May  2015, 14(3): 1169-1182. doi: 10.3934/cpaa.2015.14.1169

An elliptic system and the critical hyperbola

1. 

Universidade Estadual de Campinas, Campinas, CEP 13083-970

2. 

Universidade Federal da Paríba, Departamento de Matemática, João Pessoa-PB, CEP 58051-900, Brazil

3. 

Universidade Estadual de Campinas, IMECC, Departamento de Matemática, Caixa Postal 6065, CEP 13083-970, Campinas, SP

Received  June 2014 Revised  November 2014 Published  March 2015

We consider a nonlinear elliptic system of Lane-Emden type in the whole space $\mathbb{R}^{n}$, namely \begin{eqnarray} \Delta u+v| v| ^{p-1}=0, \quad x\in\mathbb{R}^{n},\\ \Delta v+u| u| ^{q-1}+f=0, \quad x\in\mathbb{R}^{n}. \end{eqnarray} Our region for $(p,q)$ covers in particular the critical and supercritical cases with respect to the critical hyperbola $\frac{1}{p+1}+\frac{1} {q+1}=\frac{n-2}{n}.$ We prove existence of solutions for $f\in L^d (\mathbb{R}^n)$, by means of a fixed point technique in the Lebesgue space $L^{r_1}\times L^{r_2}$. Our results allow unbounded solutions without $H^{s}$-regularity. The solutions are shown to be classical and positive when $f$ is smooth enough and positive. Moreover, if $f$ is radial or odd (or even), we prove that the solutions preserve these properties. Also, it is shown that the solutions $(u,v)$ are nonradial when $f$ is nonradial.
Citation: Lucas C. F. Ferreira, Everaldo Medeiros, Marcelo Montenegro. An elliptic system and the critical hyperbola. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1169-1182. doi: 10.3934/cpaa.2015.14.1169
References:
[1]

G. Bernard, An inhomogeneous semilinear equation in entire space,, \emph{J. Differential Equations}, 125 (1996), 184.  doi: 10.1006/jdeq.1996.0029.  Google Scholar

[2]

J. Busca and R. Manásevich, A Liouville type theorem for Lane Emden systems,, \emph{Indiana Univ. Math. J.}, 51 (2002), 37.   Google Scholar

[3]

G. Caristi, L. D'Ambrosio and E. Mitidieri, Liouville theorems for some nonlinear inequalities,, Tr. Mat. Inst. Steklova 260 (2008), 260 (2008), 97.  doi: 10.1134/S0081543808010070.  Google Scholar

[4]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, \emph{Discrete Contin. Dyn. Syst.}, 24 (2009), 1167.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[5]

Ph. Clément, D. G. de Figueiredo and E. Mitidieri, Positive solutions of semilinear elliptic systems,, \emph{Comm. Partial Differential Equations}, 17 (1992), 923.  doi: 10.1080/03605309208820869.  Google Scholar

[6]

Q. Dai, Entire positive solutions for inhomogeneous semilinear elliptic systems,, \emph{Glasg. Math. J.}, 47 (2005), 97.  doi: 10.1017/S0017089504002101.  Google Scholar

[7]

D. G. de Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 21 (1994), 387.   Google Scholar

[8]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, \emph{Comm. Partial Differential Equations}, 6 (1981), 883.  doi: 10.1080/03605308108820196.  Google Scholar

[9]

L. Grafakos, Classical and Modern Fourier Analysis,, Pearson Education, (2004).   Google Scholar

[10]

C. Jin and C. Li, Quantitative analysis of some system of integral equations,, \emph{Cal. Var. PDEs, 26 (2006), 447.  doi: 10.1007/s00526-006-0013-5.  Google Scholar

[11]

L. Ma and B. Liu, Symmetry results for decay solutions of elliptic systems in the whole space,, \emph{Adv. Math.}, 225 (2010), 3052.  doi: 10.1016/j.aim.2010.05.022.  Google Scholar

[12]

E. Mitidieri, A Rellich type identity and applications,, \emph{Commun. Partial Differential Equations}, 18 (1993), 125.  doi: 10.1080/03605309308820923.  Google Scholar

[13]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbbR^n$,, \emph{Differential Integral Equations}, 9 (1996), 465.   Google Scholar

[14]

S. I. Pokhozhaev, Elliptic problems in $\mathbf{\mathbbR}^N$ with a supercritical exponent of nonlinearity,, \emph{Mat. Sb.}, 182 (1991), 467.   Google Scholar

[15]

P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems,, \emph{Duke Math. J.}, 139 (2007), 555.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[16]

P. Quittner and Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces,, \emph{Arch. Ration. Mech. Anal.}, 174 (2004), 49.  doi: 10.1007/s00205-004-0323-8.  Google Scholar

[17]

J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Dedicated to Prof. C. Vinti (Italian) (Perugia, 1996)., Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 369.   Google Scholar

[18]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems,, \emph{Differential Integral Equations}, 9 (1996), 635.   Google Scholar

[19]

P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, \emph{Adv. Math.}, 221 (2009), 1409.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar

[20]

M. A. S. Souto, A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems,, \emph{Differential Integral Equations}, 8 (1995), 1245.   Google Scholar

[21]

R. Van Der Vorst, Variational identities and applications to differential systems,, \emph{Arch. Rational Mech. Anal.}, 116 (1991), 375.  doi: 10.1007/BF00375674.  Google Scholar

[22]

H. Zou, Symmetry of ground states for a semilinear elliptic system,, \emph{Trans. Amer. Math. Soc.}, 352 (2000), 1217.  doi: 10.1090/S0002-9947-99-02526-X.  Google Scholar

[23]

H. Zou, Symmetry of positive solutions of $\Delta u+u^p=0$ in $\mathbf{\mathbbR}^n$,, \emph{J. Differential Equations}, 120 (1995), 46.  doi: 10.1006/jdeq.1995.1105.  Google Scholar

show all references

References:
[1]

G. Bernard, An inhomogeneous semilinear equation in entire space,, \emph{J. Differential Equations}, 125 (1996), 184.  doi: 10.1006/jdeq.1996.0029.  Google Scholar

[2]

J. Busca and R. Manásevich, A Liouville type theorem for Lane Emden systems,, \emph{Indiana Univ. Math. J.}, 51 (2002), 37.   Google Scholar

[3]

G. Caristi, L. D'Ambrosio and E. Mitidieri, Liouville theorems for some nonlinear inequalities,, Tr. Mat. Inst. Steklova 260 (2008), 260 (2008), 97.  doi: 10.1134/S0081543808010070.  Google Scholar

[4]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, \emph{Discrete Contin. Dyn. Syst.}, 24 (2009), 1167.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[5]

Ph. Clément, D. G. de Figueiredo and E. Mitidieri, Positive solutions of semilinear elliptic systems,, \emph{Comm. Partial Differential Equations}, 17 (1992), 923.  doi: 10.1080/03605309208820869.  Google Scholar

[6]

Q. Dai, Entire positive solutions for inhomogeneous semilinear elliptic systems,, \emph{Glasg. Math. J.}, 47 (2005), 97.  doi: 10.1017/S0017089504002101.  Google Scholar

[7]

D. G. de Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 21 (1994), 387.   Google Scholar

[8]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, \emph{Comm. Partial Differential Equations}, 6 (1981), 883.  doi: 10.1080/03605308108820196.  Google Scholar

[9]

L. Grafakos, Classical and Modern Fourier Analysis,, Pearson Education, (2004).   Google Scholar

[10]

C. Jin and C. Li, Quantitative analysis of some system of integral equations,, \emph{Cal. Var. PDEs, 26 (2006), 447.  doi: 10.1007/s00526-006-0013-5.  Google Scholar

[11]

L. Ma and B. Liu, Symmetry results for decay solutions of elliptic systems in the whole space,, \emph{Adv. Math.}, 225 (2010), 3052.  doi: 10.1016/j.aim.2010.05.022.  Google Scholar

[12]

E. Mitidieri, A Rellich type identity and applications,, \emph{Commun. Partial Differential Equations}, 18 (1993), 125.  doi: 10.1080/03605309308820923.  Google Scholar

[13]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbbR^n$,, \emph{Differential Integral Equations}, 9 (1996), 465.   Google Scholar

[14]

S. I. Pokhozhaev, Elliptic problems in $\mathbf{\mathbbR}^N$ with a supercritical exponent of nonlinearity,, \emph{Mat. Sb.}, 182 (1991), 467.   Google Scholar

[15]

P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems,, \emph{Duke Math. J.}, 139 (2007), 555.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[16]

P. Quittner and Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces,, \emph{Arch. Ration. Mech. Anal.}, 174 (2004), 49.  doi: 10.1007/s00205-004-0323-8.  Google Scholar

[17]

J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Dedicated to Prof. C. Vinti (Italian) (Perugia, 1996)., Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 369.   Google Scholar

[18]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems,, \emph{Differential Integral Equations}, 9 (1996), 635.   Google Scholar

[19]

P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, \emph{Adv. Math.}, 221 (2009), 1409.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar

[20]

M. A. S. Souto, A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems,, \emph{Differential Integral Equations}, 8 (1995), 1245.   Google Scholar

[21]

R. Van Der Vorst, Variational identities and applications to differential systems,, \emph{Arch. Rational Mech. Anal.}, 116 (1991), 375.  doi: 10.1007/BF00375674.  Google Scholar

[22]

H. Zou, Symmetry of ground states for a semilinear elliptic system,, \emph{Trans. Amer. Math. Soc.}, 352 (2000), 1217.  doi: 10.1090/S0002-9947-99-02526-X.  Google Scholar

[23]

H. Zou, Symmetry of positive solutions of $\Delta u+u^p=0$ in $\mathbf{\mathbbR}^n$,, \emph{J. Differential Equations}, 120 (1995), 46.  doi: 10.1006/jdeq.1995.1105.  Google Scholar

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