique. Les Karellis, France (1994) A12-A24.

[16] A. Prignet, Remarks on existence and uniqueness of solutions of elliptic problems with right-hand side measures. Rend. Mat. Appl. 15 (1995) 321-337. | MR | Zbl

[17] J. Serrin, Pathological solutions of elliptic differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 (1964) 385-387. | Numdam | MR | Zbl

[18] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189-258. | Numdam | MR | Zbl