• Previous Article
    On special regularity properties of solutions of the benjamin-ono-zakharov-kuznetsov (bo-zk) equation
  • CPAA Home
  • This Issue
  • Next Article
    On nonexistence of extremals for the Trudinger-Moser functionals involving $ L^p $ norms
September  2020, 19(9): 4269-4284. doi: 10.3934/cpaa.2020193

Existence of infinitely many solutions for semilinear problems on exterior domains

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 $.

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

[14]

K. McLeodW. 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. SankarS. 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. CastroL. 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. LeeR. 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. LeeL. Sankar and R. Shivaji, Positive solutions for infinite semipositone problems on exterior domains, Differ. Integral Equ., 24 (2011), 861-875.   Google Scholar

[14]

K. McLeodW. 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. SankarS. 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

[1]

Dagny Butler, Eunkyung Ko, Eun Kyoung Lee, R. Shivaji. Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2713-2731. doi: 10.3934/cpaa.2014.13.2713

[2]

João Marcos do Ó, Sebastián Lorca, Justino Sánchez, Pedro Ubilla. Positive radial solutions for some quasilinear elliptic systems in exterior domains. Communications on Pure & Applied Analysis, 2006, 5 (3) : 571-581. doi: 10.3934/cpaa.2006.5.571

[3]

Xiaotao Huang, Lihe Wang. Radial symmetry results for Bessel potential integral equations in exterior domains and in annular domains. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1121-1134. doi: 10.3934/cpaa.2017054

[4]

Alireza Khatib, Liliane A. Maia. A positive bound state for an asymptotically linear or superlinear Schrödinger equation in exterior domains. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2789-2812. doi: 10.3934/cpaa.2018132

[5]

Sara Barile, Addolorata Salvatore. Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains. Conference Publications, 2013, 2013 (special) : 41-49. doi: 10.3934/proc.2013.2013.41

[6]

Francesca De Marchis, Isabella Ianni. Blow up of solutions of semilinear heat equations in non radial domains of $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 891-907. doi: 10.3934/dcds.2015.35.891

[7]

Paolo Maremonti. On the Stokes problem in exterior domains: The maximum modulus theorem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2135-2171. doi: 10.3934/dcds.2014.34.2135

[8]

Kai Yang. The focusing NLS on exterior domains in three dimensions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2269-2297. doi: 10.3934/cpaa.2017112

[9]

Lassaad Aloui, Moez Khenissi. Boundary stabilization of the wave and Schrödinger equations in exterior domains. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 919-934. doi: 10.3934/dcds.2010.27.919

[10]

Hongxia Zhang, Ying Wang. Liouville results for fully nonlinear integral elliptic equations in exterior domains. Communications on Pure & Applied Analysis, 2018, 17 (1) : 85-112. doi: 10.3934/cpaa.2018006

[11]

Dorina Mitrea, Marius Mitrea, Sylvie Monniaux. The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1295-1333. doi: 10.3934/cpaa.2008.7.1295

[12]

Riccardo Molle, Donato Passaseo. On the behaviour of the solutions for a class of nonlinear elliptic problems in exterior domains. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 445-454. doi: 10.3934/dcds.1998.4.445

[13]

Dagny Butler, Eunkyung Ko, R. Shivaji. Alternate steady states for classes of reaction diffusion models on exterior domains. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1181-1191. doi: 10.3934/dcdss.2014.7.1181

[14]

Marcio V. Ferreira, Gustavo Alberto Perla Menzala. Uniform stabilization of an electromagnetic-elasticity problem in exterior domains. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 719-746. doi: 10.3934/dcds.2007.18.719

[15]

Satoshi Hashimoto, Mitsuharu Ôtani. Existence of nontrivial solutions for some elliptic equations with supercritical nonlinearity in exterior domains. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 323-333. doi: 10.3934/dcds.2007.19.323

[16]

Matthias Hieber. Remarks on the theory of Oldroyd-B fluids in exterior domains. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1307-1313. doi: 10.3934/dcdss.2013.6.1307

[17]

Kazuhiro Ishige, Michinori Ishiwata. Global solutions for a semilinear heat equation in the exterior domain of a compact set. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 847-865. doi: 10.3934/dcds.2012.32.847

[18]

Trad Alotaibi, D. D. Hai, R. Shivaji. Existence and nonexistence of positive radial solutions for a class of $p$-Laplacian superlinear problems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4655-4666. doi: 10.3934/cpaa.2020131

[19]

Chia-Yu Hsieh. Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2657-2681. doi: 10.3934/dcdsb.2018269

[20]

Zhuoran Du. Some properties of positive radial solutions for some semilinear elliptic equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 943-953. doi: 10.3934/cpaa.2010.9.943

2019 Impact Factor: 1.105

Article outline

[Back to Top]