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September  2020, 19(9): 4269-4284. doi: 10.3934/cpaa.2020193

Existence of infinitely many solutions for semilinear problems on exterior domains

Joseph Iaia

University of North Texas, Denton, TX 76203-1430, USA

Received  September 2019 Revised  March 2020 Published  June 2020

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 $.

Keywords: Exterior domains, semilinear, superlinear, radial.
Mathematics Subject Classification: Primary: 34B40; Secondary: 35B05.
Citation: Joseph Iaia. Existence of infinitely many solutions for semilinear problems on exterior domains. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4269-4284. doi: 10.3934/cpaa.2020193
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.  Google Scholar

[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.  Google Scholar

[3]

M. Berger, Nonlinearity and Functional Analysis, Academic Press, New York, 1977.  Google Scholar

[4]

G. Birkhoff and G. C. Rota, Ordinary Differential Equations, Ginn and Company, 1962.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Iaia, Existence and nonexistence for semilinear equations on exterior domains, J. Partial Differ. Equ., 30 (2017), 1-17.   Google Scholar

[8]

J. Iaia, Existence and nonexistence of solutions for sublinear equations on exterior domains, Electron. J. Differ. Equ., 181 (2018), 1-14.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[13]

E. Lee, L. Sankar and R. Shivaji, Positive solutions for infinite semipositone problems on exterior domains, Differ. Integral Equ., 24 (2011), 861-875.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

W. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

M. Berger, Nonlinearity and Functional Analysis, Academic Press, New York, 1977.  Google Scholar

[4]

G. Birkhoff and G. C. Rota, Ordinary Differential Equations, Ginn and Company, 1962.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Iaia, Existence and nonexistence for semilinear equations on exterior domains, J. Partial Differ. Equ., 30 (2017), 1-17.   Google Scholar

[8]

J. Iaia, Existence and nonexistence of solutions for sublinear equations on exterior domains, Electron. J. Differ. Equ., 181 (2018), 1-14.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[13]

E. Lee, L. Sankar and R. Shivaji, Positive solutions for infinite semipositone problems on exterior domains, Differ. Integral Equ., 24 (2011), 861-875.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

W. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.   Google Scholar

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