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May  2019, 18(3): 1023-1048. doi: 10.3934/cpaa.2019050

Uniqueness for Neumann problems for nonlinear elliptic equations

Maria Francesca Betta 1, , Olivier Guibé 2, and  Anna Mercaldo 3,,

1. 

Dipartimento di Ingegneria, Università degli Studi di Napoli Parthenope, Centro Direzionale, Isola C4 80143 Napoli, Italy
2. 

Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS-Université de Rouen, Avenue de l'Université, BP.12 76801 Saint-Étienne-du-Rouvray, France
3. 

Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli Federico Ⅱ, Complesso Monte S. Angelo, Via Cintia, 80126 Napoli, Italy

* Corresponding author

Received  October 2017 Revised  September 2018 Published  November 2018

In the present paper we prove uniqueness results for solutions to a class of Neumann boundary value problems whose prototype is
$\left\{ \begin{align} & -\text{div}({{(1+|\nabla u{{|}^{2}})}^{(p-2)/2}}\nabla u)-\text{div}(c(x)|u{{|}^{p-2}}u)=f\ \ \ \text{in}\ \Omega , \\ & \left( {{(1+|\nabla u{{|}^{2}})}^{(p-2)/2}}\nabla u+c(x)|u{{|}^{p-2}}u \right)\cdot \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n}=0\ \ \ \text{on}\ \partial \Omega , \\ \end{align} \right.$
where
$Ω$
 is a bounded domain of
$\mathbb{R}^{N}$
,
$N≥ 2$
, with Lipschitz boundary,
$ 1 < p < N$
,
$\underline n$
 is the outer unit normal to
$\partial Ω$
, the datum
$f$
 belongs to
$L^{(p^{*})'}(Ω)$
 or to
$L^{1}(Ω)$
 and satisfies the compatibility condition
$\int{{}}_Ω f \, dx = 0$
. Finally the coefficient
$c(x)$
 belongs to an appropriate Lebesgue space.
Keywords: Nonlinear elliptic equations, Neumann problems, renormalized solutions, weak solutions, uniqueness results.
Mathematics Subject Classification: Primary: 35J25; Secondary: 35J60.
Citation: Maria Francesca Betta, Olivier Guibé, Anna Mercaldo. Uniqueness for Neumann problems for nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1023-1048. doi: 10.3934/cpaa.2019050
References:
[1]

A. Alvino, A. Cianchi, V. G. Maz'ya and A. Mercaldo, Well-posed elliptic Neumann problems involving irregular data and domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1017-1054.  doi: 10.1016/j.anihpc.2010.01.010.  Google Scholar

[2]

A. Alvino and A. Mercaldo, Nonlinear elliptic problems with $L^1$ data: an approach via symmetrization methods, Mediterr. J. Math., 5 (2008), 173-185.  doi: 10.1007/s00009-008-0142-5.  Google Scholar

[3]

F. Andreu, J. M. Mazón, S. Segura de León and J. Toledo, Quasi-linear elliptic and parabolic equations in $L^1$ with nonlinear boundary conditions, Adv. Math. Sci. Appl., 7 (1997), 183-213.   Google Scholar

[4]

M. Artola, Sur une classe de problémes paraboliques quasi-linéaires, Boll. Un. Mat. Ital. B (6), 5 (1986), 51-70.   Google Scholar

[5]

G. Barles, G. Diaz and J. I. Diaz, Uniqueness and continuum of foliated solutions for a quasilinear elliptic equation with a non-Lipschitz nonlinearity, Comm. Partial Differential Equations, 17 (1992), 1037-1050.  doi: 10.1080/03605309208820876.  Google Scholar

[6]

M. Ben Cheikh Ali and O. Guibé, Nonlinear and non-coercive elliptic problems with integrable data, Adv. Math. Sci. Appl., 16 (2006), 275-297.   Google Scholar

[7]

P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 22 (1995), 241-273.   Google Scholar

[8]

M. F. Betta, O. Guibé and A. Mercaldo, Neumann problems for nonlinear elliptic equations with $L^1$ data, J. Differential Equations, 259 (2015), 898-924.  doi: 10.1016/j.jde.2015.02.031.  Google Scholar

[9]

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. Math. Acad. Sci. Paris, 334 (2002), 757-762.  doi: 10.1016/S1631-073X(02)02338-5.  Google Scholar

[10]

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 a measure, J. Math. Pures Appl. (9), 82 (2003), 90–124. Corrected reprint of J. Math. Pures Appl. (9), 8 (2002), 533–566. doi: 10.1016/S0021-7824(03)00006-0.  Google Scholar

[11]

L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169.  doi: 10.1016/0022-1236(89)90005-0.  Google Scholar

[12]

L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations, 17 (1992), 641-655.  doi: 10.1080/03605309208820857.  Google Scholar

[13]

L. Boccardo, T. Gallouët and F. Murat, Unicité de la solution de certaines équations elliptiques non linéaires, C. R. Acad. Sci. Paris S´er. I Math., 315 (1992), 1159-1164.   Google Scholar

[14]

J. Chabrowski, On the Neumann problem with $L^1$ data, Colloq. Math., 107 (2007), 301-316.  doi: 10.4064/cm107-2-10.  Google Scholar

[15]

M. Chipot and G. Michaille, Uniqueness results and monotonicity properties for strongly nonlinear elliptic variational inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 16 (1989), 137-166.   Google Scholar

[16]

G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 741-808.   Google Scholar

[17]

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. (4), 170 (1996), 207-240.  doi: 10.1007/BF01758989.  Google Scholar

[18]

A. Decarreau, J. Liang and J.-M. Rakotoson, Trace imbeddings for $T$-sets and application to Neumann-Dirichlet problems with measures included in the boundary data, Ann. Fac. Sci. Toulouse Math. (6), 5 (1996), 443-470.   Google Scholar

[19]

J. Droniou, Solving convection-diffusion equations with mixed, Neumann and Fourier boundary conditions and measures as data, by a duality method, Adv. Differential Equations, 5 (2000), 1341-1396.   Google Scholar

[20]

J. Droniou and J.-L. Vázquez, Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions, Calc. Var. Partial Differential Equations, 34 (2009), 413-434.  doi: 10.1007/s00526-008-0189-y.  Google Scholar

[21]

V. Ferone and A. Mercaldo, A second order derivation formula for functions defined by integrals, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 549-554.  doi: 10.1016/S0764-4442(98)85005-2.  Google Scholar

[22]

V. Ferone and A. Mercaldo, Neumann problems and Steiner symmetrization, Comm. Partial Differential Equations, 30 (2005), 1537-1553.  doi: 10.1080/03605300500299596.  Google Scholar

[23]

O. Guibé and A. Mercaldo, Existence and stability results for renormalized solutions to noncoercive nonlinear elliptic equations with measure data, Potential Anal., 25 (2006), 223-258.  doi: 10.1007/s11118-006-9011-7.  Google Scholar

[24]

O. Guibé and A. Mercaldo, Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data, Trans. Amer. Math. Soc., 360 (2008), 643-669.  doi: 10.1090/S0002-9947-07-04139-6.  Google Scholar

[25]

J. Leray and J.-L. Lions, Quelques résulatats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107.   Google Scholar

[26]

J.-L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, 1969.  Google Scholar

[27]

P. L. Lions and F. Murat, Sur les solutions renormalisées d'équations elliptiques non linéaires, In manuscript. Google Scholar

[28]

F. Murat, Equations elliptiques non linéaires avec second membre ${L}^1$ ou mesure, In Compte Rendus du 26ème Congrès d'Analyse Numérique, les Karellis, 1994. Google Scholar

[29]

A. Prignet, Conditions aux limites non homogènes pour des problèmes elliptiques avec second membre mesure, Ann. Fac. Sci. Toulouse Math. (6), 6 (1997), 297-318.   Google Scholar

[30]

W. P. Ziemer, Weakly Differentiable Functions, volume 120 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

show all references

References:
[1]

A. Alvino, A. Cianchi, V. G. Maz'ya and A. Mercaldo, Well-posed elliptic Neumann problems involving irregular data and domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1017-1054.  doi: 10.1016/j.anihpc.2010.01.010.  Google Scholar

[2]

A. Alvino and A. Mercaldo, Nonlinear elliptic problems with $L^1$ data: an approach via symmetrization methods, Mediterr. J. Math., 5 (2008), 173-185.  doi: 10.1007/s00009-008-0142-5.  Google Scholar

[3]

F. Andreu, J. M. Mazón, S. Segura de León and J. Toledo, Quasi-linear elliptic and parabolic equations in $L^1$ with nonlinear boundary conditions, Adv. Math. Sci. Appl., 7 (1997), 183-213.   Google Scholar

[4]

M. Artola, Sur une classe de problémes paraboliques quasi-linéaires, Boll. Un. Mat. Ital. B (6), 5 (1986), 51-70.   Google Scholar

[5]

G. Barles, G. Diaz and J. I. Diaz, Uniqueness and continuum of foliated solutions for a quasilinear elliptic equation with a non-Lipschitz nonlinearity, Comm. Partial Differential Equations, 17 (1992), 1037-1050.  doi: 10.1080/03605309208820876.  Google Scholar

[6]

M. Ben Cheikh Ali and O. Guibé, Nonlinear and non-coercive elliptic problems with integrable data, Adv. Math. Sci. Appl., 16 (2006), 275-297.   Google Scholar

[7]

P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 22 (1995), 241-273.   Google Scholar

[8]

M. F. Betta, O. Guibé and A. Mercaldo, Neumann problems for nonlinear elliptic equations with $L^1$ data, J. Differential Equations, 259 (2015), 898-924.  doi: 10.1016/j.jde.2015.02.031.  Google Scholar

[9]

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. Math. Acad. Sci. Paris, 334 (2002), 757-762.  doi: 10.1016/S1631-073X(02)02338-5.  Google Scholar

[10]

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 a measure, J. Math. Pures Appl. (9), 82 (2003), 90–124. Corrected reprint of J. Math. Pures Appl. (9), 8 (2002), 533–566. doi: 10.1016/S0021-7824(03)00006-0.  Google Scholar

[11]

L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169.  doi: 10.1016/0022-1236(89)90005-0.  Google Scholar

[12]

L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations, 17 (1992), 641-655.  doi: 10.1080/03605309208820857.  Google Scholar

[13]

L. Boccardo, T. Gallouët and F. Murat, Unicité de la solution de certaines équations elliptiques non linéaires, C. R. Acad. Sci. Paris S´er. I Math., 315 (1992), 1159-1164.   Google Scholar

[14]

J. Chabrowski, On the Neumann problem with $L^1$ data, Colloq. Math., 107 (2007), 301-316.  doi: 10.4064/cm107-2-10.  Google Scholar

[15]

M. Chipot and G. Michaille, Uniqueness results and monotonicity properties for strongly nonlinear elliptic variational inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 16 (1989), 137-166.   Google Scholar

[16]

G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 741-808.   Google Scholar

[17]

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. (4), 170 (1996), 207-240.  doi: 10.1007/BF01758989.  Google Scholar

[18]

A. Decarreau, J. Liang and J.-M. Rakotoson, Trace imbeddings for $T$-sets and application to Neumann-Dirichlet problems with measures included in the boundary data, Ann. Fac. Sci. Toulouse Math. (6), 5 (1996), 443-470.   Google Scholar

[19]

J. Droniou, Solving convection-diffusion equations with mixed, Neumann and Fourier boundary conditions and measures as data, by a duality method, Adv. Differential Equations, 5 (2000), 1341-1396.   Google Scholar

[20]

J. Droniou and J.-L. Vázquez, Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions, Calc. Var. Partial Differential Equations, 34 (2009), 413-434.  doi: 10.1007/s00526-008-0189-y.  Google Scholar

[21]

V. Ferone and A. Mercaldo, A second order derivation formula for functions defined by integrals, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 549-554.  doi: 10.1016/S0764-4442(98)85005-2.  Google Scholar

[22]

V. Ferone and A. Mercaldo, Neumann problems and Steiner symmetrization, Comm. Partial Differential Equations, 30 (2005), 1537-1553.  doi: 10.1080/03605300500299596.  Google Scholar

[23]

O. Guibé and A. Mercaldo, Existence and stability results for renormalized solutions to noncoercive nonlinear elliptic equations with measure data, Potential Anal., 25 (2006), 223-258.  doi: 10.1007/s11118-006-9011-7.  Google Scholar

[24]

O. Guibé and A. Mercaldo, Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data, Trans. Amer. Math. Soc., 360 (2008), 643-669.  doi: 10.1090/S0002-9947-07-04139-6.  Google Scholar

[25]

J. Leray and J.-L. Lions, Quelques résulatats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107.   Google Scholar

[26]

J.-L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, 1969.  Google Scholar

[27]

P. L. Lions and F. Murat, Sur les solutions renormalisées d'équations elliptiques non linéaires, In manuscript. Google Scholar

[28]

F. Murat, Equations elliptiques non linéaires avec second membre ${L}^1$ ou mesure, In Compte Rendus du 26ème Congrès d'Analyse Numérique, les Karellis, 1994. Google Scholar

[29]

A. Prignet, Conditions aux limites non homogènes pour des problèmes elliptiques avec second membre mesure, Ann. Fac. Sci. Toulouse Math. (6), 6 (1997), 297-318.   Google Scholar

[30]

W. P. Ziemer, Weakly Differentiable Functions, volume 120 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

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