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

[2]

B. BougheraraJ. Giacomoni and 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.   Google Scholar

[3]

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

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

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A. Callegari and 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.  Google Scholar

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Y. ChuY. Gao and 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.   Google Scholar

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F. CîrsteaM. Ghergu and 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.  Google Scholar

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

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

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

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

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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. Google Scholar

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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.  Google Scholar

[14]

J. I. Díaz and 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.   Google Scholar

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

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L. DupaigneM. Ghergu and 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.  Google Scholar

[17]

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

[18]

L. Gasiński and 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.  Google Scholar

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M. GherguV. Liskevich and 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.  Google Scholar

[20]

M. Ghergu and 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.  Google Scholar

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

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

[23]

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

[24]

A. C. Lazer and 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.  Google Scholar

[25]

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

[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.  Google Scholar

[27]

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

show all references

References:
[1]

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

[2]

B. BougheraraJ. Giacomoni and 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.   Google Scholar

[3]

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

[4]

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

[5]

A. Callegari and 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.  Google Scholar

[6]

Y. ChuY. Gao and 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.   Google Scholar

[7]

F. CîrsteaM. Ghergu and 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.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[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. Google Scholar

[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.  Google Scholar

[14]

J. I. Díaz and 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.   Google Scholar

[15]

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

[16]

L. DupaigneM. Ghergu and 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.  Google Scholar

[17]

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

[18]

L. Gasiński and 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.  Google Scholar

[19]

M. GherguV. Liskevich and 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.  Google Scholar

[20]

M. Ghergu and 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.  Google Scholar

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

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

[23]

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

[24]

A. C. Lazer and 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.  Google Scholar

[25]

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

[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.  Google Scholar

[27]

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

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