# American Institute of Mathematical Sciences

July  2013, 12(4): 1569-1585. doi: 10.3934/cpaa.2013.12.1569

## Uniqueness for elliptic problems with Hölder--type dependence on the solution

 1 Dipartimento di Matematica, Università di Roma 1, Piazza A. Moro 2, 00185 Roma 2 Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scienti ca 1, 00133 Roma

Received  May 2011 Revised  June 2012 Published  November 2012

We prove uniqueness of weak (or entropy) solutions for nonmonotone elliptic equations of the type \begin{eqnarray} -div (a(x,u)\nabla u)=f \end{eqnarray} in a bounded set $\Omega\subset R^N$ with Dirichlet boundary conditions. The novelty of our results consists in the possibility to deal with cases when $a(x,u)$ is only Hölder continuous with respect to $u$.
Citation: Lucio Boccardo, Alessio Porretta. Uniqueness for elliptic problems with Hölder--type dependence on the solution. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1569-1585. doi: 10.3934/cpaa.2013.12.1569
##### References:
 [1] M. Artola, Sur une classe de problèmes paraboliques quasi-linéaires,, Boll. U.M.I. B., 5 (1986), 51.   Google Scholar [2] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vàzquez, An $L^1$ theory of existence and uniqueness of nonlinear elliptic equations,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1995), 240.   Google Scholar [3] D. Blanchard, F. Désir and O. Guibé, Quasi-linear degenerate elliptic problems with $L^1$ data,, Nonlinear Anal., 60 (2005), 557.  doi: 10.1016/S0362-546X(04)00395-5.  Google Scholar [4] L. Boccardo, Some nonlinear Dirichlet problems in $L^1$ involving lower order terms in divergence form,, Progress in elliptic and parabolic partial differential equations (Capri, (1994), 43.   Google Scholar [5] L. Boccardo, Uniqueness of solutions for some nonlinear Dirichlet problems,, dedicated to M. Artola, ().   Google Scholar [6] L. Boccardo, A remark on some nonlinear elliptic problems,, 2001-Luminy Conference on Quasilinear Elliptic and Parabolic Equations and Systems, Conf. 08 (2002), 47.   Google Scholar [7] L. Boccardo and B. Dacorogna, Monotonicity of certain differential operators in divergence form,, Manuscripta Math., 64 (1989), 253.  doi: 10.1007/BF01160123.  Google Scholar [8] L. Boccardo, I. Diaz, D. Giachetti and F. Murat, Existence of a solution for a weaker form of a nonlinear elliptic equation,, in, 208 (1988), 229.   Google Scholar [9] L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right hand side measures,, Comm. P.D.E., 17 (1992), 641.  doi: 10.1080/03605309208820857.  Google Scholar [10] L. Boccardo, T. Gallouët and F. Murat, Unicité de la solution pour des equations elliptiques non linéaires,, C. R. Acad. Sc. Paris, 315 (1992), 1159.   Google Scholar [11] J. Carrillo and M. Chipot, On some elliptic equations involving derivatives of the nonlinearity,, Proc. Roy. Soc. Edinburgh, 100 (1985), 281.  doi: 10.1017/S0308210500013822.  Google Scholar [12] J. Casado Diaz, F. Murat and A. Porretta, Uniqueness results for pseudomonotone problems with $p>2$,, C. R. Math. Acad. Sci. Paris, 344 (2007), 487.  doi: 10.1016/j.crma.2007.02.007.  Google Scholar [13] M. Chipot and G. Michaille, Uniqueness results and monotonicity properties for strongly nonlinear elliptic variational inequalities,, Ann. Sc. Norm. Sup. Pisa, 16 (1989), 137.   Google Scholar [14] 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.  doi: 10.1007/BF01758989.  Google Scholar [15] O. Guibé, Uniqueness of the solution to quasilinear elliptic equations under a local condition on the diffusion matrix,, Adv. Math. Sci. Appl., 17 (2007), 357.   Google Scholar [16] O. Guibé, Uniqueness of the renormalized solution to a class of nonlinear elliptic equations,, in, 23 (2008), 459.   Google Scholar [17] A. G. Kartsatos and I. V. Skrypnik, The index of a critical point for nonlinear elliptic operators with strong coefficient growth,, J. Math. Soc. Japan, 52 (2000), 109.  doi: 10.2969/jmsj/05210109.  Google Scholar [18] C. Leone and A. Porretta, Entropy solutions for nonlinear elliptic equations in $L^1$,, Nonlinear Anal., 32 (1998), 325.  doi: 10.1016/S0362-546X(96)00323-9.  Google Scholar [19] M. Marcus and V. J. Mizel, Every superposition operator mapping one Sobolev space into another is continuous,, J. Funct. Anal., 33 (1979), 217.  doi: 10.1016/0022-1236(79)90113-7.  Google Scholar [20] A. Porretta, Uniqueness and homogenization for a class of noncoercive operators in divergence form,, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 915.   Google Scholar [21] A. Porretta, Uniqueness of solutions for some nonlinear Dirichlet problems,, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 407.  doi: 10.1007/s00030-004-0031-y.  Google Scholar [22] A. Porretta, Some remarks on the regularity of solutions for a class of elliptic equations with measure data,, Houston J. Math., 26 (2000), 183.   Google Scholar [23] M. M. Porzio, A uniqueness result for monotone elliptic problems,, C. R. Math. Acad. Sci. Paris, 337 (2003), 313.  doi: 10.1016/S1631-073X(03)00347-9.  Google Scholar [24] N. Trudinger, On the comparison principle for quasilinear divergence structure equations,, Arch. for Rat. Mech. Anal., 57 (1975), 128.  doi: 10.1007/BF00248414.  Google Scholar

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##### References:
 [1] M. Artola, Sur une classe de problèmes paraboliques quasi-linéaires,, Boll. U.M.I. B., 5 (1986), 51.   Google Scholar [2] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vàzquez, An $L^1$ theory of existence and uniqueness of nonlinear elliptic equations,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1995), 240.   Google Scholar [3] D. Blanchard, F. Désir and O. Guibé, Quasi-linear degenerate elliptic problems with $L^1$ data,, Nonlinear Anal., 60 (2005), 557.  doi: 10.1016/S0362-546X(04)00395-5.  Google Scholar [4] L. Boccardo, Some nonlinear Dirichlet problems in $L^1$ involving lower order terms in divergence form,, Progress in elliptic and parabolic partial differential equations (Capri, (1994), 43.   Google Scholar [5] L. Boccardo, Uniqueness of solutions for some nonlinear Dirichlet problems,, dedicated to M. Artola, ().   Google Scholar [6] L. Boccardo, A remark on some nonlinear elliptic problems,, 2001-Luminy Conference on Quasilinear Elliptic and Parabolic Equations and Systems, Conf. 08 (2002), 47.   Google Scholar [7] L. Boccardo and B. Dacorogna, Monotonicity of certain differential operators in divergence form,, Manuscripta Math., 64 (1989), 253.  doi: 10.1007/BF01160123.  Google Scholar [8] L. Boccardo, I. Diaz, D. Giachetti and F. Murat, Existence of a solution for a weaker form of a nonlinear elliptic equation,, in, 208 (1988), 229.   Google Scholar [9] L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right hand side measures,, Comm. P.D.E., 17 (1992), 641.  doi: 10.1080/03605309208820857.  Google Scholar [10] L. Boccardo, T. Gallouët and F. Murat, Unicité de la solution pour des equations elliptiques non linéaires,, C. R. Acad. Sc. Paris, 315 (1992), 1159.   Google Scholar [11] J. Carrillo and M. Chipot, On some elliptic equations involving derivatives of the nonlinearity,, Proc. Roy. Soc. Edinburgh, 100 (1985), 281.  doi: 10.1017/S0308210500013822.  Google Scholar [12] J. Casado Diaz, F. Murat and A. Porretta, Uniqueness results for pseudomonotone problems with $p>2$,, C. R. Math. Acad. Sci. Paris, 344 (2007), 487.  doi: 10.1016/j.crma.2007.02.007.  Google Scholar [13] M. Chipot and G. Michaille, Uniqueness results and monotonicity properties for strongly nonlinear elliptic variational inequalities,, Ann. Sc. Norm. Sup. Pisa, 16 (1989), 137.   Google Scholar [14] 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.  doi: 10.1007/BF01758989.  Google Scholar [15] O. Guibé, Uniqueness of the solution to quasilinear elliptic equations under a local condition on the diffusion matrix,, Adv. Math. Sci. Appl., 17 (2007), 357.   Google Scholar [16] O. Guibé, Uniqueness of the renormalized solution to a class of nonlinear elliptic equations,, in, 23 (2008), 459.   Google Scholar [17] A. G. Kartsatos and I. V. Skrypnik, The index of a critical point for nonlinear elliptic operators with strong coefficient growth,, J. Math. Soc. Japan, 52 (2000), 109.  doi: 10.2969/jmsj/05210109.  Google Scholar [18] C. Leone and A. Porretta, Entropy solutions for nonlinear elliptic equations in $L^1$,, Nonlinear Anal., 32 (1998), 325.  doi: 10.1016/S0362-546X(96)00323-9.  Google Scholar [19] M. Marcus and V. J. Mizel, Every superposition operator mapping one Sobolev space into another is continuous,, J. Funct. Anal., 33 (1979), 217.  doi: 10.1016/0022-1236(79)90113-7.  Google Scholar [20] A. Porretta, Uniqueness and homogenization for a class of noncoercive operators in divergence form,, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 915.   Google Scholar [21] A. Porretta, Uniqueness of solutions for some nonlinear Dirichlet problems,, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 407.  doi: 10.1007/s00030-004-0031-y.  Google Scholar [22] A. Porretta, Some remarks on the regularity of solutions for a class of elliptic equations with measure data,, Houston J. Math., 26 (2000), 183.   Google Scholar [23] M. M. Porzio, A uniqueness result for monotone elliptic problems,, C. R. Math. Acad. Sci. Paris, 337 (2003), 313.  doi: 10.1016/S1631-073X(03)00347-9.  Google Scholar [24] N. Trudinger, On the comparison principle for quasilinear divergence structure equations,, Arch. for Rat. Mech. Anal., 57 (1975), 128.  doi: 10.1007/BF00248414.  Google Scholar
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