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Nonexistence results on the space or the half space of $ -\Delta u+\lambda u = |u|^{p-1}u $ via the Morse index
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Connecting orbits in Hilbert spaces and applications to P.D.E
Ground state solution and multiple solutions to elliptic equations with exponential growth and singular term
1. | School of Mathematical Sciences, Nankai University, Tianjin, 300071, China |
2. | School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, P. R. China |
$ -\texttt{div}(|\nabla u|^{N-2}\nabla u)+V(x)|u|^{N-2}u = \frac{f(x, u)}{|x|^{\eta}}\; \; \operatorname{in}\; \; \mathbb{R}^{N} $ |
$ N\geq 2 $ |
$ 0<\eta<N $ |
$ V(x) \geq V_{0 }> 0 $ |
$ f(x, t) $ |
References:
[1] |
A. Adimurthi and Y. Yang,
An interpolation of Hardy inequality and Trudinger-Moser inequality in $\mathbb{R}^{N}$ and its applications, Int. Math. Res. Notices, 13 (2010), 2394-2426.
doi: 10.1093/imrn/rnp194. |
[2] |
C. O. Alves and G. M. Figueiredo,
On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in $\mathbb{R}^{N}$, J. Differ. Equ., 246 (2009), 1288-1311.
doi: 10.1016/j.jde.2008.08.004. |
[3] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations, I. Existence of ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[4] |
D. M. Cao,
Nontrivial solution of semilinear elliptic equations with critical exponent in $\mathbb{R}^{2}$, Commun. Partial Differ. Equ., 17 (1992), 407-435.
doi: 10.1080/03605309208820848. |
[5] |
G. Cerami,
An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A, 112 (1978), 332-336.
|
[6] |
G. Cerami,
On the existence of eigenvalues for a nonlinear boundary value problem, Ann. Mat. Pura Appl., 124 (1980), 161-179.
doi: 10.1007/BF01795391. |
[7] |
D. G. de Figueiredo, O. H. Miyagaki and B. Ruf,
Elliptic equations in $\mathbb{R}^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differ. Equ., 3 (1995), 139-153.
doi: 10.1007/BF01205003. |
[8] |
M. de Souza and J. M. do Ó,
On singular Trudinger-Moser type inequalities for unbounded domains and their best exponents, Potential Anal., 38 (2013), 1091-1101.
doi: 10.1007/s11118-012-9308-7. |
[9] |
J. M. do Ó,
N-Laplacian equations in $\mathbb{R}^{N}$ with critical growth, Abstr. Appl. Anal., 2 (1997), 301-315.
doi: 10.1155/S1085337597000419. |
[10] |
J. M. do Ó, E. de Medeiros and U. Severo,
On a quasilinear nonhomogeneous elliptic equation with critical growth in $\mathbb{R}^{N}$, J. Differ. Equ., 246 (2009), 1363-1386.
doi: 10.1016/j.jde.2008.11.020. |
[11] |
J. M. do Ó, M. de Souza, E. de Medeiros and U. Severo,
An improvement for the Trudinger-Moser inequality and applications, J. Differ. Equ., 256 (2014), 1317-1349.
doi: 10.1016/j.jde.2013.10.016. |
[12] |
N. Lam and G. Lu,
N-Laplacian equations in $\mathbb{R}^{N}$ with subcritical and critical growth without the Ambrosetti-Rabinowitz condition, Adv. Nonlinear Stud., 13 (2013), 289-308.
doi: 10.1515/ans-2013-0203. |
[13] |
N. Lam and G. Lu,
Existence and multiplicity of solutions to equations of $n$-Laplacian type with critical exponential growth in $\mathbb{R}^{N}$, J. Funct. Anal., 262 (2012), 1132-1165.
doi: 10.1016/j.jfa.2011.10.012. |
[14] |
N. Lam and G. Lu,
Elliptic equations and systems with subscritical and critical exponential growth without the Ambrosetti-Rabinowitz condition, J. Geom. Anal., 24 (2014), 118-143.
doi: 10.1007/s12220-012-9330-4. |
[15] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The limit case I, Rev. Mat. Iberoam, 1 (1985), 145-201.
doi: 10.4171/RMI/6. |
[16] |
R. Panda,
Nontrivial solution of a quasilinear elliptic equation with critical growth in $\mathbb{R}^{N}$, Proc. Indian Acad. Sci. Math. Sci., 105 (1995), 425-444.
doi: 10.1007/BF02836879. |
[17] |
Y. Yang,
Adams type inequalities and related elliptic partial differential equations in dimension four, J. Differ. Equ., 252 (2012), 2266-2295.
doi: 10.1016/j.jde.2011.08.027. |
[18] |
Y. Yang,
Existence of positive solutions to quasilinear elliptic equations with exponential growth in the whole Euclidean space, J. Funct. Anal., 262 (2012), 1679-1704.
doi: 10.1016/j.jfa.2011.11.018. |
[19] |
J. M. do Ó, M. de Souza, E. de Medeiros and U. Severo,
Critical points for a functional involving critical growth of Trudinger-Moser type, Potential Anal., 42 (2015), 229-246.
doi: 10.1007/s11118-014-9431-8. |
[20] |
C. Zhang and L. Chen,
Concentration-compactness principle of singular Trudinger-Moser inequalities in $\mathbb{R}^{N}$ and $n$-Laplace equations, Adv. Nonlinear Stud., 18 (2018), 567-585.
doi: 10.1515/ans-2017-6041. |
show all references
References:
[1] |
A. Adimurthi and Y. Yang,
An interpolation of Hardy inequality and Trudinger-Moser inequality in $\mathbb{R}^{N}$ and its applications, Int. Math. Res. Notices, 13 (2010), 2394-2426.
doi: 10.1093/imrn/rnp194. |
[2] |
C. O. Alves and G. M. Figueiredo,
On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in $\mathbb{R}^{N}$, J. Differ. Equ., 246 (2009), 1288-1311.
doi: 10.1016/j.jde.2008.08.004. |
[3] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations, I. Existence of ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[4] |
D. M. Cao,
Nontrivial solution of semilinear elliptic equations with critical exponent in $\mathbb{R}^{2}$, Commun. Partial Differ. Equ., 17 (1992), 407-435.
doi: 10.1080/03605309208820848. |
[5] |
G. Cerami,
An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A, 112 (1978), 332-336.
|
[6] |
G. Cerami,
On the existence of eigenvalues for a nonlinear boundary value problem, Ann. Mat. Pura Appl., 124 (1980), 161-179.
doi: 10.1007/BF01795391. |
[7] |
D. G. de Figueiredo, O. H. Miyagaki and B. Ruf,
Elliptic equations in $\mathbb{R}^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differ. Equ., 3 (1995), 139-153.
doi: 10.1007/BF01205003. |
[8] |
M. de Souza and J. M. do Ó,
On singular Trudinger-Moser type inequalities for unbounded domains and their best exponents, Potential Anal., 38 (2013), 1091-1101.
doi: 10.1007/s11118-012-9308-7. |
[9] |
J. M. do Ó,
N-Laplacian equations in $\mathbb{R}^{N}$ with critical growth, Abstr. Appl. Anal., 2 (1997), 301-315.
doi: 10.1155/S1085337597000419. |
[10] |
J. M. do Ó, E. de Medeiros and U. Severo,
On a quasilinear nonhomogeneous elliptic equation with critical growth in $\mathbb{R}^{N}$, J. Differ. Equ., 246 (2009), 1363-1386.
doi: 10.1016/j.jde.2008.11.020. |
[11] |
J. M. do Ó, M. de Souza, E. de Medeiros and U. Severo,
An improvement for the Trudinger-Moser inequality and applications, J. Differ. Equ., 256 (2014), 1317-1349.
doi: 10.1016/j.jde.2013.10.016. |
[12] |
N. Lam and G. Lu,
N-Laplacian equations in $\mathbb{R}^{N}$ with subcritical and critical growth without the Ambrosetti-Rabinowitz condition, Adv. Nonlinear Stud., 13 (2013), 289-308.
doi: 10.1515/ans-2013-0203. |
[13] |
N. Lam and G. Lu,
Existence and multiplicity of solutions to equations of $n$-Laplacian type with critical exponential growth in $\mathbb{R}^{N}$, J. Funct. Anal., 262 (2012), 1132-1165.
doi: 10.1016/j.jfa.2011.10.012. |
[14] |
N. Lam and G. Lu,
Elliptic equations and systems with subscritical and critical exponential growth without the Ambrosetti-Rabinowitz condition, J. Geom. Anal., 24 (2014), 118-143.
doi: 10.1007/s12220-012-9330-4. |
[15] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The limit case I, Rev. Mat. Iberoam, 1 (1985), 145-201.
doi: 10.4171/RMI/6. |
[16] |
R. Panda,
Nontrivial solution of a quasilinear elliptic equation with critical growth in $\mathbb{R}^{N}$, Proc. Indian Acad. Sci. Math. Sci., 105 (1995), 425-444.
doi: 10.1007/BF02836879. |
[17] |
Y. Yang,
Adams type inequalities and related elliptic partial differential equations in dimension four, J. Differ. Equ., 252 (2012), 2266-2295.
doi: 10.1016/j.jde.2011.08.027. |
[18] |
Y. Yang,
Existence of positive solutions to quasilinear elliptic equations with exponential growth in the whole Euclidean space, J. Funct. Anal., 262 (2012), 1679-1704.
doi: 10.1016/j.jfa.2011.11.018. |
[19] |
J. M. do Ó, M. de Souza, E. de Medeiros and U. Severo,
Critical points for a functional involving critical growth of Trudinger-Moser type, Potential Anal., 42 (2015), 229-246.
doi: 10.1007/s11118-014-9431-8. |
[20] |
C. Zhang and L. Chen,
Concentration-compactness principle of singular Trudinger-Moser inequalities in $\mathbb{R}^{N}$ and $n$-Laplace equations, Adv. Nonlinear Stud., 18 (2018), 567-585.
doi: 10.1515/ans-2017-6041. |
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