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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, Classe di Scienze, arXiv:1001.4489. [3] S. N. Armstrong, B. Sirakov and C. K. Smart, Fundamental solutions of homogeneous fully nonlinear elliptic equations, preprint, arXiv:0910.4002. [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, Classe di Scienze, arXiv:1001.4489. [3] S. N. Armstrong, B. Sirakov and C. K. Smart, Fundamental solutions of homogeneous fully nonlinear elliptic equations, preprint, arXiv:0910.4002. [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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