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Multi-bump solutions for Schrödinger equation involving critical growth and potential wells
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On weak interaction between a ground state and a trapping potential
On a fractional harmonic replacement
1. | Maxwell Institute for Mathematical Sciences and School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom |
2. | Weierstraß Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany |
  We investigate how the energy changes in dependence of such set. In particular, under mild regularity conditions, we show that adding a set $A$ to $K$ increases the energy of at most the measure of $A$ (this may be seen as a perturbation result for small sets $A$).
  Also, we point out a monotonicity feature of the energy with respect to the prescribed sets and the boundary conditions.
References:
[1] |
Commun. Pure Appl. Math., 54 (2001), 479-499.
doi: 10.1002/1097-0312(200104)54:4<479::AID-CPA3>3.0.CO;2-2. |
[2] |
L. Caffarelli, O. Savin and E. Valdinoci, Minimization of a fractional perimeter-Dirichlet integral functional,, Ann. Inst. H. Poincaré Anal. Non Linéaire, ().
doi: 10.1016/j.anihpc.2014.04.004. |
[3] |
Bull. Sci. math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[4] |
Comm. Partial Differential Equations, 39 (2014), 2351-2387.
doi: 10.1080/03605302.2014.914536. |
[5] |
Graduate Texts in Mathematics, 214. Springer, New York, 2013. xiv+410 pp.
doi: 10.1007/978-1-4614-4809-9. |
[6] |
J. Chungcheong Math. Soc., 24 (2011), 267-271. Google Scholar |
[7] |
J. Math. Pures Appl. (9), 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[8] |
Publ. Mat., 58 (2014), 133-154.
doi: 10.5565/PUBLMAT_58114_06. |
[9] |
Bol. Soc. Esp. Mat. Apl., S$\vec e$MA, 49 (2009), 33-44. |
show all references
References:
[1] |
Commun. Pure Appl. Math., 54 (2001), 479-499.
doi: 10.1002/1097-0312(200104)54:4<479::AID-CPA3>3.0.CO;2-2. |
[2] |
L. Caffarelli, O. Savin and E. Valdinoci, Minimization of a fractional perimeter-Dirichlet integral functional,, Ann. Inst. H. Poincaré Anal. Non Linéaire, ().
doi: 10.1016/j.anihpc.2014.04.004. |
[3] |
Bull. Sci. math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[4] |
Comm. Partial Differential Equations, 39 (2014), 2351-2387.
doi: 10.1080/03605302.2014.914536. |
[5] |
Graduate Texts in Mathematics, 214. Springer, New York, 2013. xiv+410 pp.
doi: 10.1007/978-1-4614-4809-9. |
[6] |
J. Chungcheong Math. Soc., 24 (2011), 267-271. Google Scholar |
[7] |
J. Math. Pures Appl. (9), 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[8] |
Publ. Mat., 58 (2014), 133-154.
doi: 10.5565/PUBLMAT_58114_06. |
[9] |
Bol. Soc. Esp. Mat. Apl., S$\vec e$MA, 49 (2009), 33-44. |
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