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Existence and multiplicity of semiclassical states for a quasilinear Schrödinger equation in $\mathbb{R}^N$
Multiplicity solutions for fully nonlinear equation involving nonlinearity with zeros
| 1. | Institute for Advanced Study, Shenzhen University, Shenzhen Guangdong, 518060, China |
$ -\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.
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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