doi: 10.3934/dcds.2021125
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A log–exp elliptic equation in the plane

1. 

Universidade de Brasília, Departamento de Matemática, Campus Darcy Ribeiro, 01, CEP 70910-900, Brasília, DF, Brasil

2. 

Universidade Estadual de Campinas, IMECC, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, CEP 13083-859, Campinas, SP, Brasil

* Corresponding author

Received  December 2020 Revised  June 2021 Early access September 2021

Fund Project: The author G.F. is supported by CNPq and FAP-DF, M.M. is supported by CNPq and FAPESP and M.F.S. is supported by CAPES

In this paper we show the existence of a nonnegative solution for a singular problem with logarithmic and exponential nonlinearity, namely $ -\Delta u = \log(u)\chi_{\{u>0\}} + \lambda f(u) $ in $ \Omega $ with $ u = 0 $ on $ \partial\Omega $, where $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^{2} $. We replace the singular function $ \log(u) $ by a function $ g_\epsilon(u) $ which pointwisely converges to -$ \log(u) $ as $ \epsilon \rightarrow 0 $. When the parameter $ \lambda>0 $ is small enough, the corresponding energy functional to the perturbed equation $ -\Delta u + g_\epsilon(u) = \lambda f(u) $ has a critical point $ u_\epsilon $ in $ H_0^1(\Omega) $, which converges to a nontrivial nonnegative solution of the original problem as $ \epsilon \rightarrow 0 $.

Citation: Giovany Figueiredo, Marcelo Montenegro, Matheus F. Stapenhorst. A log–exp elliptic equation in the plane. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021125
References:
[1]

F. S. B. AlbuquerqueC. O. Alves and E. S. Medeiros, Nonlinear Schrödinger equation with unbounded or decaying radial potentials involving exponential critical growth in $\mathbb{R}^2$, J. Math. Anal. Appl., 409 (2014), 1021-1031.  doi: 10.1016/j.jmaa.2013.07.005.  Google Scholar

[2]

A. AmbrosettiH. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.  Google Scholar

[3]

S. Aouaoui, A new Trudinger–Moser type inequality and an application to some elliptic equation with doubly exponential nonlinearity in the whole space $\mathbb{R}^2$, Arch. Math., 114 (2020), 199-214.  doi: 10.1007/s00013-019-01386-7.  Google Scholar

[4]

Y. BaiL. Gasiński and N. S. Papageorgiou, Positive solutions for nonlinear singular superlinear elliptic equations, Positivity, 23 (2019), 761-778.  doi: 10.1007/s11117-018-0636-8.  Google Scholar

[5]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar

[6]

M. I. M. Copetti and C. M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math., 63 (1992), 39-65.  doi: 10.1007/BF01385847.  Google Scholar

[7]

R. Dal PassoL. Giacomelli and A. Novick-Cohen, Existence for an Allen-Cahn/Cahn-Hilliard system with degenerate mobility, Interfaces Free Bound., 1 (1999), 199-226.  doi: 10.4171/IFB/9.  Google Scholar

[8]

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

[9]

J. Dávila and M. Montenegro, Concentration for an elliptic equation with singular nonlinearity, J. Math. Pures Appl., 97 (2012), 545-578.  doi: 10.1016/j.matpur.2011.02.001.  Google Scholar

[10]

A. L. A. de Araujo and M. Montenegro, Existence of solution for a general class of elliptic equations with exponential growth, Ann. Mat. Pura Appl., 195 (2016), 1737-1748.  doi: 10.1007/s10231-015-0545-4.  Google Scholar

[11]

D. G. de FigueiredoO. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbb{R}^{2}$ with nonlinearities in the critical growth range, Calc. Var., 3 (1995), 139-153.  doi: 10.1007/BF01205003.  Google Scholar

[12]

C. M. Elliott and and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423.  doi: 10.1137/S0036141094267662.  Google Scholar

[13]

G. Figueiredo and M. Montenegro, A class of elliptic equations with singular and critical nonlinearities, Acta Appl. Math., 143 (2016), 63-89.  doi: 10.1007/s10440-015-0028-z.  Google Scholar

[14]

A. Friedman and D. Phillips, The free boundary of a semilinear elliptic equation, Trans. AMS, 282 (1984), 153-182.  doi: 10.1090/S0002-9947-1984-0728708-4.  Google Scholar

[15]

M. F. FurtadoE. S. Medeiros and U. B. Severo, A Trudinger–Moser inequality in a weighted Sobolev space and applications, Math. Nachr., 287 (2014), 1255-1273.  doi: 10.1002/mana.201200315.  Google Scholar

[16]

G. Gilardi and E. Rocca, Well-posedness and long-time behaviour for a singular phase field system of conserved type, IMA J. Appl. Math., 72 (2007), 498-530.  doi: 10.1093/imamat/hxm015.  Google Scholar

[17]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, Rev. Mat. Iberoamericana, 1 (1985), 145-201.  doi: 10.4171/RMI/6.  Google Scholar

[18]

S. Lorca and M. Montenegro, Free boundary solutions to a log-singular elliptic equation, Asymptot. Anal., 82 (2013), 91-107.  doi: 10.3233/ASY-2012-1138.  Google Scholar

[19]

M. Montenegro and O. Queiroz, Existence and regularity to an elliptic equation with logarithmic nonlinearity, J. Differential Equations, 246 (2009), 482-511.  doi: 10.1016/j.jde.2008.06.035.  Google Scholar

[20]

M. Montenegro and E. A. B. Silva, Two solutions for a singular elliptic equation by variational methods, Ann. Sc. Norm. Sup. Pisa Cl. Sci., 11 (2012), 143-165.   Google Scholar

[21]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.  Google Scholar

[22]

D. Phillips, A minimization problem and the regularity of solutions in the presence of a free boundary, Indiana Univ. Math. J., 32 (1983), 1-17.  doi: 10.1512/iumj.1983.32.32001.  Google Scholar

[23]

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

[24]

M. Willem, Minimax Theorems, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

show all references

References:
[1]

F. S. B. AlbuquerqueC. O. Alves and E. S. Medeiros, Nonlinear Schrödinger equation with unbounded or decaying radial potentials involving exponential critical growth in $\mathbb{R}^2$, J. Math. Anal. Appl., 409 (2014), 1021-1031.  doi: 10.1016/j.jmaa.2013.07.005.  Google Scholar

[2]

A. AmbrosettiH. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.  Google Scholar

[3]

S. Aouaoui, A new Trudinger–Moser type inequality and an application to some elliptic equation with doubly exponential nonlinearity in the whole space $\mathbb{R}^2$, Arch. Math., 114 (2020), 199-214.  doi: 10.1007/s00013-019-01386-7.  Google Scholar

[4]

Y. BaiL. Gasiński and N. S. Papageorgiou, Positive solutions for nonlinear singular superlinear elliptic equations, Positivity, 23 (2019), 761-778.  doi: 10.1007/s11117-018-0636-8.  Google Scholar

[5]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar

[6]

M. I. M. Copetti and C. M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math., 63 (1992), 39-65.  doi: 10.1007/BF01385847.  Google Scholar

[7]

R. Dal PassoL. Giacomelli and A. Novick-Cohen, Existence for an Allen-Cahn/Cahn-Hilliard system with degenerate mobility, Interfaces Free Bound., 1 (1999), 199-226.  doi: 10.4171/IFB/9.  Google Scholar

[8]

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

[9]

J. Dávila and M. Montenegro, Concentration for an elliptic equation with singular nonlinearity, J. Math. Pures Appl., 97 (2012), 545-578.  doi: 10.1016/j.matpur.2011.02.001.  Google Scholar

[10]

A. L. A. de Araujo and M. Montenegro, Existence of solution for a general class of elliptic equations with exponential growth, Ann. Mat. Pura Appl., 195 (2016), 1737-1748.  doi: 10.1007/s10231-015-0545-4.  Google Scholar

[11]

D. G. de FigueiredoO. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbb{R}^{2}$ with nonlinearities in the critical growth range, Calc. Var., 3 (1995), 139-153.  doi: 10.1007/BF01205003.  Google Scholar

[12]

C. M. Elliott and and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423.  doi: 10.1137/S0036141094267662.  Google Scholar

[13]

G. Figueiredo and M. Montenegro, A class of elliptic equations with singular and critical nonlinearities, Acta Appl. Math., 143 (2016), 63-89.  doi: 10.1007/s10440-015-0028-z.  Google Scholar

[14]

A. Friedman and D. Phillips, The free boundary of a semilinear elliptic equation, Trans. AMS, 282 (1984), 153-182.  doi: 10.1090/S0002-9947-1984-0728708-4.  Google Scholar

[15]

M. F. FurtadoE. S. Medeiros and U. B. Severo, A Trudinger–Moser inequality in a weighted Sobolev space and applications, Math. Nachr., 287 (2014), 1255-1273.  doi: 10.1002/mana.201200315.  Google Scholar

[16]

G. Gilardi and E. Rocca, Well-posedness and long-time behaviour for a singular phase field system of conserved type, IMA J. Appl. Math., 72 (2007), 498-530.  doi: 10.1093/imamat/hxm015.  Google Scholar

[17]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, Rev. Mat. Iberoamericana, 1 (1985), 145-201.  doi: 10.4171/RMI/6.  Google Scholar

[18]

S. Lorca and M. Montenegro, Free boundary solutions to a log-singular elliptic equation, Asymptot. Anal., 82 (2013), 91-107.  doi: 10.3233/ASY-2012-1138.  Google Scholar

[19]

M. Montenegro and O. Queiroz, Existence and regularity to an elliptic equation with logarithmic nonlinearity, J. Differential Equations, 246 (2009), 482-511.  doi: 10.1016/j.jde.2008.06.035.  Google Scholar

[20]

M. Montenegro and E. A. B. Silva, Two solutions for a singular elliptic equation by variational methods, Ann. Sc. Norm. Sup. Pisa Cl. Sci., 11 (2012), 143-165.   Google Scholar

[21]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.  Google Scholar

[22]

D. Phillips, A minimization problem and the regularity of solutions in the presence of a free boundary, Indiana Univ. Math. J., 32 (1983), 1-17.  doi: 10.1512/iumj.1983.32.32001.  Google Scholar

[23]

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

[24]

M. Willem, Minimax Theorems, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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