A variational approach to semilinear elliptic equations with measure data

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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

Abstract
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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 and 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.
[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.
[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.
[4]	L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right hand side measures, Comm. Partial Differential Equations, 17 (1992), 641-655.
[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.
[6]	H. Brezis and F. Browder, A property of Sobolev spaces, Comm. Partial Differential Equations, 4 (1979), 1077-1083.
[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.
[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.
[9]	A. Canino and M. Degiovanni, A variational approach to a class of singular semilinear elliptic equations, J. Convex Anal., 11 (2004), 147-162.
[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.
[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.
[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.
[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.
[14]	A. Ferrero and C. Saccon, Existence and multiplicity results for semilinear equations with measure data, Topol. Methods Nonlinear Anal., 28 (2006), 285-318.
[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.
[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.
[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.
[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.
[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.
[20]	J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591. doi: 10.1002/cpa.3160140329.
[21]	J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954. 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), 207-238.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[4]	L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right hand side measures, Comm. Partial Differential Equations, 17 (1992), 641-655.
[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.
[6]	H. Brezis and F. Browder, A property of Sobolev spaces, Comm. Partial Differential Equations, 4 (1979), 1077-1083.
[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.
[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.
[9]	A. Canino and M. Degiovanni, A variational approach to a class of singular semilinear elliptic equations, J. Convex Anal., 11 (2004), 147-162.
[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.
[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.
[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.
[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.
[14]	A. Ferrero and C. Saccon, Existence and multiplicity results for semilinear equations with measure data, Topol. Methods Nonlinear Anal., 28 (2006), 285-318.
[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.
[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.
[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.
[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.
[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.
[20]	J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591. doi: 10.1002/cpa.3160140329.
[21]	J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954. 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), 207-238.
[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.
[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.
[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.
[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.
[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.

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