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On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion
Focusing nlkg equation with singular potential
1. | Department of Mathematics, University of Pisa, Largo B. Pontecorvo 5 Pisa, 56127 Italy |
2. | Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan |
3. | Department of Mathematics, University of Bari, Via E. Orabona 4 I-70125 Bari, Italy |
$u_{tt}-Δ u+m^2 u=|x|^{-a}|u|^{p-1}u$ |
$0 < a < 2$ |
$d≥3$ |
$a$ |
References:
[1] |
V. Combet and F. Genoud, Classification of minimal mass blow-up solutions for an L2 critical inhomogeneous NLS, J-Evol. Eq., 16 (2016), 483-500. Google Scholar |
[2] |
Z. Gan and J. Zhang, Standing waves of the inhomogeneous Klein-Gordon equations with critical exponent, Acta Math. Sin. (Engl. Ser.), 22 (2006), 357-366. Google Scholar |
[3] |
Z. Gan and J. Zhang, Cross-constrained variational problem and the non-linear Klein-Gordon equations, Glasg. Math. J., 50 (2008), 467-481. Google Scholar |
[4] |
V. Georgiev and S. Lucente, Breaking Symmetry in focusing NLKG with potential, submitted Google Scholar |
[5] |
S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE, 4 (2011), 405-460. Errata arXiv: 1506.06248. Google Scholar |
[6] |
S. Ibrahim, N. Masmoudi and K. Nakanishi, Threshold solutions in the case of mass-shift for the critical Klein-Gordon equation, Transactions of the American Mathematical Society, 366 (2014), 5653-5669. Google Scholar |
[7] |
C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-upup for the energycritical focusing non-linear wave equation, Acta Mathematica, 201 (2008), 147-212. Google Scholar |
[8] |
E. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, 2nd edition, AMS, 2001. Google Scholar |
[9] |
L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303. Google Scholar |
[10] |
W. A. Strauss, Existence of solitary waves in higher dimension, Comm. Math. Phys., 55 (1977), 149-162. Google Scholar |
[11] |
J. Su, Z. Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differ. Equ., 238 (2007), 201-219. Google Scholar |
[12] |
E. Yanagida, Uniqueness of positive radial solutions of ∆u+g(r)u+h(r)up = 0 in ${{\mathbb{R}}^{n}}$, Arch. Rational Mech. Anal., 155 (1991), 257-274. Google Scholar |
[13] |
J. Zhang, Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations, Nonlinear Anal. Ser. A: Theory Methods, 48 (2002), 191-207. Google Scholar |
show all references
References:
[1] |
V. Combet and F. Genoud, Classification of minimal mass blow-up solutions for an L2 critical inhomogeneous NLS, J-Evol. Eq., 16 (2016), 483-500. Google Scholar |
[2] |
Z. Gan and J. Zhang, Standing waves of the inhomogeneous Klein-Gordon equations with critical exponent, Acta Math. Sin. (Engl. Ser.), 22 (2006), 357-366. Google Scholar |
[3] |
Z. Gan and J. Zhang, Cross-constrained variational problem and the non-linear Klein-Gordon equations, Glasg. Math. J., 50 (2008), 467-481. Google Scholar |
[4] |
V. Georgiev and S. Lucente, Breaking Symmetry in focusing NLKG with potential, submitted Google Scholar |
[5] |
S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE, 4 (2011), 405-460. Errata arXiv: 1506.06248. Google Scholar |
[6] |
S. Ibrahim, N. Masmoudi and K. Nakanishi, Threshold solutions in the case of mass-shift for the critical Klein-Gordon equation, Transactions of the American Mathematical Society, 366 (2014), 5653-5669. Google Scholar |
[7] |
C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-upup for the energycritical focusing non-linear wave equation, Acta Mathematica, 201 (2008), 147-212. Google Scholar |
[8] |
E. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, 2nd edition, AMS, 2001. Google Scholar |
[9] |
L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303. Google Scholar |
[10] |
W. A. Strauss, Existence of solitary waves in higher dimension, Comm. Math. Phys., 55 (1977), 149-162. Google Scholar |
[11] |
J. Su, Z. Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differ. Equ., 238 (2007), 201-219. Google Scholar |
[12] |
E. Yanagida, Uniqueness of positive radial solutions of ∆u+g(r)u+h(r)up = 0 in ${{\mathbb{R}}^{n}}$, Arch. Rational Mech. Anal., 155 (1991), 257-274. Google Scholar |
[13] |
J. Zhang, Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations, Nonlinear Anal. Ser. A: Theory Methods, 48 (2002), 191-207. Google Scholar |
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