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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

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, 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), 185-206. doi: 10.5802/aif.956.  Google Scholar

[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), 241-273. Google Scholar

[3]

P. Bénilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation, Dedicated to Philippe Bénilan, J. Evol. Equ., 3 (2003), 673-770.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

H. Brezis and F. Browder, A property of Sobolev spaces, Comm. Partial Differential Equations, 4 (1979), 1077-1083.  Google Scholar

[7]

H. Brezis, M. Marcus and A. C. Ponce, Nonlinear elliptic equations with measures revisited, in "Mathematical Aspects of Nonlinear Dispersive Equations" (eds. J. Bourgain, C. E. Kenig and S. Klainerman), Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ, (2007), 55-109.  Google Scholar

[8]

H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590. doi: 10.2969/jmsj/02540565.  Google Scholar

[9]

A. Canino and M. Degiovanni, A variational approach to a class of singular semilinear elliptic equations, J. Convex Anal., 11 (2004), 147-162.  Google Scholar

[10]

K.-C. Chang, "Infinite-Dimensional Morse Theory and Multiple Solution Problems," Progress in Nonlinear Differential Equations and their Applications, 6, Birkhäuser Boston, Inc., Boston, MA, 1993.  Google Scholar

[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), 741-808.  Google Scholar

[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), 25-43.  Google Scholar

[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), 2857-2870. doi: 10.1137/090747968.  Google Scholar

[14]

A. Ferrero and C. Saccon, Existence and multiplicity results for semilinear equations with measure data, Topol. Methods Nonlinear Anal., 28 (2006), 285-318.  Google Scholar

[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), 37-65.  Google Scholar

[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), 433-479.  Google Scholar

[17]

T. Gallouët and J.-M. Morel, Resolution of a semilinear equation in $L^1$, Proc. Roy. Soc. Edinburgh Sect. A, 96 (1984), 275-288.  Google Scholar

[18]

T. Gallouët and J.-M. Morel, Corrigenda: "Resolution of a semilinear equation in $L^1$," Proc. Roy. Soc. Edinburgh Sect. A, 99 (1985), 399. Google Scholar

[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), 457-468. doi: 10.1002/cpa.3160130308.  Google Scholar

[20]

J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591. doi: 10.1002/cpa.3160140329.  Google Scholar

[21]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954. doi: 10.2307/2372841.  Google Scholar

[22]

L. Orsina, Solvability of linear and semilinear eigenvalue problems with $L\^1$ data, Rend. Sem. Mat. Univ. Padova, 90 (1993), 207-238.  Google Scholar

[23]

L. Orsina and A. Ponce, Semilinear elliptic equations and systems with diffuse measures, J. Evol. Equ., 8 (2008), 781-812. doi: 10.1007/s00028-008-0446-32.  Google Scholar

[24]

G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. doi: 10.5802/aif.204.  Google Scholar

[25]

G. Stampacchia, "Équations Elliptiques du Second Ordre à Coefficients Discontinus," Séminaire de Mathématiques Supérieures, 16, Les Presses de l'Université de Montréal, Montreal, Que., 1966.  Google Scholar

[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), 77-109.  Google Scholar

[27]

N. S. Trudinger and X.-J. Wang, Quasilinear elliptic equations with signed measure data, Discrete Contin. Dyn. Syst., 23 (2009), 477-494. doi: 10.3934/dcds.2009.23.477.  Google Scholar

show all references

References:
[1]

P. Baras and M. Pierre, Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier (Grenoble), 34 (1984), 185-206. doi: 10.5802/aif.956.  Google Scholar

[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), 241-273. Google Scholar

[3]

P. Bénilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation, Dedicated to Philippe Bénilan, J. Evol. Equ., 3 (2003), 673-770.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

H. Brezis and F. Browder, A property of Sobolev spaces, Comm. Partial Differential Equations, 4 (1979), 1077-1083.  Google Scholar

[7]

H. Brezis, M. Marcus and A. C. Ponce, Nonlinear elliptic equations with measures revisited, in "Mathematical Aspects of Nonlinear Dispersive Equations" (eds. J. Bourgain, C. E. Kenig and S. Klainerman), Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ, (2007), 55-109.  Google Scholar

[8]

H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590. doi: 10.2969/jmsj/02540565.  Google Scholar

[9]

A. Canino and M. Degiovanni, A variational approach to a class of singular semilinear elliptic equations, J. Convex Anal., 11 (2004), 147-162.  Google Scholar

[10]

K.-C. Chang, "Infinite-Dimensional Morse Theory and Multiple Solution Problems," Progress in Nonlinear Differential Equations and their Applications, 6, Birkhäuser Boston, Inc., Boston, MA, 1993.  Google Scholar

[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), 741-808.  Google Scholar

[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), 25-43.  Google Scholar

[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), 2857-2870. doi: 10.1137/090747968.  Google Scholar

[14]

A. Ferrero and C. Saccon, Existence and multiplicity results for semilinear equations with measure data, Topol. Methods Nonlinear Anal., 28 (2006), 285-318.  Google Scholar

[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), 37-65.  Google Scholar

[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), 433-479.  Google Scholar

[17]

T. Gallouët and J.-M. Morel, Resolution of a semilinear equation in $L^1$, Proc. Roy. Soc. Edinburgh Sect. A, 96 (1984), 275-288.  Google Scholar

[18]

T. Gallouët and J.-M. Morel, Corrigenda: "Resolution of a semilinear equation in $L^1$," Proc. Roy. Soc. Edinburgh Sect. A, 99 (1985), 399. Google Scholar

[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), 457-468. doi: 10.1002/cpa.3160130308.  Google Scholar

[20]

J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591. doi: 10.1002/cpa.3160140329.  Google Scholar

[21]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954. doi: 10.2307/2372841.  Google Scholar

[22]

L. Orsina, Solvability of linear and semilinear eigenvalue problems with $L\^1$ data, Rend. Sem. Mat. Univ. Padova, 90 (1993), 207-238.  Google Scholar

[23]

L. Orsina and A. Ponce, Semilinear elliptic equations and systems with diffuse measures, J. Evol. Equ., 8 (2008), 781-812. doi: 10.1007/s00028-008-0446-32.  Google Scholar

[24]

G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. doi: 10.5802/aif.204.  Google Scholar

[25]

G. Stampacchia, "Équations Elliptiques du Second Ordre à Coefficients Discontinus," Séminaire de Mathématiques Supérieures, 16, Les Presses de l'Université de Montréal, Montreal, Que., 1966.  Google Scholar

[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), 77-109.  Google Scholar

[27]

N. S. Trudinger and X.-J. Wang, Quasilinear elliptic equations with signed measure data, Discrete Contin. Dyn. Syst., 23 (2009), 477-494. doi: 10.3934/dcds.2009.23.477.  Google Scholar

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