On Ground States and Compactly Supported Solutions of Elliptic Equations with Non-Lipschitz Nonlinearities
In a bounded domain Ω ⊂ ℝ N , we consider the Dirichlet boundary-value problem for an elliptic equation with a non-Lipschitz nonlinearity of the form Δ u = λ u − u α − 1 u , λ ∈ ℝ , 0 < α < 1 . The problem of the existence of a solution of the ground-state-type with compact support is examined...
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| Published in | Journal of mathematical sciences (New York, N.Y.) Vol. 258; no. 1; pp. 110 - 114 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.10.2021
Springer Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1072-3374 1573-8795 |
| DOI | 10.1007/s10958-021-05539-4 |
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| Abstract | In a bounded domain Ω ⊂ ℝ
N
, we consider the Dirichlet boundary-value problem for an elliptic equation with a non-Lipschitz nonlinearity of the form
Δ
u
=
λ
u
−
u
α
−
1
u
,
λ
∈
ℝ
,
0
<
α
<
1
.
The problem of the existence of a solution of the ground-state-type with compact support is examined. |
|---|---|
| AbstractList | In a bounded domain [OMEGA] [subset] [??], we consider the Dirichlet boundary-value problem for an elliptic equation with a non-Lipschitz nonlinearity of the form In a bounded domain Ω ⊂ ℝ N , we consider the Dirichlet boundary-value problem for an elliptic equation with a non-Lipschitz nonlinearity of the form Δ u = λ u − u α − 1 u , λ ∈ ℝ , 0 < α < 1 . The problem of the existence of a solution of the ground-state-type with compact support is examined. In a bounded domain Ω ⊂ ℝN, we consider the Dirichlet boundary-value problem for an elliptic equation with a non-Lipschitz nonlinearity of the formΔu=λu−uα−1u,λ∈ℝ,0<α<1.The problem of the existence of a solution of the ground-state-type with compact support is examined. In a bounded domain [OMEGA] [subset] [??], we consider the Dirichlet boundary-value problem for an elliptic equation with a non-Lipschitz nonlinearity of the form [DELTA]u = [lambda]u - [|u|.sup.[alpha]-1] u, [lambda] [member of] [??], 0 <[alpha] < 1. The problem of the existence of a solution of the ground-state-type with compact support is examined. Keywords and phrases: elliptic equation, solution with compact support, non-Lipschitz nonlinearity. AMS Subject Classification: 35B44, 35B32, 35K59, 35J60, 35J70 |
| Audience | Academic |
| Author | Kholodnov, E. E. |
| Author_xml | – sequence: 1 givenname: E. E. surname: Kholodnov fullname: Kholodnov, E. E. email: emil.kholod@gmail.com organization: Institute of Mathematics with Computing Center, Ufa Federal Research Center of the Russian Academy of Sciences |
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| Cites_doi | 10.1007/s002050050162 10.1016/j.na.2009.12.015 10.13108/2017-9-4-44 10.57262/die/1370021911 10.1016/j.na.2014.11.019 10.1007/s11401-016-1073-2 10.57262/die/1371586149 |
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| Keywords | solution with compact support 35J60 35J70 35B44 35B32 non-Lipschitz nonlinearity 35K59 elliptic equation |
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| References | Ya. Sh. Il’yasov and Yu. V. Egorov, “Hopf maximum principle violation for elliptic equations with non-Lipschitz nonlinearity,” Nonlin. Anal., 72, 3346–3355 (2010). Ya. Sh. Il’yasov, “On critical exponent for an elliptic equation with non-Lipschitz nonlinearity,” Dynam. Syst., 698–706 (2011). KaperHKwongMKLiYSymmetry results for reaction-diffusion equationsDiffer. Integr. Equations199361045105612304800799.35083 J. I. Díaz, J. Hernández, and Ya. Il’yasov, “Flat solutions of some non-Lipschitz autonomous semilinear equations may be stable for N ≥ 3,” Chin. Ann. Math. Ser. B., 38 (1), 345–378 (2017). J. I. Díaz, J. Hernández, and Ya. Il’yasov, “On the existence of positive solutions and solutions with compact support for a spectral nonlinear elliptic problem with strong absorption,” Nonlin. Anal. Theory Meth. Appl., 119, 484–500 (2015). SerrinJZouHSymmetry of ground states of quasilinear elliptic equationsArch. Rat. Mech. Anal.19991484265290171666510.1007/s002050050162 Ya. Sh. Il’yasov and E. E. Kholodnov, “On the global instability of solutions of hyperbolic equations with non-Lipschitzian nonlinearities,” Ufim. Mat. Zh., 9, No. 4, 45–54 (2017). KaperHGKwongMKFree boundary problems for Emden–Fowler equationsDiffer. Integr. Equations19903235336210251840726.34024 HG Kaper (5539_CR6) 1990; 3 5539_CR5 5539_CR4 5539_CR3 5539_CR2 5539_CR1 H Kaper (5539_CR7) 1993; 6 J Serrin (5539_CR8) 1999; 148 |
| References_xml | – reference: Ya. Sh. Il’yasov and Yu. V. Egorov, “Hopf maximum principle violation for elliptic equations with non-Lipschitz nonlinearity,” Nonlin. Anal., 72, 3346–3355 (2010). – reference: SerrinJZouHSymmetry of ground states of quasilinear elliptic equationsArch. Rat. Mech. Anal.19991484265290171666510.1007/s002050050162 – reference: Ya. Sh. Il’yasov, “On critical exponent for an elliptic equation with non-Lipschitz nonlinearity,” Dynam. Syst., 698–706 (2011). – reference: KaperHGKwongMKFree boundary problems for Emden–Fowler equationsDiffer. Integr. Equations19903235336210251840726.34024 – reference: Ya. Sh. Il’yasov and E. E. Kholodnov, “On the global instability of solutions of hyperbolic equations with non-Lipschitzian nonlinearities,” Ufim. Mat. Zh., 9, No. 4, 45–54 (2017). – reference: J. I. Díaz, J. Hernández, and Ya. Il’yasov, “Flat solutions of some non-Lipschitz autonomous semilinear equations may be stable for N ≥ 3,” Chin. Ann. Math. Ser. B., 38 (1), 345–378 (2017). – reference: KaperHKwongMKLiYSymmetry results for reaction-diffusion equationsDiffer. Integr. Equations199361045105612304800799.35083 – reference: J. I. Díaz, J. Hernández, and Ya. Il’yasov, “On the existence of positive solutions and solutions with compact support for a spectral nonlinear elliptic problem with strong absorption,” Nonlin. Anal. Theory Meth. Appl., 119, 484–500 (2015). – volume: 148 start-page: 265 issue: 4 year: 1999 ident: 5539_CR8 publication-title: Arch. Rat. Mech. Anal. doi: 10.1007/s002050050162 – ident: 5539_CR4 doi: 10.1016/j.na.2009.12.015 – ident: 5539_CR5 doi: 10.13108/2017-9-4-44 – volume: 6 start-page: 1045 year: 1993 ident: 5539_CR7 publication-title: Differ. Integr. Equations doi: 10.57262/die/1370021911 – ident: 5539_CR2 doi: 10.1016/j.na.2014.11.019 – ident: 5539_CR1 doi: 10.1007/s11401-016-1073-2 – ident: 5539_CR3 – volume: 3 start-page: 353 issue: 2 year: 1990 ident: 5539_CR6 publication-title: Differ. Integr. Equations doi: 10.57262/die/1371586149 |
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| Snippet | In a bounded domain Ω ⊂ ℝ
N
, we consider the Dirichlet boundary-value problem for an elliptic equation with a non-Lipschitz nonlinearity of the form
Δ
u
=
λ
u... In a bounded domain [OMEGA] [subset] [??], we consider the Dirichlet boundary-value problem for an elliptic equation with a non-Lipschitz nonlinearity of the... In a bounded domain Ω ⊂ ℝN, we consider the Dirichlet boundary-value problem for an elliptic equation with a non-Lipschitz nonlinearity of the... |
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| SubjectTerms | Boundary value problems Dirichlet problem Elliptic functions Mathematics Mathematics and Statistics Nonlinearity |
| Title | On Ground States and Compactly Supported Solutions of Elliptic Equations with Non-Lipschitz Nonlinearities |
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