Concentrating Bound States for Kirchhoff Type Problems in ℝ 3 Involving Critical Sobolev Exponents
We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth, where ε is a small positive parameter and a, b > 0 are constants, f ∈ C 1 (ℝ + ,ℝ) is subcritical, V : ℝ 3 → ℝ is a locally Hölder continuous function. We first prove that fo...
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| Published in | Advanced nonlinear studies Vol. 14; no. 2; pp. 483 - 510 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
01.05.2014
|
| Online Access | Get full text |
| ISSN | 1536-1365 2169-0375 |
| DOI | 10.1515/ans-2014-0214 |
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| Abstract | We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth,
where ε is a small positive parameter and a, b > 0 are constants, f ∈ C
1
(ℝ
+
,ℝ) is subcritical, V : ℝ
3
→ ℝ is a locally Hölder continuous function. We first prove that for ε
0
> 0 sufficiently small, the above problem has a weak solution u
ε
with exponential decay at infinity. Moreover, u
ε
concentrates around a local minimum point of V in Λ as ε → 0. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple solutions by employing the topological construction of the set where the potential V(z) attains its minimum. |
|---|---|
| AbstractList | We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth,
where ε is a small positive parameter and a, b > 0 are constants, f ∈ C
1
(ℝ
+
,ℝ) is subcritical, V : ℝ
3
→ ℝ is a locally Hölder continuous function. We first prove that for ε
0
> 0 sufficiently small, the above problem has a weak solution u
ε
with exponential decay at infinity. Moreover, u
ε
concentrates around a local minimum point of V in Λ as ε → 0. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple solutions by employing the topological construction of the set where the potential V(z) attains its minimum. |
| Author | He, Yi Li, Gongbao Peng, Shuangjie |
| Author_xml | – sequence: 1 givenname: Yi surname: He fullname: He, Yi organization: Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics Central China Normal University, Wuhan, 430079 P. R. China – sequence: 2 givenname: Gongbao surname: Li fullname: Li, Gongbao organization: Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics Central China Normal University, Wuhan, 430079 P. R. China – sequence: 3 givenname: Shuangjie surname: Peng fullname: Peng, Shuangjie organization: Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics Central China Normal University, Wuhan, 430079 P. R. China |
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| Cites_doi | 10.1016/S0893-9659(03)80038-1 10.1016/0022-1236(73)90051-7 10.1007/BF01189950 10.5186/aasfm.1990.1521 |
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| References | Lions (ref151) 1985 Ma (ref171) 2003; 16 Arosio (ref11) 1996 Felmer (ref211) 1996; 4 Li (ref131) 1990; 15 Wang (ref271) 1993 Ekeland (ref91) 1974 Floer (ref101) 1986 Ding (ref81) 1986 Strauss (ref251) 1977 Lions (ref141) 1984; 2 Rabinowitz (ref241) 1992 Oh (ref201) 1990 Ambrosetti (ref21) 1973; 14 Berestycki (ref41) 1983 Berestycki (ref51) 1983 Ancona (ref01) 1992 |
| References_xml | – start-page: 108 year: 1992 ident: ref01 article-title: Global solvability for the degenerate Kirchhoff equation with real analytic data publication-title: Invent Math – volume: 16 start-page: 243 year: 2003 ident: ref171 article-title: Positive solutions for a nonlinear nonlocal elliptic transmission problem publication-title: Appl Math Lett doi: 10.1016/S0893-9659(03)80038-1 – start-page: 55 year: 1977 ident: ref251 article-title: Existence of solitary waves in higher dimensions publication-title: Commun Math Phys – start-page: 91 year: 1986 ident: ref81 article-title: On the existence of positive entire solutions of a semilinear elliptic equation publication-title: Arch Ration Mech Anal – volume: 14 start-page: 349 year: 1973 ident: ref21 article-title: Dual variational methods in critical points theory and applications publication-title: Funct Anal doi: 10.1016/0022-1236(73)90051-7 – volume: 2 start-page: 223 year: 1984 ident: ref141 article-title: The concentration - compactness principle in the calculus of variations The locally compact case part publication-title: Inst Anal – start-page: 47 year: 1974 ident: ref91 article-title: On the variational principle publication-title: Math Anal Appl – start-page: 1 year: 1985 ident: ref151 article-title: The concentration - compactness principle in the calculus of variations The limit case part publication-title: Rev Mat – start-page: 43 year: 1992 ident: ref241 article-title: On a class of nonlinear Schro dinger equations publication-title: Angew Math Phys – start-page: 69 year: 1986 ident: ref101 article-title: Nonspreading wave packets for the cubic Schro dinger equation with a bounded potential publication-title: Funct Anal – start-page: 82 year: 1983 ident: ref41 article-title: Nonlinear scalar field equations I existence of a ground state publication-title: Arch Ration Mech Anal – volume: 4 start-page: 121 year: 1996 ident: ref211 article-title: del Pino Local mountain pass for semilinear elliptic problems in unbounded domains Partial Differential Equations publication-title: Calc doi: 10.1007/BF01189950 – start-page: 153 year: 1993 ident: ref271 article-title: On concentration of positive bound states of nonlinear Schro dinger equations publication-title: Math Phys – volume: 15 start-page: 27 year: 1990 ident: ref131 article-title: Some properties of weak solutions of nonlinear scalar field equations A publication-title: Acad Sci Fenn Math doi: 10.5186/aasfm.1990.1521 – start-page: 131 year: 1990 ident: ref201 article-title: On positive multi - lump bound states of nonlinear Schro dinger equations under multiple well potential publication-title: Math Phys – start-page: 348 year: 1996 ident: ref11 article-title: On the well - posedness of the Kirchhoff string publication-title: Trans Amer Math Soc – start-page: 82 year: 1983 ident: ref51 article-title: Nonlinear scalar field equations II existence of infinitely many solutions publication-title: Arch Ration Mech Anal |
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