A variational approach to semilinear elliptic equations with measure data

Previous Article
Crack growth with non-interpenetration: A simplified proof for the pure Neumann problem
DCDS Home
This Issue
Next Article
Center manifold: A case study

December 2011, 31(4): 1233-1248. doi: 10.3934/dcds.2011.31.1233

A variational approach to semilinear elliptic equations with measure data

Marco Degiovanni ^1, and Michele Scaglia ^2,

1.	Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41 - 25121 Brescia, Italy
2.	Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Via Cozzi 53 - 20125 Milano, Italy

Received October 2010 Revised December 2010 Published September 2011

Abstract
Related Papers

We describe a direct variational approach to a class of semilinear elliptic equations with measure data. Using a typical variational argument, we show the existence of multiple solutions.

Keywords: measure data., semilinear elliptic equations, Variational methods.

Mathematics Subject Classification: Primary: 49J40; Secondary: 35J2.

Citation: Marco Degiovanni, Michele Scaglia. A variational approach to semilinear elliptic equations with measure data. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1233-1248. doi: 10.3934/dcds.2011.31.1233

References:

[1]	P. Baras and M. Pierre, Singularités éliminables pour des équations semi-linéaires,, Ann. Inst. Fourier (Grenoble), 34 (1984), 185. doi: 10.5802/aif.956.
[2]	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.
[3]	P. Bénilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation,, Dedicated to Philippe Bénilan, 3 (2003), 673.
[4]	L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right hand side measures,, Comm. Partial Differential Equations, 17 (1992), 641.
[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.
[6]	H. Brezis and F. Browder, A property of Sobolev spaces,, Comm. Partial Differential Equations, 4 (1979), 1077.
[7]	H. Brezis, M. Marcus and A. C. Ponce, Nonlinear elliptic equations with measures revisited,, in, 163 (2007), 55.
[8]	H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$,, J. Math. Soc. Japan, 25 (1973), 565. doi: 10.2969/jmsj/02540565.
[9]	A. Canino and M. Degiovanni, A variational approach to a class of singular semilinear elliptic equations,, J. Convex Anal., 11 (2004), 147.
[10]	K.-C. Chang, "Infinite-Dimensional Morse Theory and Multiple Solution Problems,", Progress in Nonlinear Differential Equations and their Applications, 6 (1993).
[11]	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.
[12]	E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari,, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3), 3 (1957), 25.
[13]	M. Degiovanni and M. Marzocchi, On the Euler-Lagrange equation for functionals of the calculus of variations without upper growth conditions,, SIAM J. Control Optim., 48 (2009), 2857. doi: 10.1137/090747968.
[14]	A. Ferrero and C. Saccon, Existence and multiplicity results for semilinear equations with measure data,, Topol. Methods Nonlinear Anal., 28 (2006), 285.
[15]	A. Ferrero and C. Saccon, Existence and multiplicity results for semilinear elliptic equations with measure data and jumping nonlinearities,, Topol. Methods Nonlinear Anal., 30 (2007), 37.
[16]	A. Ferrero and C. Saccon, Multiplicity results for a class of asymptotically linear elliptic problems with resonance and applications to problems with measure data,, Adv. Nonlinear Stud., 10 (2010), 433.
[17]	T. Gallouët and J.-M. Morel, Resolution of a semilinear equation in $L^1$,, Proc. Roy. Soc. Edinburgh Sect. A, 96 (1984), 275.
[18]	T. Gallouët and J.-M. Morel, Corrigenda: "Resolution of a semilinear equation in $L^1$,", Proc. Roy. Soc. Edinburgh Sect. A, 99 (1985).
[19]	J. Moser, A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations,, Comm. Pure Appl. Math., 13 (1960), 457. doi: 10.1002/cpa.3160130308.
[20]	J. Moser, On Harnack's theorem for elliptic differential equations,, Comm. Pure Appl. Math., 14 (1961), 577. doi: 10.1002/cpa.3160140329.
[21]	J. Nash, Continuity of solutions of parabolic and elliptic equations,, Amer. J. Math., 80 (1958), 931. doi: 10.2307/2372841.
[22]	L. Orsina, Solvability of linear and semilinear eigenvalue problems with $L\^1$ data,, Rend. Sem. Mat. Univ. Padova, 90 (1993), 207.
[23]	L. Orsina and A. Ponce, Semilinear elliptic equations and systems with diffuse measures,, J. Evol. Equ., 8 (2008), 781. doi: 10.1007/s00028-008-0446-32.
[24]	G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus,, Ann. Inst. Fourier (Grenoble), 15 (1965), 189. doi: 10.5802/aif.204.
[25]	G. Stampacchia, "Équations Elliptiques du Second Ordre à Coefficients Discontinus,", Séminaire de Mathématiques Supérieures, 16 (1966).
[26]	A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 77.
[27]	N. S. Trudinger and X.-J. Wang, Quasilinear elliptic equations with signed measure data,, Discrete Contin. Dyn. Syst., 23 (2009), 477. doi: 10.3934/dcds.2009.23.477.

show all references

References:

[1]	P. Baras and M. Pierre, Singularités éliminables pour des équations semi-linéaires,, Ann. Inst. Fourier (Grenoble), 34 (1984), 185. doi: 10.5802/aif.956.
[2]	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.
[3]	P. Bénilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation,, Dedicated to Philippe Bénilan, 3 (2003), 673.
[4]	L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right hand side measures,, Comm. Partial Differential Equations, 17 (1992), 641.
[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.
[6]	H. Brezis and F. Browder, A property of Sobolev spaces,, Comm. Partial Differential Equations, 4 (1979), 1077.
[7]	H. Brezis, M. Marcus and A. C. Ponce, Nonlinear elliptic equations with measures revisited,, in, 163 (2007), 55.
[8]	H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$,, J. Math. Soc. Japan, 25 (1973), 565. doi: 10.2969/jmsj/02540565.
[9]	A. Canino and M. Degiovanni, A variational approach to a class of singular semilinear elliptic equations,, J. Convex Anal., 11 (2004), 147.
[10]	K.-C. Chang, "Infinite-Dimensional Morse Theory and Multiple Solution Problems,", Progress in Nonlinear Differential Equations and their Applications, 6 (1993).
[11]	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.
[12]	E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari,, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3), 3 (1957), 25.
[13]	M. Degiovanni and M. Marzocchi, On the Euler-Lagrange equation for functionals of the calculus of variations without upper growth conditions,, SIAM J. Control Optim., 48 (2009), 2857. doi: 10.1137/090747968.
[14]	A. Ferrero and C. Saccon, Existence and multiplicity results for semilinear equations with measure data,, Topol. Methods Nonlinear Anal., 28 (2006), 285.
[15]	A. Ferrero and C. Saccon, Existence and multiplicity results for semilinear elliptic equations with measure data and jumping nonlinearities,, Topol. Methods Nonlinear Anal., 30 (2007), 37.
[16]	A. Ferrero and C. Saccon, Multiplicity results for a class of asymptotically linear elliptic problems with resonance and applications to problems with measure data,, Adv. Nonlinear Stud., 10 (2010), 433.
[17]	T. Gallouët and J.-M. Morel, Resolution of a semilinear equation in $L^1$,, Proc. Roy. Soc. Edinburgh Sect. A, 96 (1984), 275.
[18]	T. Gallouët and J.-M. Morel, Corrigenda: "Resolution of a semilinear equation in $L^1$,", Proc. Roy. Soc. Edinburgh Sect. A, 99 (1985).
[19]	J. Moser, A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations,, Comm. Pure Appl. Math., 13 (1960), 457. doi: 10.1002/cpa.3160130308.
[20]	J. Moser, On Harnack's theorem for elliptic differential equations,, Comm. Pure Appl. Math., 14 (1961), 577. doi: 10.1002/cpa.3160140329.
[21]	J. Nash, Continuity of solutions of parabolic and elliptic equations,, Amer. J. Math., 80 (1958), 931. doi: 10.2307/2372841.
[22]	L. Orsina, Solvability of linear and semilinear eigenvalue problems with $L\^1$ data,, Rend. Sem. Mat. Univ. Padova, 90 (1993), 207.
[23]	L. Orsina and A. Ponce, Semilinear elliptic equations and systems with diffuse measures,, J. Evol. Equ., 8 (2008), 781. doi: 10.1007/s00028-008-0446-32.
[24]	G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus,, Ann. Inst. Fourier (Grenoble), 15 (1965), 189. doi: 10.5802/aif.204.
[25]	G. Stampacchia, "Équations Elliptiques du Second Ordre à Coefficients Discontinus,", Séminaire de Mathématiques Supérieures, 16 (1966).
[26]	A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 77.
[27]	N. S. Trudinger and X.-J. Wang, Quasilinear elliptic equations with signed measure data,, Discrete Contin. Dyn. Syst., 23 (2009), 477. doi: 10.3934/dcds.2009.23.477.

[1]	Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477
[2]	G. R. Cirmi, S. Leonardi. Higher differentiability for solutions of linear elliptic systems with measure data. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 89-104. doi: 10.3934/dcds.2010.26.89
[3]	Lei Wei, Zhaosheng Feng. Isolated singularity for semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3239-3252. doi: 10.3934/dcds.2015.35.3239
[4]	Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133
[5]	Wolf-Jüergen Beyn, Janosch Rieger. Galerkin finite element methods for semilinear elliptic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 295-312. doi: 10.3934/dcdsb.2013.18.295
[6]	Francis Ribaud. Semilinear parabolic equations with distributions as initial data. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 305-316. doi: 10.3934/dcds.1997.3.305
[7]	Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105
[8]	Rong Xiao, Yuying Zhou. Multiple solutions for a class of semilinear elliptic variational inclusion problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 991-1002. doi: 10.3934/jimo.2011.7.991
[9]	Paul H. Rabinowitz. A new variational characterization of spatially heteroclinic solutions of a semilinear elliptic PDE. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 507-515. doi: 10.3934/dcds.2004.10.507
[10]	Verena Bögelein, Frank Duzaar, Ugo Gianazza. Very weak solutions of singular porous medium equations with measure data. Communications on Pure & Applied Analysis, 2015, 14 (1) : 23-49. doi: 10.3934/cpaa.2015.14.23
[11]	Alberto Fiorenza, Anna Mercaldo, Jean Michel Rakotoson. Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 893-906. doi: 10.3934/dcds.2002.8.893
[12]	Junping Shi, R. Shivaji. Semilinear elliptic equations with generalized cubic nonlinearities. Conference Publications, 2005, 2005 (Special) : 798-805. doi: 10.3934/proc.2005.2005.798
[13]	Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601
[14]	Hwai-Chiuan Wang. On domains and their indexes with applications to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 447-467. doi: 10.3934/dcds.2007.19.447
[15]	Diane Denny. A unique positive solution to a system of semilinear elliptic equations. Conference Publications, 2013, 2013 (special) : 193-195. doi: 10.3934/proc.2013.2013.193
[16]	Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801
[17]	Antonio Greco, Marcello Lucia. Gamma-star-shapedness for semilinear elliptic equations. Communications on Pure & Applied Analysis, 2005, 4 (1) : 93-99. doi: 10.3934/cpaa.2005.4.93
[18]	Jiabao Su, Zhaoli Liu. A bounded resonance problem for semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 431-445. doi: 10.3934/dcds.2007.19.431
[19]	Mousomi Bhakta, Debangana Mukherjee. Semilinear nonlocal elliptic equations with critical and supercritical exponents. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1741-1766. doi: 10.3934/cpaa.2017085
[20]	Hwai-Chiuan Wang. Stability and symmetry breaking of solutions of semilinear elliptic equations. Conference Publications, 2005, 2005 (Special) : 886-894. doi: 10.3934/proc.2005.2005.886

2017 Impact Factor: 1.179

American Institute of Mathematical Sciences