# 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 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.
 [1] Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2010, 26 (1) : 89-104. doi: 10.3934/dcds.2010.26.89 [3] Lei Wei, Zhaosheng Feng. Isolated singularity for semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 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 and Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133 [5] Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105 [6] Rong Xiao, Yuying Zhou. Multiple solutions for a class of semilinear elliptic variational inclusion problems. Journal of Industrial and Management Optimization, 2011, 7 (4) : 991-1002. doi: 10.3934/jimo.2011.7.991 [7] Paul H. Rabinowitz. A new variational characterization of spatially heteroclinic solutions of a semilinear elliptic PDE. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 507-515. doi: 10.3934/dcds.2004.10.507 [8] Wolf-Jüergen Beyn, Janosch Rieger. Galerkin finite element methods for semilinear elliptic differential inclusions. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 295-312. doi: 10.3934/dcdsb.2013.18.295 [9] Sandra Lucente. Large data solutions for semilinear higher order equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3525-3533. doi: 10.3934/dcdss.2020247 [10] Francis Ribaud. Semilinear parabolic equations with distributions as initial data. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 305-316. doi: 10.3934/dcds.1997.3.305 [11] Verena Bögelein, Frank Duzaar, Ugo Gianazza. Very weak solutions of singular porous medium equations with measure data. Communications on Pure and Applied Analysis, 2015, 14 (1) : 23-49. doi: 10.3934/cpaa.2015.14.23 [12] Alberto Fiorenza, Anna Mercaldo, Jean Michel Rakotoson. Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 893-906. doi: 10.3934/dcds.2002.8.893 [13] 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 [14] Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601 [15] Hwai-Chiuan Wang. On domains and their indexes with applications to semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 447-467. doi: 10.3934/dcds.2007.19.447 [16] Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801 [17] 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 [18] Antonio Greco, Marcello Lucia. Gamma-star-shapedness for semilinear elliptic equations. Communications on Pure and Applied Analysis, 2005, 4 (1) : 93-99. doi: 10.3934/cpaa.2005.4.93 [19] Ying-Chieh Lin, Tsung-Fang Wu. On the semilinear fractional elliptic equations with singular weight functions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2067-2084. doi: 10.3934/dcdsb.2020325 [20] Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439

2020 Impact Factor: 1.392