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February  2020, 40(2): 817-845. doi: 10.3934/dcds.2020063

A curve of positive solutions for an indefinite sublinear Dirichlet problem

Uriel Kaufmann 1,, , Humberto Ramos Quoirin 2, and  Kenichiro Umezu 3,

1. 

FaMAF-CIEM (CONICET), Universidad Nacional de Córdoba, Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba, Argentina
2. 

Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile
3. 

Department of Mathematics, Faculty of Education, Ibaraki University, Mito 310-8512, Japan

* Corresponding author: Uriel Kaufmann

Received  February 2019 Revised  July 2019 Published  November 2019

Fund Project: U. Kaufmann was partially supported by Secyt-UNC 33620180100016CB; H. Ramos Quoirin was supported by Fondecyt grants 1161635, 1171532, 1171691, and 1181125; K. Umezu was supported by JSPS KAKENHI Grant Numbers JP15K04945 and JP18K03353

We investigate the existence of a curve
$ q\mapsto u_{q} $
, with
$ q\in(0, 1) $
, of positive solutions for the problem
$(P_{a, q}) \qquad \qquad \begin{cases} -\Delta u = a(x)u^{q} & \mbox{ in }\Omega, \\ u = 0 & \mbox{ on }\partial\Omega, \end{cases} $
where
$ \Omega $
 is a bounded and smooth domain of
$ \mathbb{R}^{N} $
 and
$ a:\Omega\rightarrow\mathbb{R} $
 is a sign-changing function (in which case the strong maximum principle does not hold). In addition, we analyze the asymptotic behavior of
$ u_{q} $
 as
$ q\rightarrow0^{+} $
 and
$ q\rightarrow1^{-} $
. We also show that in some cases
$ u_{q} $
 is the ground state solution of
$ (P_{a, q}) $
. As a byproduct, we obtain existence results for a singular and indefinite Dirichlet problem. Our results are mainly based on bifurcation and sub-supersolutions methods.
Keywords: Elliptic problem, indefinite nonlinearity, sublinear, positive solution.
Mathematics Subject Classification: Primary: 35J25, 35J61; Secondary: 35B32.
Citation: Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu. A curve of positive solutions for an indefinite sublinear Dirichlet problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 817-845. doi: 10.3934/dcds.2020063
References:
[1]

S. Alama and Q. Lu, Compactly supported solutions to stationary degenerate diffusion equations,, J. Differential Equations, 246 (2009), 3214-3240.  doi: 10.1016/j.jde.2009.01.029.  Google Scholar

[2]

C. Bandle, M. Pozio and A. Tesei, The asymptotic behavior of the solutions of degenerate parabolic equations, Trans. Amer. Math. Soc., 303 (1987), 487-501.  doi: 10.1090/S0002-9947-1987-0902780-3.  Google Scholar

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C. Bandle, M. A. Pozio and A. Tesei, Existence and uniqueness of solutions of nonlinear Neumann problems, Math. Z., 199 (1988), 257-278.  doi: 10.1007/BF01159655.  Google Scholar

[4]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^{n}$, Manuscripta Math., 74 (1992), 87-106.  doi: 10.1007/BF02567660.  Google Scholar

[5]

M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-9982-5.  Google Scholar

[6]

M. Crandall, P. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1977), 193-222.  doi: 10.1080/03605307708820029.  Google Scholar

[7]

M. Cuesta and P. Takáč, A strong comparison principle for positive solutions of degenerate elliptic equations, Differential Integral Equations, 13 (2000), 721-746.   Google Scholar

[8]

D. De Figueiredo, Positive solutions of semilinear elliptic equations, Lect. Notes Math. Springer, 957 (1982), 34-87.   Google Scholar

[9]

M. Delgado and A. Suárez, On the uniqueness of positive solution of an elliptic equation, Appl. Math. Lett., 18 (2005), 1089-1093.  doi: 10.1016/j.aml.2004.09.020.  Google Scholar

[10]

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

[11]

P. Drábek and J. Milota, Methods of Nonlinear Analysis, Applications to Differential Equations, Second edition, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser/ Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0387-8.  Google Scholar

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Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations. Vol. 1. Maximum Principles and Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812774446.  Google Scholar

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J. Hernández, F. Mancebo and J. Vega, On the linearization of some singular, nonlinear elliptic problems and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 777-813.  doi: 10.1016/S0294-1449(02)00102-6.  Google Scholar

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J. Hernández, F. Mancebo and J. Vega, Positive solutions for singular nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 41-62.  doi: 10.1017/S030821050500065X.  Google Scholar

[15]

T. Godoy and U. Kaufmann, On strictly positive solutions for some semilinear elliptic problems, NoDEA Nonlinear Differ. Equ. Appl., 20 (2013), 779-795.  doi: 10.1007/s00030-012-0179-9.  Google Scholar

[16]

T. Godoy and U. Kaufmann, On Dirichlet problems with singular nonlinearity of indefinite sign, J. Math. Anal. Appl., 428 (2015), 1239-1251.  doi: 10.1016/j.jmaa.2015.03.069.  Google Scholar

[17]

T. Godoy and U. Kaufmann, Existence of strictly positive solutions for sublinear elliptic problems in bounded domains, Adv. Nonlinear Stud., 14 (2014), 353-359.  doi: 10.1515/ans-2014-0207.  Google Scholar

[18]

S. Gomes, On a singular nonlinear elliptic problem, SIAM J. Math. Anal., 17 (1986), 1359-1369.  doi: 10.1137/0517096.  Google Scholar

[19]

L. Jeanjean, Some continuation properties via minimax arguments, Electron. J. Differential Equations, 2011 (2011), Paper No. 48, 10 pp.  Google Scholar

[20]

R. Kajikiya, Positive solutions of semilinear elliptic equations with small perturbations, Proc. Amer. Math. Soc., 141 (2013), 1335-1342.  doi: 10.1090/S0002-9939-2012-11569-2.  Google Scholar

[21]

U. Kaufmann and I. Medri, One-dimensional singular problems involving the $p$-Laplacian and nonlinearities indefinite in sign, Adv. Nonlinear Anal., 5 (2016), 251-259.  doi: 10.1515/anona-2015-0116.  Google Scholar

[22]

U. Kaufmann, H. Ramos Quoirin and K. Umezu, Positivity results for indefinite sublinear elliptic problems via a continuity argument, J. Differential Equations, 263 (2017), 4481-4502.  doi: 10.1016/j.jde.2017.05.021.  Google Scholar

[23]

U. Kaufmann, H. Ramos Quoirin and K. Umezu, Positive solutions of an elliptic Neumann problem with a sublinear indefinite nonlinearity, NoDEA Nonlinear Differ. Equ. Appl., 25 (2018), Art. 12, 34 pp. doi: 10.1007/s00030-018-0502-1.  Google Scholar

[24]

B. Kawohl, M. Lucia and S. Prashanth, Simplicity of the principal eigenvalue for indefinite quasilinear problems, Adv. Differential Equations, 12 (2007), 407-434.   Google Scholar

[25]

A. Lair and A. Shaker, Classical and weak solutions of a singular semilinear elliptic problem, J. Math. Anal. Appl., 211 (1997), 371-385.  doi: 10.1006/jmaa.1997.5470.  Google Scholar

[26]

A. Lazer and P. 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

[27]

M. A. Pozio and A. Tesei, Support properties of solution for a class of degenerate parabolic problems, Comm. Partial Differential Equations, 12 (1987), 47-75.  doi: 10.1080/03605308708820484.  Google Scholar

[28]

J. Spruck, Uniqueness in a diffusion model of population biology, Comm. Partial Differential Equations, 8 (1983), 1605-1620.  doi: 10.1080/03605308308820317.  Google Scholar

[29]

E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed-point Theorems, Translated from the German by Peter R. Wadsack, Springer-Verlag, New York, 1986.  Google Scholar

show all references

References:
[1]

S. Alama and Q. Lu, Compactly supported solutions to stationary degenerate diffusion equations,, J. Differential Equations, 246 (2009), 3214-3240.  doi: 10.1016/j.jde.2009.01.029.  Google Scholar

[2]

C. Bandle, M. Pozio and A. Tesei, The asymptotic behavior of the solutions of degenerate parabolic equations, Trans. Amer. Math. Soc., 303 (1987), 487-501.  doi: 10.1090/S0002-9947-1987-0902780-3.  Google Scholar

[3]

C. Bandle, M. A. Pozio and A. Tesei, Existence and uniqueness of solutions of nonlinear Neumann problems, Math. Z., 199 (1988), 257-278.  doi: 10.1007/BF01159655.  Google Scholar

[4]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^{n}$, Manuscripta Math., 74 (1992), 87-106.  doi: 10.1007/BF02567660.  Google Scholar

[5]

M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-9982-5.  Google Scholar

[6]

M. Crandall, P. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1977), 193-222.  doi: 10.1080/03605307708820029.  Google Scholar

[7]

M. Cuesta and P. Takáč, A strong comparison principle for positive solutions of degenerate elliptic equations, Differential Integral Equations, 13 (2000), 721-746.   Google Scholar

[8]

D. De Figueiredo, Positive solutions of semilinear elliptic equations, Lect. Notes Math. Springer, 957 (1982), 34-87.   Google Scholar

[9]

M. Delgado and A. Suárez, On the uniqueness of positive solution of an elliptic equation, Appl. Math. Lett., 18 (2005), 1089-1093.  doi: 10.1016/j.aml.2004.09.020.  Google Scholar

[10]

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

[11]

P. Drábek and J. Milota, Methods of Nonlinear Analysis, Applications to Differential Equations, Second edition, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser/ Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0387-8.  Google Scholar

[12]

Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations. Vol. 1. Maximum Principles and Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812774446.  Google Scholar

[13]

J. Hernández, F. Mancebo and J. Vega, On the linearization of some singular, nonlinear elliptic problems and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 777-813.  doi: 10.1016/S0294-1449(02)00102-6.  Google Scholar

[14]

J. Hernández, F. Mancebo and J. Vega, Positive solutions for singular nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 41-62.  doi: 10.1017/S030821050500065X.  Google Scholar

[15]

T. Godoy and U. Kaufmann, On strictly positive solutions for some semilinear elliptic problems, NoDEA Nonlinear Differ. Equ. Appl., 20 (2013), 779-795.  doi: 10.1007/s00030-012-0179-9.  Google Scholar

[16]

T. Godoy and U. Kaufmann, On Dirichlet problems with singular nonlinearity of indefinite sign, J. Math. Anal. Appl., 428 (2015), 1239-1251.  doi: 10.1016/j.jmaa.2015.03.069.  Google Scholar

[17]

T. Godoy and U. Kaufmann, Existence of strictly positive solutions for sublinear elliptic problems in bounded domains, Adv. Nonlinear Stud., 14 (2014), 353-359.  doi: 10.1515/ans-2014-0207.  Google Scholar

[18]

S. Gomes, On a singular nonlinear elliptic problem, SIAM J. Math. Anal., 17 (1986), 1359-1369.  doi: 10.1137/0517096.  Google Scholar

[19]

L. Jeanjean, Some continuation properties via minimax arguments, Electron. J. Differential Equations, 2011 (2011), Paper No. 48, 10 pp.  Google Scholar

[20]

R. Kajikiya, Positive solutions of semilinear elliptic equations with small perturbations, Proc. Amer. Math. Soc., 141 (2013), 1335-1342.  doi: 10.1090/S0002-9939-2012-11569-2.  Google Scholar

[21]

U. Kaufmann and I. Medri, One-dimensional singular problems involving the $p$-Laplacian and nonlinearities indefinite in sign, Adv. Nonlinear Anal., 5 (2016), 251-259.  doi: 10.1515/anona-2015-0116.  Google Scholar

[22]

U. Kaufmann, H. Ramos Quoirin and K. Umezu, Positivity results for indefinite sublinear elliptic problems via a continuity argument, J. Differential Equations, 263 (2017), 4481-4502.  doi: 10.1016/j.jde.2017.05.021.  Google Scholar

[23]

U. Kaufmann, H. Ramos Quoirin and K. Umezu, Positive solutions of an elliptic Neumann problem with a sublinear indefinite nonlinearity, NoDEA Nonlinear Differ. Equ. Appl., 25 (2018), Art. 12, 34 pp. doi: 10.1007/s00030-018-0502-1.  Google Scholar

[24]

B. Kawohl, M. Lucia and S. Prashanth, Simplicity of the principal eigenvalue for indefinite quasilinear problems, Adv. Differential Equations, 12 (2007), 407-434.   Google Scholar

[25]

A. Lair and A. Shaker, Classical and weak solutions of a singular semilinear elliptic problem, J. Math. Anal. Appl., 211 (1997), 371-385.  doi: 10.1006/jmaa.1997.5470.  Google Scholar

[26]

A. Lazer and P. 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

[27]

M. A. Pozio and A. Tesei, Support properties of solution for a class of degenerate parabolic problems, Comm. Partial Differential Equations, 12 (1987), 47-75.  doi: 10.1080/03605308708820484.  Google Scholar

[28]

J. Spruck, Uniqueness in a diffusion model of population biology, Comm. Partial Differential Equations, 8 (1983), 1605-1620.  doi: 10.1080/03605308308820317.  Google Scholar

[29]

E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed-point Theorems, Translated from the German by Peter R. Wadsack, Springer-Verlag, New York, 1986.  Google Scholar

Figure 1.  The curve of positive solutions in the case $ \lambda_{1}(a) = 1 $
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Figure 2.  The curve of positive solutions emanating from $ (0, \mathcal{S}(a)) $: (ⅰ) The case $ \lambda_{1}(a)>1 $. (ⅱ) The case $ \lambda_{1}(a)<1 $
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Figure 3.  The situation of $ \underline{u} $ and $ u_{*} $
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Figure 4.  The curve of positive solutions in the case $ \lambda_{1}(a) = 1 $
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Figure 5.  The indefinite weight $ a $ in the case $ q = \frac{1}{3} $
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Figure 6.  The nontrivial nonnegative solutions $ u_{1}, u_{2} $ with dead cores and a possible solution $ u_{3} $ in $ \mathcal{P}^{\circ} $ (which is even, by uniqueness) in the case $ q = \frac{1}{3} $
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