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On nonexistence of extremals for the Trudinger-Moser functionals involving $ L^p $ norms
Existence of infinitely many solutions for semilinear problems on exterior domains
University of North Texas, Denton, TX 76203-1430, USA |
In this paper we prove the existence of infinitely many radial solutions of $ \Delta u + K(r)f(u) = 0 $ on the exterior of the ball of radius $ R>0 $, $ B_{R} $, centered at the origin in $ {\mathbb R}^{N} $ with $ u = 0 $ on $ \partial B_{R} $ and $ \lim_{r \to \infty} u(r) = 0 $ where $ N>2 $, $ f $ is odd with $ f<0 $ on $ (0, \beta) $, $ f>0 $ on $ (\beta, \infty), $ $ f $ superlinear for large $ u $ and $ 0< K(r) \leq \frac{K_{1}}{r^{\alpha}} $ with $ 2<\alpha <2(N-1) $ for large $ r $.
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H. Berestycki and P. L. Lions,
Non-linear scalar field equations Ⅰ, Arch. Ration. Mech. Anal., 82 (1983), 313-347.
doi: 10.1007/BF00250555. |
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H. Berestycki and P.L. Lions,
Non-linear scalar field equations Ⅱ, Arch. Rational Mech. Anal., 82 (1983), 347-375.
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M. Berger, Nonlinearity and Functional Analysis, Academic Press, New York, 1977. |
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A. Castro, L. Sankar and R. Shivaji,
Uniqueness of nonnegative solutions for semipositone problems on exterior domains, J. Math. Anal. Appl., 394 (2012), 432-437.
doi: 10.1016/j.jmaa.2012.04.005. |
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M. Chhetri, L. Sankar and R. Shivaji, Positive solutions for a class of superlinear semipositone systems on exterior domains, Bound. Value Probl., (2014), 198–207.
doi: 10.1186/s13661-014-0198-z. |
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J. Iaia,
Existence and nonexistence for semilinear equations on exterior domains, J. Partial Differ. Equ., 30 (2017), 1-17.
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J. Iaia,
Existence and nonexistence of solutions for sublinear equations on exterior domains, Electron. J. Differ. Equ., 181 (2018), 1-14.
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| [9] |
J. Iaia,
Existence of solutions for semilinear problems with prescribed number of zeros on exterior domains, J. Math. Anal. Appl., 446 (2017), 591-604.
doi: 10.1016/j.jmaa.2016.08.063. |
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C. K. R. T. Jones and T. Kupper,
On the infinitely many solutions of a semi-linear equation, SIAM J. Math. Anal., 17 (1986), 803-835.
doi: 10.1137/0517059. |
| [11] |
J. Joshi,
Existence and nonexistence of solutions of sublinear problems with prescribed number of zeros on exterior domains, Electron. J. Differ. Equ., 133 (2017), 1-10.
|
| [12] |
E. K. Lee, R. Shivaji and B. Son,
Positive radial solutions to classes of singular problems on the exterior of a ball, J. Math. Anal. Appl., 434 (2016), 1597-1611.
doi: 10.1016/j.jmaa.2015.09.072. |
| [13] |
E. Lee, L. Sankar and R. Shivaji,
Positive solutions for infinite semipositone problems on exterior domains, Differ. Integral Equ., 24 (2011), 861-875.
|
| [14] |
K. McLeod, W. C. Troy and F. B. Weissler,
Radial solutions of $\Delta u + f(u) = 0$ with prescribed numbers of zeros, J. Differ. Equ., 83 (1990), 368-373.
doi: 10.1016/0022-0396(90)90063-U. |
| [15] |
L. Sankar, S. Sasi and R. Shivaji,
Semipositone problems with falling zeros on exterior domains, J. Math. Anal. Appl., 401 (2013), 146-153.
doi: 10.1016/j.jmaa.2012.11.031. |
| [16] |
W. Strauss,
Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.
|
show all references
References:
| [1] |
H. Berestycki and P. L. Lions,
Non-linear scalar field equations Ⅰ, Arch. Ration. Mech. Anal., 82 (1983), 313-347.
doi: 10.1007/BF00250555. |
| [2] |
H. Berestycki and P.L. Lions,
Non-linear scalar field equations Ⅱ, Arch. Rational Mech. Anal., 82 (1983), 347-375.
doi: 10.1007/BF00250556. |
| [3] |
M. Berger, Nonlinearity and Functional Analysis, Academic Press, New York, 1977. |
| [4] |
G. Birkhoff and G. C. Rota, Ordinary Differential Equations, Ginn and Company, 1962. |
| [5] |
A. Castro, L. Sankar and R. Shivaji,
Uniqueness of nonnegative solutions for semipositone problems on exterior domains, J. Math. Anal. Appl., 394 (2012), 432-437.
doi: 10.1016/j.jmaa.2012.04.005. |
| [6] |
M. Chhetri, L. Sankar and R. Shivaji, Positive solutions for a class of superlinear semipositone systems on exterior domains, Bound. Value Probl., (2014), 198–207.
doi: 10.1186/s13661-014-0198-z. |
| [7] |
J. Iaia,
Existence and nonexistence for semilinear equations on exterior domains, J. Partial Differ. Equ., 30 (2017), 1-17.
|
| [8] |
J. Iaia,
Existence and nonexistence of solutions for sublinear equations on exterior domains, Electron. J. Differ. Equ., 181 (2018), 1-14.
|
| [9] |
J. Iaia,
Existence of solutions for semilinear problems with prescribed number of zeros on exterior domains, J. Math. Anal. Appl., 446 (2017), 591-604.
doi: 10.1016/j.jmaa.2016.08.063. |
| [10] |
C. K. R. T. Jones and T. Kupper,
On the infinitely many solutions of a semi-linear equation, SIAM J. Math. Anal., 17 (1986), 803-835.
doi: 10.1137/0517059. |
| [11] |
J. Joshi,
Existence and nonexistence of solutions of sublinear problems with prescribed number of zeros on exterior domains, Electron. J. Differ. Equ., 133 (2017), 1-10.
|
| [12] |
E. K. Lee, R. Shivaji and B. Son,
Positive radial solutions to classes of singular problems on the exterior of a ball, J. Math. Anal. Appl., 434 (2016), 1597-1611.
doi: 10.1016/j.jmaa.2015.09.072. |
| [13] |
E. Lee, L. Sankar and R. Shivaji,
Positive solutions for infinite semipositone problems on exterior domains, Differ. Integral Equ., 24 (2011), 861-875.
|
| [14] |
K. McLeod, W. C. Troy and F. B. Weissler,
Radial solutions of $\Delta u + f(u) = 0$ with prescribed numbers of zeros, J. Differ. Equ., 83 (1990), 368-373.
doi: 10.1016/0022-0396(90)90063-U. |
| [15] |
L. Sankar, S. Sasi and R. Shivaji,
Semipositone problems with falling zeros on exterior domains, J. Math. Anal. Appl., 401 (2013), 146-153.
doi: 10.1016/j.jmaa.2012.11.031. |
| [16] |
W. Strauss,
Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.
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