# American Institute of Mathematical Sciences

• Previous Article
Entire solutions in nonlocal monostable equations: Asymmetric case
• CPAA Home
• This Issue
• Next Article
Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator
May  2019, 18(3): 1023-1048. doi: 10.3934/cpaa.2019050

## Uniqueness for Neumann problems for nonlinear elliptic equations

 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.
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
 [1] Xia Huang. Stable weak solutions of weighted nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 293-305. doi: 10.3934/cpaa.2014.13.293 [2] Zhifu Xie. General uniqueness results and examples for blow-up solutions of elliptic equations. Conference Publications, 2009, 2009 (Special) : 828-837. doi: 10.3934/proc.2009.2009.828 [3] Peiying Chen. Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations. Electronic Research Announcements, 2017, 24: 38-52. doi: 10.3934/era.2017.24.005 [4] Monica Musso, Donato Passaseo. Multiple solutions of Neumann elliptic problems with critical nonlinearity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 301-320. doi: 10.3934/dcds.1999.5.301 [5] Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771 [6] Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Multiple solutions for nonlinear coercive Neumann problems. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1957-1974. doi: 10.3934/cpaa.2009.8.1957 [7] Shouchuan Hu, Nikolaos S. Papageorgiou. Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2889-2922. doi: 10.3934/cpaa.2013.12.2889 [8] Leszek Gasiński, Nikolaos S. Papageorgiou. Multiple solutions for a class of nonlinear Neumann eigenvalue problems. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1491-1512. doi: 10.3934/cpaa.2014.13.1491 [9] Pierpaolo Soravia. Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients. Communications on Pure & Applied Analysis, 2006, 5 (1) : 213-240. doi: 10.3934/cpaa.2006.5.213 [10] Olivier Guibé, Anna Mercaldo. Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms. Communications on Pure & Applied Analysis, 2008, 7 (1) : 163-192. doi: 10.3934/cpaa.2008.7.163 [11] Sabri Bensid, Jesús Ildefonso Díaz. Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1757-1778. doi: 10.3934/dcdsb.2017105 [12] Wanwan Wang, Hongxia Zhang, Huyuan Chen. Remarks on weak solutions of fractional elliptic equations. Communications on Pure & Applied Analysis, 2016, 15 (2) : 335-340. doi: 10.3934/cpaa.2016.15.335 [13] Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu. Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5003-5036. doi: 10.3934/dcds.2015.35.5003 [14] Jiahong Wu. Regularity results for weak solutions of the 3D MHD equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 543-556. doi: 10.3934/dcds.2004.10.543 [15] M. Chuaqui, C. Cortázar, M. Elgueta, J. García-Melián. Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights. Communications on Pure & Applied Analysis, 2004, 3 (4) : 653-662. doi: 10.3934/cpaa.2004.3.653 [16] Phuong Le. Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator. Communications on Pure & Applied Analysis, 2020, 19 (1) : 511-525. doi: 10.3934/cpaa.2020025 [17] Shiren Zhu, Xiaoli Chen, Jianfu Yang. Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2685-2696. doi: 10.3934/cpaa.2013.12.2685 [18] Robert Jensen, Andrzej Świech. Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE. Communications on Pure & Applied Analysis, 2005, 4 (1) : 199-207. doi: 10.3934/cpaa.2005.4.187 [19] Rushun Tian, Zhi-Qiang Wang. Bifurcation results on positive solutions of an indefinite nonlinear elliptic system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 335-344. doi: 10.3934/dcds.2013.33.335 [20] Gabriele Bonanno, Pasquale Candito, Roberto Livrea, Nikolaos S. Papageorgiou. Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1169-1188. doi: 10.3934/cpaa.2017057

2018 Impact Factor: 0.925