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Stabilizing blow up solutions to nonlinear schrÖdinger equations
Multiple positive solutions of a sturm-liouville boundary value problem with conflicting nonlinearities
SISSA -International School for Advanced Studies, via Bonomea 265, 34136 Trieste, Italy |
$ u'' + \sum\limits_{i = 1}^m {} {\alpha _i}{a_i}(x){g_i}(u) - \sum\limits_{j = 1}^{m + 1} {} {\beta _j}{b_j}(x){k_j}(u) = 0,{\rm{ }} $ |
References:
[1] |
S. Alama and G. Tarantello,
Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., 141 (1996), 159-215.
doi: 10.1006/jfan.1996.0125. |
[2] |
A. Ambrosetti, H. Brezis and G. Cerami,
Combined effects of concave, convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[3] |
D. L. T. Anderson,
Stability of time-dependent particlelike solutions in nonlinear field theories. Ⅱ, J. Math. Phys., 12 (1971), 945-952.
|
[4] |
H. Berestycki and P.-L. Lions,
Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[5] |
D. Bonheure, J. M. Gomes and P. Habets,
Multiple positive solutions of superlinear elliptic problems with sign-changing weight, J. Differential Equations, 214 (2005), 36-64.
doi: 10.1016/j.jde.2004.08.009. |
[6] |
A. Boscaggin, G. Feltrin and F. Zanolin, Positive solutions for super-sublinear indefinite problems: high multiplicity results via coincidence degree, Trans. Amer. Math. Soc., to appear. |
[7] |
L. H. Erbe, S. C. Hu and H. Wang,
Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl., 184 (1994), 640-648.
doi: 10.1006/jmaa.1994.1227. |
[8] |
L. H. Erbe and H. Wang,
On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120 (1994), 743-748.
doi: 10.2307/2160465. |
[9] |
G. Feltrin and F. Zanolin,
Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems, Adv. Differential Equations, 20 (2015), 937-982.
|
[10] |
G. Feltrin and F. Zanolin,
Multiple positive solutions for a superlinear problem: a topological approach, J. Differential Equations, 259 (2015), 925-963.
doi: 10.1016/j.jde.2015.02.032. |
[11] |
G. Feltrin and F. Zanolin,
Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree, J. Differential Equations, 262 (2017), 4255-4291.
doi: 10.1016/j.jde.2017.01.009. |
[12] |
M. Gaudenzi, P. Habets and F. Zanolin,
An example of a superlinear problem with multiple positive solutions, Atti Sem. Mat. Fis. Univ. Modena, 51 (2003), 259-272.
|
[13] |
M. Gaudenzi, P. Habets and F. Zanolin,
Positive solutions of superlinear boundary value problems with singular indefinite weight, Commun. Pure Appl. Anal., 2 (2003), 411-423.
doi: 10.3934/cpaa.2003.2.411. |
[14] |
P. M. Girão and J. M. Gomes,
Multi-bump nodal solutions for an indefinite non-homogeneous elliptic problem, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 797-817.
doi: 10.1017/S0308210508000474. |
[15] |
R. Gómez-Reñasco and J. López-Gómez,
The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction-diffusion equations, J. Differential Equations, 167 (2000), 36-72.
|
[16] |
K. Lan and J. R. L. Webb,
Positive solutions of semilinear differential equations with singularities, J. Differential Equations, 148 (1998), 407-421.
doi: 10.1006/jdeq.1998.3475. |
[17] |
R. Manásevich, F. I. Njoku and F. Zanolin,
Positive solutions for the one-dimensional pLaplacian, Differential Integral Equations, 8 (1995), 213-222.
|
[18] |
R. D. Nussbaum, The fixed point index and some applications, vol. 94 of Séminaire de Mathématiques Suprieures [Seminar on Higher Mathematics], Presses de l'Université de Montréal, Montreal, QC, 1985. |
[19] |
R. D. Nussbaum, The fixed point index and fixed point theorems, in Topological methods for ordinary differential equations (Montecatini Terme, 1991), vol. 1537 of Lecture Notes in Math. , Springer, Berlin, 1993, pp. 143-205.
doi: 10.1007/BFb0085077. |
[20] |
H.-J. Ruppen,
Multiplicity results for a semilinear. elliptic differential equation with conflicting nonlinearities, J. Differential Equations, 147 (1998), 79-122.
doi: 10.1006/jdeq.1998.3419. |
show all references
References:
[1] |
S. Alama and G. Tarantello,
Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., 141 (1996), 159-215.
doi: 10.1006/jfan.1996.0125. |
[2] |
A. Ambrosetti, H. Brezis and G. Cerami,
Combined effects of concave, convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[3] |
D. L. T. Anderson,
Stability of time-dependent particlelike solutions in nonlinear field theories. Ⅱ, J. Math. Phys., 12 (1971), 945-952.
|
[4] |
H. Berestycki and P.-L. Lions,
Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[5] |
D. Bonheure, J. M. Gomes and P. Habets,
Multiple positive solutions of superlinear elliptic problems with sign-changing weight, J. Differential Equations, 214 (2005), 36-64.
doi: 10.1016/j.jde.2004.08.009. |
[6] |
A. Boscaggin, G. Feltrin and F. Zanolin, Positive solutions for super-sublinear indefinite problems: high multiplicity results via coincidence degree, Trans. Amer. Math. Soc., to appear. |
[7] |
L. H. Erbe, S. C. Hu and H. Wang,
Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl., 184 (1994), 640-648.
doi: 10.1006/jmaa.1994.1227. |
[8] |
L. H. Erbe and H. Wang,
On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120 (1994), 743-748.
doi: 10.2307/2160465. |
[9] |
G. Feltrin and F. Zanolin,
Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems, Adv. Differential Equations, 20 (2015), 937-982.
|
[10] |
G. Feltrin and F. Zanolin,
Multiple positive solutions for a superlinear problem: a topological approach, J. Differential Equations, 259 (2015), 925-963.
doi: 10.1016/j.jde.2015.02.032. |
[11] |
G. Feltrin and F. Zanolin,
Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree, J. Differential Equations, 262 (2017), 4255-4291.
doi: 10.1016/j.jde.2017.01.009. |
[12] |
M. Gaudenzi, P. Habets and F. Zanolin,
An example of a superlinear problem with multiple positive solutions, Atti Sem. Mat. Fis. Univ. Modena, 51 (2003), 259-272.
|
[13] |
M. Gaudenzi, P. Habets and F. Zanolin,
Positive solutions of superlinear boundary value problems with singular indefinite weight, Commun. Pure Appl. Anal., 2 (2003), 411-423.
doi: 10.3934/cpaa.2003.2.411. |
[14] |
P. M. Girão and J. M. Gomes,
Multi-bump nodal solutions for an indefinite non-homogeneous elliptic problem, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 797-817.
doi: 10.1017/S0308210508000474. |
[15] |
R. Gómez-Reñasco and J. López-Gómez,
The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction-diffusion equations, J. Differential Equations, 167 (2000), 36-72.
|
[16] |
K. Lan and J. R. L. Webb,
Positive solutions of semilinear differential equations with singularities, J. Differential Equations, 148 (1998), 407-421.
doi: 10.1006/jdeq.1998.3475. |
[17] |
R. Manásevich, F. I. Njoku and F. Zanolin,
Positive solutions for the one-dimensional pLaplacian, Differential Integral Equations, 8 (1995), 213-222.
|
[18] |
R. D. Nussbaum, The fixed point index and some applications, vol. 94 of Séminaire de Mathématiques Suprieures [Seminar on Higher Mathematics], Presses de l'Université de Montréal, Montreal, QC, 1985. |
[19] |
R. D. Nussbaum, The fixed point index and fixed point theorems, in Topological methods for ordinary differential equations (Montecatini Terme, 1991), vol. 1537 of Lecture Notes in Math. , Springer, Berlin, 1993, pp. 143-205.
doi: 10.1007/BFb0085077. |
[20] |
H.-J. Ruppen,
Multiplicity results for a semilinear. elliptic differential equation with conflicting nonlinearities, J. Differential Equations, 147 (1998), 79-122.
doi: 10.1006/jdeq.1998.3419. |


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