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January  2013, 12(1): 451-459. doi: 10.3934/cpaa.2013.12.451

Multiplicity solutions for fully nonlinear equation involving nonlinearity with zeros

1. 

Institute for Advanced Study, Shenzhen University, Shenzhen Guangdong, 518060, China

Received  May 2011 Revised  December 2011 Published  September 2012

We study the multiplicity solutions for the nonlinear elliptic equation

$ -\mathcal{M}_{\lambda,\Lambda}^+ (D^2u)=f(u) $ in $\Omega$,

$ u=0 $ on $\partial \Omega$

and a more general fully nonlinear elliptic equation

$ F(D^2u)=f(u) $ in $\Omega$,

$ u=0 $ on $\partial \Omega$,

where $\Omega$ is a bounded domain in $\mathbb{R}^N, N\geq 3$, $f$ is a locally Lipschitz continuous function with superlinear growth at infinity. We will show that the equation has at least two positive solutions under some assumptions.

Citation: Xiaohui Yu. Multiplicity solutions for fully nonlinear equation involving nonlinearity with zeros. Communications on Pure and Applied Analysis, 2013, 12 (1) : 451-459. doi: 10.3934/cpaa.2013.12.451
References:
[1]

A. Allendes and A. Quaas, Multiplicity results for extremal operators through bifurcation, Discrete Continuous Dynamical Systems, 29 (2011), 51-65.

[2]

S. N. Armstrong and B. Sirakov, Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities,, to appear in Annali della Scuola Normale Superiore di Pisa, (). 

[3]

S. N. Armstrong, B. Sirakov and C. K. Smart, Fundamental solutions of homogeneous fully nonlinear elliptic equations,, preprint, (). 

[4]

J. Busca, M. Esteban and A. Quaas, Nonlinear eigenvalues and bifurcation problems for Puccis operator, Ann. Inst. Henri Poincare, 22 (2005), 187-206. doi: 10.1016/j.anihpc.2004.05.004.

[5]

L. Caffarelli and X. Cabre, "Fully Nonlinear Elliptic Equations," Colloquium Publication 43, American Mathematical Society, (1995).

[6]

A. Cutri and F. Leoni, On the Liouville property for fully nonlinear equations, Ann. Inst. Henri. Poincare, 17 (2000), 219-245. doi: 10.1016/S0294-1449(00)00109-8.

[7]

D. G de Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equation, J. Math. Pures. Appl., 61 (1982), 41-63.

[8]

P. Felmer, A. Quaas and B. Sirakov, Landesman-Lazer type results for second order Hamilton-Jacobi-Bellman equations, Journal of Functional Analysis, 258 (2010), 4154-4182. doi: 10.1016/j.jfa.2010.03.012.

[9]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[10]

B. Gidas and J. Spruck, A priori bounds of positive solutions of nonlinear elliptic equations, Comm. in P.D.E, 6 (1981), 883-901. doi: 10.1080/03605308108820196.

[11]

L. Iturriaga, E. Massa, J. Sanchez and P. Ubilla, Positive solutions for the p-Laplacian with a nonlinear term with zeros, J. Differential Equations, 248 (2010), 309-327. doi: 10.1016/j.jde.2009.08.008.

[12]

L. Iturriaga, S. Lorca and E. Massa, Positive solutions for the p-Laplacian involving critical and supercritical nonlinearities with zeros, Ann. Inst. Henri. Poincare, 27 (2010), 763-771. doi: 10.1016/j.anihpc.2009.11.003.

[13]

A. Krasnoselskii, "Positive Solutions of Operator Equations," P. Noordhiff, Groningon, 1964.

[14]

P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.

[15]

A. Quaas, Existence of a positive solution to a "semilinear" equation involving Pucci's operator in a convex domain, Diff. Int. Eq., 17 (2004), 481-494.

[16]

A. Quaas and B. Sirakov, Existence results for nonproper elliptic equation involving the Pucci operator, Comm. in P.D.E. 31 (2006), 987-1003. doi: 10.1080/03605300500394421.

[17]

A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators, Advances in Mathematics, 218 (2008), 105-135. doi: 10.1016/j.aim.2007.12.002.

[18]

B. Sirakov, Non uniqueness for the Dirichlet problem for fully nonlinear elliptic operators and the Ambrosetti-Prodi phenomenon,, preprint., (). 

show all references

References:
[1]

A. Allendes and A. Quaas, Multiplicity results for extremal operators through bifurcation, Discrete Continuous Dynamical Systems, 29 (2011), 51-65.

[2]

S. N. Armstrong and B. Sirakov, Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities,, to appear in Annali della Scuola Normale Superiore di Pisa, (). 

[3]

S. N. Armstrong, B. Sirakov and C. K. Smart, Fundamental solutions of homogeneous fully nonlinear elliptic equations,, preprint, (). 

[4]

J. Busca, M. Esteban and A. Quaas, Nonlinear eigenvalues and bifurcation problems for Puccis operator, Ann. Inst. Henri Poincare, 22 (2005), 187-206. doi: 10.1016/j.anihpc.2004.05.004.

[5]

L. Caffarelli and X. Cabre, "Fully Nonlinear Elliptic Equations," Colloquium Publication 43, American Mathematical Society, (1995).

[6]

A. Cutri and F. Leoni, On the Liouville property for fully nonlinear equations, Ann. Inst. Henri. Poincare, 17 (2000), 219-245. doi: 10.1016/S0294-1449(00)00109-8.

[7]

D. G de Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equation, J. Math. Pures. Appl., 61 (1982), 41-63.

[8]

P. Felmer, A. Quaas and B. Sirakov, Landesman-Lazer type results for second order Hamilton-Jacobi-Bellman equations, Journal of Functional Analysis, 258 (2010), 4154-4182. doi: 10.1016/j.jfa.2010.03.012.

[9]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[10]

B. Gidas and J. Spruck, A priori bounds of positive solutions of nonlinear elliptic equations, Comm. in P.D.E, 6 (1981), 883-901. doi: 10.1080/03605308108820196.

[11]

L. Iturriaga, E. Massa, J. Sanchez and P. Ubilla, Positive solutions for the p-Laplacian with a nonlinear term with zeros, J. Differential Equations, 248 (2010), 309-327. doi: 10.1016/j.jde.2009.08.008.

[12]

L. Iturriaga, S. Lorca and E. Massa, Positive solutions for the p-Laplacian involving critical and supercritical nonlinearities with zeros, Ann. Inst. Henri. Poincare, 27 (2010), 763-771. doi: 10.1016/j.anihpc.2009.11.003.

[13]

A. Krasnoselskii, "Positive Solutions of Operator Equations," P. Noordhiff, Groningon, 1964.

[14]

P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.

[15]

A. Quaas, Existence of a positive solution to a "semilinear" equation involving Pucci's operator in a convex domain, Diff. Int. Eq., 17 (2004), 481-494.

[16]

A. Quaas and B. Sirakov, Existence results for nonproper elliptic equation involving the Pucci operator, Comm. in P.D.E. 31 (2006), 987-1003. doi: 10.1080/03605300500394421.

[17]

A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators, Advances in Mathematics, 218 (2008), 105-135. doi: 10.1016/j.aim.2007.12.002.

[18]

B. Sirakov, Non uniqueness for the Dirichlet problem for fully nonlinear elliptic operators and the Ambrosetti-Prodi phenomenon,, preprint., (). 

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