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A generalization of Kátai's orthogonality criterion with applications
Positive radial solutions involving nonlinearities with zeros
1. | Departamento de Matemática, Universidad Técnica Federico Santa María, Av. España 1680, Valparaíso, Chile |
2. | Dipartimento di Scienze Matematiche e Ingegneria Industriale, Università Politecnica delle Marche, Via Brecce Bianche 1, 60131 Ancona, Italy |
3. | Departamento de Matemática, Universidad Técnica Federico Santa María, Av. España 1680, Valparaíso, Chile |
$ \begin{cases} -\Delta_p u = \lambda |x|^{\delta} f(u) &\mbox{in }B_1(0)\\ u = 0 &\mbox{in }\partial B_1(0), \end{cases} $ |
$ f:\mathbb{R}\to[0,\infty) $ |
$ C^1- $ |
$ f(0) = 0 $ |
$ f(U) = 0 $ |
$ U>0 $ |
$ f $ |
$ 0 $ |
$ U $ |
$ U $ |
$ \lambda $ |
$ \lambda $ |
References:
[1] |
A. Ambrosetti, H. Brezis and G. Cerami,
Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[2] |
A. Ambrosetti, J. Garcia Azorero and I. Peral,
Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal., 137 (1996), 219-242.
doi: 10.1006/jfan.1996.0045. |
[3] |
B. Barrios, J. Garca-Melin and L. Iturriaga,
Semilinear elliptic equations and nonlinearities with zeros, Nonlinear Anal., 134 (2016), 117-126.
doi: 10.1016/j.na.2015.12.025. |
[4] |
M. F. Bidaut-Véron,
Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch. Rational Mech. Anal., 107 (1989), 293-324.
doi: 10.1007/BF00251552. |
[5] |
E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955. |
[6] |
F. Dalbono and M. Franca,
Nodal solutions for supercritical Laplace equations, Commun in Math. Phys., 347 (2016), 875-901.
doi: 10.1007/s00220-015-2546-y. |
[7] |
D. G. De Figueiredo, J. P. Gossez and P. Ubilla,
Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.
doi: 10.1016/S0022-1236(02)00060-5. |
[8] |
D. G. de Figueiredo, P.-L. Lions and R. D. Nussbaum,
A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63.
|
[9] |
O. Došlý and P. Řehák, Half-linear Differential Equations, North-Holland Mathematics Studies, 202. Elsevier Science B.V., Amsterdam, 2005. |
[10] |
I. Flores and M. Franca,
Phase plane analysis for radial solutions to supercritical quasilinear elliptic equations in a ball, Nonlinear Anal., 125 (2015), 128-149.
doi: 10.1016/j.na.2015.04.015. |
[11] |
R. H. Fowler,
Further studies of Emden's and similar differential equations, Quart. Jl. Math., 2 (1931), 259-288.
doi: 10.1093/qmath/os-2.1.259. |
[12] |
M. Franca,
Classification of positive solution of $p$-Laplace equation with a growth term, Arch. Math. (Brno), 40 (2004), 415-434.
|
[13] |
M. Franca,
A dynamical approach to the study of radial solutions for $p$-Laplace equation, Rend. Semin. Mat. Univ. Politec. Torino, 65 (2007), 53-88.
|
[14] |
M. Franca,
Radial ground states and singular ground states for a spatial dependent $p$-Laplace equation, J. Differential Equations, 248 (2010), 2629-2656.
doi: 10.1016/j.jde.2010.02.012. |
[15] |
M. Franca,
Fowler transformation and radial solutions for quasilinear elliptic equations. Part 2: nonlinearities of mixed type, Ann. Mat. Pura Appl., 189 (2010), 67-94.
doi: 10.1007/s10231-009-0101-1. |
[16] |
M. Franca and A. Sfecci,
Entire solutions of superlinear problems with indefinite weights and Hardy potentials, J. Dyn. Differential Equations, 30 (2018), 1081-1118.
doi: 10.1007/s10884-017-9589-z. |
[17] |
B. Franchi, E. Lanconelli and J. Serrin,
Existence and uniqueness of nonnegative solutions of quasilinear equations in ${\mathbb{R}}^n$, Adv. in Math., 118 (1996), 177-243.
doi: 10.1006/aima.1996.0021. |
[18] |
J. Garca-Melin and L. Iturriaga,
Multiplicity of solutions for some semilinear problems involving nonlinearities with zeros, Israel J. Math., 210 (2015), 233-344.
doi: 10.1007/s11856-015-1251-z. |
[19] |
M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astron. and Astroph., 24 (1973), 229-238. |
[20] |
L. Iturriaga, S. Lorca and E. Massa,
Positive solutions for the $p$-Laplacian involving critical and supercritical nonlinearities with zeros, Ann. Inst. H. Poincare Anal. Non Linaire, 27 (2010), 763-771.
doi: 10.1016/j.anihpc.2009.11.003. |
[21] |
L. Iturriaga, S. Lorca and E. Massa,
Multiple positive solutions for the m-Laplacian and a nonlinearity with many zeros, Differential Integral Equations, 30 (2017), 145-159.
|
[22] |
L. Iturriaga, E. Massa, J. Snchez and P. Ubilla,
Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros, J. Differential Equations, 248 (2010), 309-327.
doi: 10.1016/j.jde.2009.08.008. |
[23] |
R. Johnson, X. B. Pan and Y. F. Yi,
Singular ground states of semilinear elliptic equations via invariant manifold theory, Nonlinear Anal., 20 (1993), 1279-1302.
doi: 10.1016/0362-546X(93)90132-C. |
[24] |
R. Johnson, X. B. Pan and Y. F. Yi,
The Melnikov method and elliptic equation with critical exponent, Indiana Univ. Math. J., 43 (1994), 1045-1077.
doi: 10.1512/iumj.1994.43.43046. |
[25] |
N. Kawano, W. M. Ni and S. Yotsutani,
A generalized Pohozaev identity and its applications, J. Math. Soc. Japan, 42 (1990), 541-564.
doi: 10.2969/jmsj/04230541. |
[26] |
P. L. Lions,
On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.
doi: 10.1137/1024101. |
[27] |
Z. Liu,
Positive solutions of superlinear elliptic equations, J. Funct. Anal., 167 (1999), 370-398.
doi: 10.1006/jfan.1999.3446. |
[28] |
W. M. Ni,
A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982), 801-807.
doi: 10.1512/iumj.1982.31.31056. |
[29] |
S. I. Pohozaev, Eigenfunctions of the equations $ \Delta u+ \lambda f(u) = 0$, Soviet Math. Dokl., 165 (1965), 1408-1411. |
[30] |
S. Prashanth and K. Sreenadh,
Multiplicity results in a ball for p-Laplace equation with positive nonlinearity, Adv. Differential Equations, (7) (2002), 877-896.
|
[31] |
P. Pucci, M. Garcia-Huidobro, R. Manasevich and J. Serrin, Qualitative properties of ground states for singular elliptic equations with weights, Ann. Mat. Pura Appl., 185 (2006), suppl., S205–S243.
doi: 10.1007/s10231-004-0143-3. |
[32] |
A. Sfecci,
On the structure of radial solutions for some quasilinear elliptic equations, J. Math. Anal. Appl., 470 (2019), 515-531.
doi: 10.1016/j.jmaa.2018.10.019. |
show all references
References:
[1] |
A. Ambrosetti, H. Brezis and G. Cerami,
Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[2] |
A. Ambrosetti, J. Garcia Azorero and I. Peral,
Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal., 137 (1996), 219-242.
doi: 10.1006/jfan.1996.0045. |
[3] |
B. Barrios, J. Garca-Melin and L. Iturriaga,
Semilinear elliptic equations and nonlinearities with zeros, Nonlinear Anal., 134 (2016), 117-126.
doi: 10.1016/j.na.2015.12.025. |
[4] |
M. F. Bidaut-Véron,
Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch. Rational Mech. Anal., 107 (1989), 293-324.
doi: 10.1007/BF00251552. |
[5] |
E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955. |
[6] |
F. Dalbono and M. Franca,
Nodal solutions for supercritical Laplace equations, Commun in Math. Phys., 347 (2016), 875-901.
doi: 10.1007/s00220-015-2546-y. |
[7] |
D. G. De Figueiredo, J. P. Gossez and P. Ubilla,
Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.
doi: 10.1016/S0022-1236(02)00060-5. |
[8] |
D. G. de Figueiredo, P.-L. Lions and R. D. Nussbaum,
A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63.
|
[9] |
O. Došlý and P. Řehák, Half-linear Differential Equations, North-Holland Mathematics Studies, 202. Elsevier Science B.V., Amsterdam, 2005. |
[10] |
I. Flores and M. Franca,
Phase plane analysis for radial solutions to supercritical quasilinear elliptic equations in a ball, Nonlinear Anal., 125 (2015), 128-149.
doi: 10.1016/j.na.2015.04.015. |
[11] |
R. H. Fowler,
Further studies of Emden's and similar differential equations, Quart. Jl. Math., 2 (1931), 259-288.
doi: 10.1093/qmath/os-2.1.259. |
[12] |
M. Franca,
Classification of positive solution of $p$-Laplace equation with a growth term, Arch. Math. (Brno), 40 (2004), 415-434.
|
[13] |
M. Franca,
A dynamical approach to the study of radial solutions for $p$-Laplace equation, Rend. Semin. Mat. Univ. Politec. Torino, 65 (2007), 53-88.
|
[14] |
M. Franca,
Radial ground states and singular ground states for a spatial dependent $p$-Laplace equation, J. Differential Equations, 248 (2010), 2629-2656.
doi: 10.1016/j.jde.2010.02.012. |
[15] |
M. Franca,
Fowler transformation and radial solutions for quasilinear elliptic equations. Part 2: nonlinearities of mixed type, Ann. Mat. Pura Appl., 189 (2010), 67-94.
doi: 10.1007/s10231-009-0101-1. |
[16] |
M. Franca and A. Sfecci,
Entire solutions of superlinear problems with indefinite weights and Hardy potentials, J. Dyn. Differential Equations, 30 (2018), 1081-1118.
doi: 10.1007/s10884-017-9589-z. |
[17] |
B. Franchi, E. Lanconelli and J. Serrin,
Existence and uniqueness of nonnegative solutions of quasilinear equations in ${\mathbb{R}}^n$, Adv. in Math., 118 (1996), 177-243.
doi: 10.1006/aima.1996.0021. |
[18] |
J. Garca-Melin and L. Iturriaga,
Multiplicity of solutions for some semilinear problems involving nonlinearities with zeros, Israel J. Math., 210 (2015), 233-344.
doi: 10.1007/s11856-015-1251-z. |
[19] |
M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astron. and Astroph., 24 (1973), 229-238. |
[20] |
L. Iturriaga, S. Lorca and E. Massa,
Positive solutions for the $p$-Laplacian involving critical and supercritical nonlinearities with zeros, Ann. Inst. H. Poincare Anal. Non Linaire, 27 (2010), 763-771.
doi: 10.1016/j.anihpc.2009.11.003. |
[21] |
L. Iturriaga, S. Lorca and E. Massa,
Multiple positive solutions for the m-Laplacian and a nonlinearity with many zeros, Differential Integral Equations, 30 (2017), 145-159.
|
[22] |
L. Iturriaga, E. Massa, J. Snchez and P. Ubilla,
Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros, J. Differential Equations, 248 (2010), 309-327.
doi: 10.1016/j.jde.2009.08.008. |
[23] |
R. Johnson, X. B. Pan and Y. F. Yi,
Singular ground states of semilinear elliptic equations via invariant manifold theory, Nonlinear Anal., 20 (1993), 1279-1302.
doi: 10.1016/0362-546X(93)90132-C. |
[24] |
R. Johnson, X. B. Pan and Y. F. Yi,
The Melnikov method and elliptic equation with critical exponent, Indiana Univ. Math. J., 43 (1994), 1045-1077.
doi: 10.1512/iumj.1994.43.43046. |
[25] |
N. Kawano, W. M. Ni and S. Yotsutani,
A generalized Pohozaev identity and its applications, J. Math. Soc. Japan, 42 (1990), 541-564.
doi: 10.2969/jmsj/04230541. |
[26] |
P. L. Lions,
On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.
doi: 10.1137/1024101. |
[27] |
Z. Liu,
Positive solutions of superlinear elliptic equations, J. Funct. Anal., 167 (1999), 370-398.
doi: 10.1006/jfan.1999.3446. |
[28] |
W. M. Ni,
A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982), 801-807.
doi: 10.1512/iumj.1982.31.31056. |
[29] |
S. I. Pohozaev, Eigenfunctions of the equations $ \Delta u+ \lambda f(u) = 0$, Soviet Math. Dokl., 165 (1965), 1408-1411. |
[30] |
S. Prashanth and K. Sreenadh,
Multiplicity results in a ball for p-Laplace equation with positive nonlinearity, Adv. Differential Equations, (7) (2002), 877-896.
|
[31] |
P. Pucci, M. Garcia-Huidobro, R. Manasevich and J. Serrin, Qualitative properties of ground states for singular elliptic equations with weights, Ann. Mat. Pura Appl., 185 (2006), suppl., S205–S243.
doi: 10.1007/s10231-004-0143-3. |
[32] |
A. Sfecci,
On the structure of radial solutions for some quasilinear elliptic equations, J. Math. Anal. Appl., 470 (2019), 515-531.
doi: 10.1016/j.jmaa.2018.10.019. |





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