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Global weak solution and boundedness in a three-dimensional competing chemotaxis
Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases
Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy |
$ - {\text{div}}\left( {\frac{{\nabla u}}{{\sqrt {1 \pm |\nabla u{|^2}} }}} \right) = g(u){\text{ }}\;\;\;\;{\text{in }}{\mathbb{R}^N},$ |
References:
[1] |
A. Azzollini,
Ground state solution for a problem with mean curvature operator in Minkowski space, J. Funct. Anal., 266 (2014), 2086-2095.
doi: 10.1016/j.jfa.2013.10.002. |
[2] |
A. Azzollini,
On a prescribed mean curvature equation in Lorentz-Minkowski space, Journal de Mathématiques Pures et Appliquées, 106 (2016), 1122-1140.
doi: 10.1016/j.matpur.2016.04.003. |
[3] |
A. Azzollini, P. d'Avenia and A. Pomponio,
Quasilinear elliptic equations in $\mathbb{R}^N$ via variational methods and Orlicz-Sobolev embeddings, Calc. Var. Partial Differential Equations, 49 (2014), 197-213.
doi: 10.1007/s00526-012-0578-0. |
[4] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[5] |
H. Berestycki, P. L. Lions and L. A. Peletier,
An ODE approach to the existence of positive solutions for semilinear problems in $\mathbb{R}^N$, Indiana Univ. Math. J., 30 (1981), 141-157.
doi: 10.1512/iumj.1981.30.30012. |
[6] |
D. Bonheure, A. Derlet and C. De Coster,
Infinitely many radial solutions of a mean curvature equation in Lorentz-Minkowski space, Rend. Istit. Mat. Univ. Trieste, 44 (2012), 259-284.
|
[7] |
D. Bonheure, P. d'Avenia and A. Pomponio,
On the electrostatic Born-Infeld equation with extended charges, Comm. Math. Phys., 346 (2016), 877-906.
doi: 10.1007/s00220-016-2586-y. |
[8] |
M. Born and L. Infeld, Foundations of the new field theory,
Nature, 132 (1933), 1004. |
[9] |
M. Born and L. Infeld,
Foundations of the new field theory, Proc. Roy. Soc. London Ser. A, 144 (1934), 425-451.
|
[10] |
M. Conti and F. Gazzola,
Existence of ground states and free-boundary problems for the prescribed mean-curvature equation, Adv. Differential Equations, 7 (2002), 667-694.
|
[11] |
C. Corsato, Mathematical analysis of some differential models involving the Euclidean or the Minkowski mean curvature operator, Università degli Studi di Trieste, Trieste, 2015. |
[12] |
M. del Pino and I. Guerra,
Ground states of a prescribed mean curvature equation, J. Differential Equations, 241 (2007), 112-129.
doi: 10.1016/j.jde.2007.06.010. |
[13] |
G. Evequoz,
A dual approach in Orlicz spaces for the nonlinear Helmholtz equation, Z. Angew. Math. Phys., 66 (2015), 2995-3015.
doi: 10.1007/s00033-015-0572-4. |
[14] |
G. Evequoz and T. Weth,
Branch continuation inside the essential spectrum for the nonlinear Schrödinger equation, Journal of Fixed Point Theory and Applications, 19 (2017), 475-502.
doi: 10.1007/s11784-016-0362-4. |
[15] |
G. Evequoz and T. Weth,
Real solutions to the nonlinear Helmholtz equation with local nonlinearity, Arch. Ration. Mech. Anal., 211 (2014), 359-388.
doi: 10.1007/s00205-013-0664-2. |
[16] |
G. Evequoz and T. Weth,
Dual variational methods and nonvanishing for the nonlinear Helmholtz equation, Adv. Math., 280 (2015), 690-728.
doi: 10.1016/j.aim.2015.04.017. |
[17] |
D. Fortunato, L. Orsina and L. Pisani,
Born-Infeld type equations for electrostatic fields, J. Math. Phys., 43 (2002), 5698-5706.
doi: 10.1063/1.1508433. |
[18] |
B. Franchi, E. Lanconelli and J. Serrin,
Existence and uniqueness of nonnegative solutions of quasilinear equations in $\mathbb{R}^n$, Adv. Math., 118 (1996), 177-243.
doi: 10.1006/aima.1996.0021. |
[19] |
N. Fukagai, M. Ito and K. Narukawa,
Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on $\mathbb{R}N$, Funkcial. Ekvac., 49 (2006), 235-267.
doi: 10.1619/fesi.49.235. |
[20] |
C. Gui and F. Zhou,
Asymptotic behavior of oscillating radial solutions to certain nonlinear equations, Methods Appl. Anal., 15 (2008), 285-295.
doi: 10.4310/MAA.2008.v15.n3.a3. |
[21] |
T. Kusano and C. A. Swanson,
Radial entire solutions of a class of quasilinear elliptic equations, J. Differential Equations, 83 (1990), 379-399.
doi: 10.1016/0022-0396(90)90064-V. |
[22] |
R. Mandel, E. Montefusco and B. Pellacci, Oscillating solutions for nonlinear Helmholtz Equations Z. Angew. Math. Phys., 68 (2017), Art. 121, 19 pp.
doi: 10.1007/s00033-017-0859-8. |
[23] |
W.-M. Ni and J. Serrin,
Non-existence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo, 5 (1985), 171-185.
|
[24] |
W.-M. Ni and J. Serrin,
Existence and Non-existence theorems for quasi-linear partial differential equations, The anomalous case, Accad. Naz. Lincei, Convegni Dei Lincei, 77 (1986), 231-257.
|
[25] |
L. A. Peletier and J. Serrin,
Ground states for the prescribed mean curvature equation, Proc. Amer. Math. Soc., 100 (1987), 694-700.
doi: 10.1090/S0002-9939-1987-0894440-8. |
[26] |
A. Pomponio and T. Watanabe, Some quasilinear elliptic equations involving multiple $p$-Laplacians, to appear on Indiana Univ. Math. J. |
[27] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
show all references
References:
[1] |
A. Azzollini,
Ground state solution for a problem with mean curvature operator in Minkowski space, J. Funct. Anal., 266 (2014), 2086-2095.
doi: 10.1016/j.jfa.2013.10.002. |
[2] |
A. Azzollini,
On a prescribed mean curvature equation in Lorentz-Minkowski space, Journal de Mathématiques Pures et Appliquées, 106 (2016), 1122-1140.
doi: 10.1016/j.matpur.2016.04.003. |
[3] |
A. Azzollini, P. d'Avenia and A. Pomponio,
Quasilinear elliptic equations in $\mathbb{R}^N$ via variational methods and Orlicz-Sobolev embeddings, Calc. Var. Partial Differential Equations, 49 (2014), 197-213.
doi: 10.1007/s00526-012-0578-0. |
[4] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[5] |
H. Berestycki, P. L. Lions and L. A. Peletier,
An ODE approach to the existence of positive solutions for semilinear problems in $\mathbb{R}^N$, Indiana Univ. Math. J., 30 (1981), 141-157.
doi: 10.1512/iumj.1981.30.30012. |
[6] |
D. Bonheure, A. Derlet and C. De Coster,
Infinitely many radial solutions of a mean curvature equation in Lorentz-Minkowski space, Rend. Istit. Mat. Univ. Trieste, 44 (2012), 259-284.
|
[7] |
D. Bonheure, P. d'Avenia and A. Pomponio,
On the electrostatic Born-Infeld equation with extended charges, Comm. Math. Phys., 346 (2016), 877-906.
doi: 10.1007/s00220-016-2586-y. |
[8] |
M. Born and L. Infeld, Foundations of the new field theory,
Nature, 132 (1933), 1004. |
[9] |
M. Born and L. Infeld,
Foundations of the new field theory, Proc. Roy. Soc. London Ser. A, 144 (1934), 425-451.
|
[10] |
M. Conti and F. Gazzola,
Existence of ground states and free-boundary problems for the prescribed mean-curvature equation, Adv. Differential Equations, 7 (2002), 667-694.
|
[11] |
C. Corsato, Mathematical analysis of some differential models involving the Euclidean or the Minkowski mean curvature operator, Università degli Studi di Trieste, Trieste, 2015. |
[12] |
M. del Pino and I. Guerra,
Ground states of a prescribed mean curvature equation, J. Differential Equations, 241 (2007), 112-129.
doi: 10.1016/j.jde.2007.06.010. |
[13] |
G. Evequoz,
A dual approach in Orlicz spaces for the nonlinear Helmholtz equation, Z. Angew. Math. Phys., 66 (2015), 2995-3015.
doi: 10.1007/s00033-015-0572-4. |
[14] |
G. Evequoz and T. Weth,
Branch continuation inside the essential spectrum for the nonlinear Schrödinger equation, Journal of Fixed Point Theory and Applications, 19 (2017), 475-502.
doi: 10.1007/s11784-016-0362-4. |
[15] |
G. Evequoz and T. Weth,
Real solutions to the nonlinear Helmholtz equation with local nonlinearity, Arch. Ration. Mech. Anal., 211 (2014), 359-388.
doi: 10.1007/s00205-013-0664-2. |
[16] |
G. Evequoz and T. Weth,
Dual variational methods and nonvanishing for the nonlinear Helmholtz equation, Adv. Math., 280 (2015), 690-728.
doi: 10.1016/j.aim.2015.04.017. |
[17] |
D. Fortunato, L. Orsina and L. Pisani,
Born-Infeld type equations for electrostatic fields, J. Math. Phys., 43 (2002), 5698-5706.
doi: 10.1063/1.1508433. |
[18] |
B. Franchi, E. Lanconelli and J. Serrin,
Existence and uniqueness of nonnegative solutions of quasilinear equations in $\mathbb{R}^n$, Adv. Math., 118 (1996), 177-243.
doi: 10.1006/aima.1996.0021. |
[19] |
N. Fukagai, M. Ito and K. Narukawa,
Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on $\mathbb{R}N$, Funkcial. Ekvac., 49 (2006), 235-267.
doi: 10.1619/fesi.49.235. |
[20] |
C. Gui and F. Zhou,
Asymptotic behavior of oscillating radial solutions to certain nonlinear equations, Methods Appl. Anal., 15 (2008), 285-295.
doi: 10.4310/MAA.2008.v15.n3.a3. |
[21] |
T. Kusano and C. A. Swanson,
Radial entire solutions of a class of quasilinear elliptic equations, J. Differential Equations, 83 (1990), 379-399.
doi: 10.1016/0022-0396(90)90064-V. |
[22] |
R. Mandel, E. Montefusco and B. Pellacci, Oscillating solutions for nonlinear Helmholtz Equations Z. Angew. Math. Phys., 68 (2017), Art. 121, 19 pp.
doi: 10.1007/s00033-017-0859-8. |
[23] |
W.-M. Ni and J. Serrin,
Non-existence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo, 5 (1985), 171-185.
|
[24] |
W.-M. Ni and J. Serrin,
Existence and Non-existence theorems for quasi-linear partial differential equations, The anomalous case, Accad. Naz. Lincei, Convegni Dei Lincei, 77 (1986), 231-257.
|
[25] |
L. A. Peletier and J. Serrin,
Ground states for the prescribed mean curvature equation, Proc. Amer. Math. Soc., 100 (1987), 694-700.
doi: 10.1090/S0002-9939-1987-0894440-8. |
[26] |
A. Pomponio and T. Watanabe, Some quasilinear elliptic equations involving multiple $p$-Laplacians, to appear on Indiana Univ. Math. J. |
[27] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
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