# American Institute of Mathematical Sciences

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

## A variational approach to semilinear elliptic equations with measure data

 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.
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.

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##### 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.
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