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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 & 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.   Google Scholar [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, ().   Google Scholar [3] S. N. Armstrong, B. Sirakov and C. K. Smart, Fundamental solutions of homogeneous fully nonlinear elliptic equations,, preprint, ().   Google Scholar [4] J. Busca, M. Esteban and A. Quaas, Nonlinear eigenvalues and bifurcation problems for Puccis operator,, Ann. Inst. Henri Poincare, 22 (2005), 187.  doi: 10.1016/j.anihpc.2004.05.004.  Google Scholar [5] L. Caffarelli and X. Cabre, "Fully Nonlinear Elliptic Equations,", Colloquium Publication 43, 43 (1995).   Google Scholar [6] A. Cutri and F. Leoni, On the Liouville property for fully nonlinear equations,, Ann. Inst. Henri. Poincare, 17 (2000), 219.  doi: 10.1016/S0294-1449(00)00109-8.  Google Scholar [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.   Google Scholar [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.  doi: 10.1016/j.jfa.2010.03.012.  Google Scholar [9] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998).   Google Scholar [10] B. Gidas and J. Spruck, A priori bounds of positive solutions of nonlinear elliptic equations,, Comm. in P.D.E, 6 (1981), 883.  doi: 10.1080/03605308108820196.  Google Scholar [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.  doi: 10.1016/j.jde.2009.08.008.  Google Scholar [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.  doi: 10.1016/j.anihpc.2009.11.003.  Google Scholar [13] A. Krasnoselskii, "Positive Solutions of Operator Equations,", P. Noordhiff, (1964).   Google Scholar [14] P. L. Lions, On the existence of positive solutions of semilinear elliptic equations,, SIAM Rev., 24 (1982), 441.   Google Scholar [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.   Google Scholar [16] A. Quaas and B. Sirakov, Existence results for nonproper elliptic equation involving the Pucci operator,, Comm. in P.D.E. 31 (2006), 31 (2006), 987.  doi: 10.1080/03605300500394421.  Google Scholar [17] A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators,, Advances in Mathematics, 218 (2008), 105.  doi: 10.1016/j.aim.2007.12.002.  Google Scholar [18] B. Sirakov, Non uniqueness for the Dirichlet problem for fully nonlinear elliptic operators and the Ambrosetti-Prodi phenomenon,, preprint., ().   Google Scholar

show all references

##### References:
 [1] A. Allendes and A. Quaas, Multiplicity results for extremal operators through bifurcation,, Discrete Continuous Dynamical Systems, 29 (2011), 51.   Google Scholar [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, ().   Google Scholar [3] S. N. Armstrong, B. Sirakov and C. K. Smart, Fundamental solutions of homogeneous fully nonlinear elliptic equations,, preprint, ().   Google Scholar [4] J. Busca, M. Esteban and A. Quaas, Nonlinear eigenvalues and bifurcation problems for Puccis operator,, Ann. Inst. Henri Poincare, 22 (2005), 187.  doi: 10.1016/j.anihpc.2004.05.004.  Google Scholar [5] L. Caffarelli and X. Cabre, "Fully Nonlinear Elliptic Equations,", Colloquium Publication 43, 43 (1995).   Google Scholar [6] A. Cutri and F. Leoni, On the Liouville property for fully nonlinear equations,, Ann. Inst. Henri. Poincare, 17 (2000), 219.  doi: 10.1016/S0294-1449(00)00109-8.  Google Scholar [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.   Google Scholar [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.  doi: 10.1016/j.jfa.2010.03.012.  Google Scholar [9] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998).   Google Scholar [10] B. Gidas and J. Spruck, A priori bounds of positive solutions of nonlinear elliptic equations,, Comm. in P.D.E, 6 (1981), 883.  doi: 10.1080/03605308108820196.  Google Scholar [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.  doi: 10.1016/j.jde.2009.08.008.  Google Scholar [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.  doi: 10.1016/j.anihpc.2009.11.003.  Google Scholar [13] A. Krasnoselskii, "Positive Solutions of Operator Equations,", P. Noordhiff, (1964).   Google Scholar [14] P. L. Lions, On the existence of positive solutions of semilinear elliptic equations,, SIAM Rev., 24 (1982), 441.   Google Scholar [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.   Google Scholar [16] A. Quaas and B. Sirakov, Existence results for nonproper elliptic equation involving the Pucci operator,, Comm. in P.D.E. 31 (2006), 31 (2006), 987.  doi: 10.1080/03605300500394421.  Google Scholar [17] A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators,, Advances in Mathematics, 218 (2008), 105.  doi: 10.1016/j.aim.2007.12.002.  Google Scholar [18] B. Sirakov, Non uniqueness for the Dirichlet problem for fully nonlinear elliptic operators and the Ambrosetti-Prodi phenomenon,, preprint., ().   Google Scholar
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