2018, 38(3): 1505-1525. doi: 10.3934/dcds.2018062

Existence of nonnegative solutions to singular elliptic problems, a variational approach

Facultad de Matematica, Astronomia y Fisica, Universidad Nacional de Cordoba, Ciudad Universitaria, 5000 Cordoba, Argentina

* Corresponding author: Tomas Godoy

Received  May 2017 Published  December 2017

We consider the problem $-Δ u = χ_{\{ u>0\} }g( .,u) +f( .,u) $ in $Ω,$ $u = 0$ on $\partialΩ,$ $ u≥0$ in $Ω,$ where $Ω$ is a bounded domain in $\mathbb{R}^{n}$, $f:Ω×[ 0,∞) →\mathbb{R}$ and $ g:Ω×( 0,∞) →[ 0,∞) $ are Carathéodory functions, with $g( x,.) $ nonnegative, nonincreasing, and singular at the origin. We establish sufficient conditions for the existence of a nonnegative weak solution $0\not \equiv u∈ H_{0}^{1}( Ω) $ to the stated problem. We also provide conditions that guarantee that the found solution is positive $a.e.$ in $ Ω$. The problem with a parameter $Δ u = χ_{\{ u>0\} }g( .,u) +λ f( .,u) $ in $Ω,$ $u = 0$ on $ \partialΩ,$ $u≥0$ in $Ω$ is also studied. For both problems, the special case when $g( x,s) : = a( x) s^{-α( x) },$ i.e., a singularity with variable exponent, is also considered.

Citation: Tomas Godoy, Alfredo Guerin. Existence of nonnegative solutions to singular elliptic problems, a variational approach. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1505-1525. doi: 10.3934/dcds.2018062
References:
[1]

B. Bougherara, J. Giacomoni, Existence of mild solutions for a singular parabolic equation and stabilization, Adv. Nonlinear Anal., 4 (2015), 123-134.

[2]

B. Bougherara, J. Giacomoni, J. Herná ndez, Existence and regularity of weak solutions for singular elliptic problems, 2014 Madrid Conference on Applied Mathematics in honor of Alfonso Casal, Electron. J. Diff. Equ., 22 (2015), 19-30.

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, 1st edition, Springer-Verlag, New York, 2011.

[4]

H. Brezis, X Cabre, Some simple nonlinear pde's without solutions, Bollettino dell'Unione Matematica Italiana, 1 (1998), 223-262.

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A. Callegari, A. Nachman, A nonlinear singular boundary-value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 38 (1980), 275-281. doi: 10.1137/0138024.

[6]

Y. Chu, Y. Gao, Y. Gao, Existence of solutions to a class of semilinear elliptic problem with nonlinear singular terms and variable exponent, Journal of Function Spaces, 2016 (2016), Art. ID 9794739, 11 pp.

[7]

F. Cîrstea, M. Ghergu, V. Rădulescu, Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane-Emden-Fowler type, Math. Pures Appl., 84 (2005), 493-508. doi: 10.1016/j.matpur.2004.09.005.

[8]

M. M. Coclite, G. Palmieri, On a singular nonlinear Dirichlet problem, Comm. Part. Differ. Equat., 14 (1989), 1315-1327. doi: 10.1080/03605308908820656.

[9]

D. S. Cohen, H. B. Keller, Some positive problems suggested by nonlinear heat generators, J. Math. Mech., 16 (1967), 1361-1376.

[10]

M. G. Crandall, P. H. Rabinowitz, L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Part. Differ. Equations, 2 (1977), 193-222. doi: 10.1080/03605307708820029.

[11]

J. Dávila, M. Montenegro, Positive versus free boundary solutions to a singular elliptic equation, J. Anal. Math., 90 (2003), 303-335. doi: 10.1007/BF02786560.

[12]

D. G. De Figueiredo, Positive solutions of semilinear elliptic equations, in Lect. Notes Math. 957, Differential Equations (eds. D. G. de Figueiredo and C. S. HÃ ¶nig), Springer-Verlag, New York, (1982), 34–87.

[13]

M. A. del Pino, A global estimate for the gradient in a singular elliptic boundary value problem, Proc. R. Soc. Edinburgh Sect. A, 122 (1992), 341-352. doi: 10.1017/S0308210500021144.

[14]

J. I. Díaz, J. Hernández, Positive and free boundary solutions to singular nonlinear elliptic problems with absorption; An overview and open problems, Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems (2012). Electron. J. Diff. Equ., 21 (2014), 31-44.

[15]

J. Díaz, M. Morel, L. Oswald, An elliptic equation with singular nonlinearity, Comm. Part. Diff. Eq., 12 (1987), 1333-1344. doi: 10.1080/03605308708820531.

[16]

L. Dupaigne, M. Ghergu, V. Rădulescu, Lane-Emden-Fowler equations with convection and singular potential, J. Math. Pures Appl., 87 (2007), 563-581. doi: 10.1016/j.matpur.2007.03.002.

[17]

W. Fulks, J. S. Maybee, A singular nonlinear equation, Osaka Math. J., 12 (1960), 1-19.

[18]

L. Gasiński, N. S. Papageorgiou, Nonlinear elliptic equations with singular terms and combined nonlinearities, Ann. Henri Poincaré, 13 (2012), 481-512. doi: 10.1007/s00023-011-0129-9.

[19]

M. Ghergu, V. Liskevich, Z. Sobol, Singular solutions for second-order non-divergence type elliptic inequalities in punctured balls, J. Anal. Math., 123 (2014), 251-279. doi: 10.1007/s11854-014-0020-y.

[20]

M. Ghergu, V. D. Rădulescu, Multi-parameter bifurcation and asymptotics for the singular Lane-Emden-Fowler equation with a convection term, Proc. Royal Soc. Edinburgh, Sect. A, 135 (2005), 61-84. doi: 10.1017/S0308210500003760.

[21] M. Ghergu, V. D. Rădulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, 1 edition, Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, 2008.
[22]

T. Godoy, A. Guerin, Nonnegative solutions of a singular elliptic problem, Electron. J. Diff. Equ., 2016 (2016), 1-16.

[23]

T. Godoy, A. Guerin, Existence of nonnegative solutions for some singular elliptic problems, Journal of Nonlinear Functional Analysis, 2017 (2017), Article ID 11, 1-23.

[24]

A. C. Lazer, P. J. McKenna, On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730. doi: 10.1090/S0002-9939-1991-1037213-9.

[25]

N. S. Papageorgiou, G. Smyrlis, Nonlinear elliptic equations with singular reaction, Osaka J. Math., 53 (2016), 489-514.

[26]

V. D. Rădulescu, Singular phenomena in nonlinear elliptic problems. From blow-up boundary solutions to equations with singular nonlinearities, in Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. 4 (ed. M. Chipot), North-Holland Elsevier Science, Amsterdam, (2007), 483–591.

[27]

J. Shi, M. Yao, On a singular nonlinear semilinear elliptic problem, Proc. R. Soc. Edinburgh, Sect A, 128 (1998), 1389-1401. doi: 10.1017/S0308210500027384.

show all references

References:
[1]

B. Bougherara, J. Giacomoni, Existence of mild solutions for a singular parabolic equation and stabilization, Adv. Nonlinear Anal., 4 (2015), 123-134.

[2]

B. Bougherara, J. Giacomoni, J. Herná ndez, Existence and regularity of weak solutions for singular elliptic problems, 2014 Madrid Conference on Applied Mathematics in honor of Alfonso Casal, Electron. J. Diff. Equ., 22 (2015), 19-30.

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, 1st edition, Springer-Verlag, New York, 2011.

[4]

H. Brezis, X Cabre, Some simple nonlinear pde's without solutions, Bollettino dell'Unione Matematica Italiana, 1 (1998), 223-262.

[5]

A. Callegari, A. Nachman, A nonlinear singular boundary-value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 38 (1980), 275-281. doi: 10.1137/0138024.

[6]

Y. Chu, Y. Gao, Y. Gao, Existence of solutions to a class of semilinear elliptic problem with nonlinear singular terms and variable exponent, Journal of Function Spaces, 2016 (2016), Art. ID 9794739, 11 pp.

[7]

F. Cîrstea, M. Ghergu, V. Rădulescu, Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane-Emden-Fowler type, Math. Pures Appl., 84 (2005), 493-508. doi: 10.1016/j.matpur.2004.09.005.

[8]

M. M. Coclite, G. Palmieri, On a singular nonlinear Dirichlet problem, Comm. Part. Differ. Equat., 14 (1989), 1315-1327. doi: 10.1080/03605308908820656.

[9]

D. S. Cohen, H. B. Keller, Some positive problems suggested by nonlinear heat generators, J. Math. Mech., 16 (1967), 1361-1376.

[10]

M. G. Crandall, P. H. Rabinowitz, L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Part. Differ. Equations, 2 (1977), 193-222. doi: 10.1080/03605307708820029.

[11]

J. Dávila, M. Montenegro, Positive versus free boundary solutions to a singular elliptic equation, J. Anal. Math., 90 (2003), 303-335. doi: 10.1007/BF02786560.

[12]

D. G. De Figueiredo, Positive solutions of semilinear elliptic equations, in Lect. Notes Math. 957, Differential Equations (eds. D. G. de Figueiredo and C. S. HÃ ¶nig), Springer-Verlag, New York, (1982), 34–87.

[13]

M. A. del Pino, A global estimate for the gradient in a singular elliptic boundary value problem, Proc. R. Soc. Edinburgh Sect. A, 122 (1992), 341-352. doi: 10.1017/S0308210500021144.

[14]

J. I. Díaz, J. Hernández, Positive and free boundary solutions to singular nonlinear elliptic problems with absorption; An overview and open problems, Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems (2012). Electron. J. Diff. Equ., 21 (2014), 31-44.

[15]

J. Díaz, M. Morel, L. Oswald, An elliptic equation with singular nonlinearity, Comm. Part. Diff. Eq., 12 (1987), 1333-1344. doi: 10.1080/03605308708820531.

[16]

L. Dupaigne, M. Ghergu, V. Rădulescu, Lane-Emden-Fowler equations with convection and singular potential, J. Math. Pures Appl., 87 (2007), 563-581. doi: 10.1016/j.matpur.2007.03.002.

[17]

W. Fulks, J. S. Maybee, A singular nonlinear equation, Osaka Math. J., 12 (1960), 1-19.

[18]

L. Gasiński, N. S. Papageorgiou, Nonlinear elliptic equations with singular terms and combined nonlinearities, Ann. Henri Poincaré, 13 (2012), 481-512. doi: 10.1007/s00023-011-0129-9.

[19]

M. Ghergu, V. Liskevich, Z. Sobol, Singular solutions for second-order non-divergence type elliptic inequalities in punctured balls, J. Anal. Math., 123 (2014), 251-279. doi: 10.1007/s11854-014-0020-y.

[20]

M. Ghergu, V. D. Rădulescu, Multi-parameter bifurcation and asymptotics for the singular Lane-Emden-Fowler equation with a convection term, Proc. Royal Soc. Edinburgh, Sect. A, 135 (2005), 61-84. doi: 10.1017/S0308210500003760.

[21] M. Ghergu, V. D. Rădulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, 1 edition, Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, 2008.
[22]

T. Godoy, A. Guerin, Nonnegative solutions of a singular elliptic problem, Electron. J. Diff. Equ., 2016 (2016), 1-16.

[23]

T. Godoy, A. Guerin, Existence of nonnegative solutions for some singular elliptic problems, Journal of Nonlinear Functional Analysis, 2017 (2017), Article ID 11, 1-23.

[24]

A. C. Lazer, P. J. McKenna, On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730. doi: 10.1090/S0002-9939-1991-1037213-9.

[25]

N. S. Papageorgiou, G. Smyrlis, Nonlinear elliptic equations with singular reaction, Osaka J. Math., 53 (2016), 489-514.

[26]

V. D. Rădulescu, Singular phenomena in nonlinear elliptic problems. From blow-up boundary solutions to equations with singular nonlinearities, in Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. 4 (ed. M. Chipot), North-Holland Elsevier Science, Amsterdam, (2007), 483–591.

[27]

J. Shi, M. Yao, On a singular nonlinear semilinear elliptic problem, Proc. R. Soc. Edinburgh, Sect A, 128 (1998), 1389-1401. doi: 10.1017/S0308210500027384.

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