June  2021, 20(6): 2313-2322. doi: 10.3934/cpaa.2021078

Uniqueness and sign properties of minimizers in a quasilinear indefinite problem

1. 

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

2. 

CIEM-FaMAF, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina

3. 

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

* Corresponding author

Received  January 2021 Revised  March 2021 Published  June 2021 Early access  May 2021

Fund Project: The first author is partially supported Secyt-UNC 33620180100016CB, the third author is supported by JSPS KAKENHI Grant Number JP18K03353

Let
$ 1<q<p $
and
$ a\in C(\overline{\Omega}) $
be sign-changing, where
$ \Omega $
is a bounded and smooth domain of
$ \mathbb{R}^{N} $
. We show that the functional
$ I_{q}(u): = \int_{\Omega}\left( \frac{1}{p}|\nabla u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) , $
has exactly one nonnegative minimizer
$ U_{q} $
(in
$ W_{0}^{1,p}(\Omega) $
or
$ W^{1,p}(\Omega) $
). In addition, we prove that
$ U_{q} $
is the only possible positive solution of the associated Euler-Lagrange equation, which shows that this equation has at most one positive solution. Furthermore, we show that if
$ q $
is close enough to
$ p $
then
$ U_{q} $
is positive, which also guarantees that minimizers of
$ I_{q} $
do not change sign. Several of these results are new even for
$ p = 2 $
.
Citation: Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu. Uniqueness and sign properties of minimizers in a quasilinear indefinite problem. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2313-2322. doi: 10.3934/cpaa.2021078
References:
[1]

W. Allegretto and Y. X. Huang, A Picone's identity for the p-Laplacian and applications, Nonlinear Anal., 32 (1998), 819-830.  doi: 10.1016/S0362-546X(97)00530-0.

[2]

C. BandleM. Pozio and A. Tesei, The asymptotic behavior of the solutions of degenerate parabolic equations, Trans. Amer. Math. Soc., 303 (1987), 487-501.  doi: 10.2307/2000679.

[3]

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

[4]

M. Belloni and B. Kawohl, A direct uniqueness proof for equations involving the$p$-Laplace operator, Manuscripta Math., 109 (2002), 229-231.  doi: 10.1007/s00229-002-0305-9.

[5]

D. BonheureJ. M. Gomes and P. Habets, Multiple positive solutions of superlinear elliptic problems with sign-changing weight, J. Differ. Equ., 214 (2005), 36-64.  doi: 10.1016/j.jde.2004.08.009.

[6]

L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J., 37 (2014), 769-799.  doi: 10.2996/kmj/1414674621.

[7]

L. Brasco and G. Franzina, An overview on constrained critical points of Dirichlet integrals, Rendiconti Sem. Mat. Univ. Pol. Torino, 78 (2020), 7-50. 

[8]

H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55-64.  doi: 10.1016/0362-546X(86)90011-8.

[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.

[10]

J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries. Vol.I. Elliptic equations, Pitman, London, 1985.

[11]

J. I. Díaz and J. E. Saa, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 521-524. 

[12]

E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.  doi: 10.1016/0362-546X(83)90061-5.

[13]

P. Drábek and J. Hernández, Existence and uniqueness of positive solutions for some quasilinear elliptic problems, Nonlinear Anal., 44 (2001), 189-204.  doi: 10.1016/S0362-546X(99)00258-8.

[14]

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.

[15]

U. Kaufmann and I. Medri, Strictly positive solutions for one-dimensional nonlinear problems involving the p-Laplacian, Bull. Aust. Math. Soc., 89 (2014), 243-251.  doi: 10.1017/S0004972713000725.

[16]

U. KaufmannH. Ramos Quoirin and K. Umezu, Positivity results for indefinite sublinear elliptic problems via a continuity argument, J. Differ. Equ., 263 (2017), 4481-4502.  doi: 10.1016/j.jde.2017.05.021.

[17]

U. KaufmannH. Ramos Quoirin and K. Umezu, A curve of positive solutions for an indefinite sublinear Dirichlet problem, Discrete Contin. Dyn. Syst., 40 (2020), 617-645.  doi: 10.3934/dcds.2020063.

[18]

B. KawohlM. Lucia and S. Prashanth, Simplicity of the principal eigenvalue for indefinite quasilinear problems, Adv. Differ. Equ., 12 (2007), 407-434. 

[19]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.

[20]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.  doi: 10.1007/BF01449041.

show all references

References:
[1]

W. Allegretto and Y. X. Huang, A Picone's identity for the p-Laplacian and applications, Nonlinear Anal., 32 (1998), 819-830.  doi: 10.1016/S0362-546X(97)00530-0.

[2]

C. BandleM. Pozio and A. Tesei, The asymptotic behavior of the solutions of degenerate parabolic equations, Trans. Amer. Math. Soc., 303 (1987), 487-501.  doi: 10.2307/2000679.

[3]

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

[4]

M. Belloni and B. Kawohl, A direct uniqueness proof for equations involving the$p$-Laplace operator, Manuscripta Math., 109 (2002), 229-231.  doi: 10.1007/s00229-002-0305-9.

[5]

D. BonheureJ. M. Gomes and P. Habets, Multiple positive solutions of superlinear elliptic problems with sign-changing weight, J. Differ. Equ., 214 (2005), 36-64.  doi: 10.1016/j.jde.2004.08.009.

[6]

L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J., 37 (2014), 769-799.  doi: 10.2996/kmj/1414674621.

[7]

L. Brasco and G. Franzina, An overview on constrained critical points of Dirichlet integrals, Rendiconti Sem. Mat. Univ. Pol. Torino, 78 (2020), 7-50. 

[8]

H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55-64.  doi: 10.1016/0362-546X(86)90011-8.

[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.

[10]

J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries. Vol.I. Elliptic equations, Pitman, London, 1985.

[11]

J. I. Díaz and J. E. Saa, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 521-524. 

[12]

E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.  doi: 10.1016/0362-546X(83)90061-5.

[13]

P. Drábek and J. Hernández, Existence and uniqueness of positive solutions for some quasilinear elliptic problems, Nonlinear Anal., 44 (2001), 189-204.  doi: 10.1016/S0362-546X(99)00258-8.

[14]

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.

[15]

U. Kaufmann and I. Medri, Strictly positive solutions for one-dimensional nonlinear problems involving the p-Laplacian, Bull. Aust. Math. Soc., 89 (2014), 243-251.  doi: 10.1017/S0004972713000725.

[16]

U. KaufmannH. Ramos Quoirin and K. Umezu, Positivity results for indefinite sublinear elliptic problems via a continuity argument, J. Differ. Equ., 263 (2017), 4481-4502.  doi: 10.1016/j.jde.2017.05.021.

[17]

U. KaufmannH. Ramos Quoirin and K. Umezu, A curve of positive solutions for an indefinite sublinear Dirichlet problem, Discrete Contin. Dyn. Syst., 40 (2020), 617-645.  doi: 10.3934/dcds.2020063.

[18]

B. KawohlM. Lucia and S. Prashanth, Simplicity of the principal eigenvalue for indefinite quasilinear problems, Adv. Differ. Equ., 12 (2007), 407-434. 

[19]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.

[20]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.  doi: 10.1007/BF01449041.

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