Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Multiplicity of asymmetric solutions for nonlinear elliptic problems

Authors: Daomin Cao, Ezzat S. Noussair and Shusen Yan
Journal: Quart. Appl. Math. 64 (2006), 463-482
MSC (2000): Primary 35J65, 35B25; Secondary 35Q60
DOI: https://doi.org/10.1090/S0033-569X-06-01026-5
Published electronically: June 21, 2006
MathSciNet review: 2259049
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the existence of multiple asymmetric positive solutions for the following symmetric problem:

\begin{displaymath} \begin{cases} -\Delta u+(\lambda-h(x))u=(1-f(x))u^p, &\quad ... ...x\in\mathbb{R}^N,\\ \qquad u\in H^1(\mathbb{R}^N), \end{cases}\end{displaymath}

where $ \lambda>0$ is a parameter, $ h(x)$ and $ f(x)$ are nonnegative radially symmetric functions in $ L^\infty(\mathbb{R}^N)$, $ h(x)$ and $ f(x)$ have compact support in $ \mathbb{R}^N$, $ f(x)\leq1$ for all $ x\in\mathbb{R}^N$, $ 1<p<+\infty$ for $ N=1,2$, $ 1<p<\frac{N+2}{N-2}$ for $ N\geq3$. We prove that for any $ k=1,2,\,\ldots\,$, if $ \lambda$ is large enough the above problem has positive solutions $ u_\lambda$ concentrating at $ k$ distinct points away from the origin as $ \lambda$ goes to $ \infty$.

References [Enhancements On Off] (What's this?)

  • 1. R.V. Akhmanov, R.V. Khokhlou and A.P. Sukhornkov, Self-focusing, self-defocusing and self-modulation of laser beams, Laser Handbook, North-Holland, Amsterdam, 1972.
  • 2. N.N. Akhmediev, Novel class of nonlinear surface waves: Asymmetric modes in a symmetric layered structure, Sov. Phys. JEPT, 56 (1982), 231-247.
  • 3. A. Ambrosetti, D. Arcoya and J.L. Gámez, Asymmetric bound states of differential equations in nonlinear optics, Rend. Sem. Mat. Univ. Padova, 100 (1998), 231-247. MR 1675283 (99m:34103)
  • 4. D. Arcoya, S. Cingolani and J.L. Gámez, Asymmetric modes in symmetric nonlinear optical wave- guides, SIAM J. Math. Anal., 30 (1999), 1391-1400. MR 1718307 (2000j:78012)
  • 5. A. Bahri and Y.Y. Li, On a Min-max procedure for the existence of a positive solution for certain scalar field equations in $ {\mathbb{R}}^N$, Revista Math. Iber., 6 (1990), 1-15. MR 1086148 (92b:35054)
  • 6. A. Bahri and P.L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 14 (1997), 365-413. MR 1450954 (98k:35047)
  • 7. H. Berestycki and P.L. Lions, Nonlinear scalar field equations I. Existence of a ground state, Arch. Rat. Mech. Anal., 82 (1983), 313-345. MR 0695535 (84h:35054a)
  • 8. H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. MR 0709644 (84h:35059)
  • 9. D. Cao, E.N. Dancer, E.S. Noussair and S. Yan, On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems, Discrete and Continuous Dynamical Systems, 2(1996), 221-236. MR 1382508 (96m:35095)
  • 10. D. Cao and E.S. Noussair, Existence of symmetric multi-peaked solutions to singularly perturbed semilinear elliptic problems, Comm. PDE, 25 (2000), 2185-2232. MR 1789925 (2001h:35047)
  • 11. D. Cao, E.S. Noussair and S. Yan, Solutions with multiple `` peaks'' for nonlinear elliptic equations, Proc. Royal. Soc. Edinburgh, 129A (1999), 235-264. MR 1686700 (2000a:35052)
  • 12. D.Cao, E.S. Noussair and S. Yan, Existence and nonexistence of interior-peaked solutions for a nonlinear Neumann problem, Pacific J. Math., 200 (2001), 19-41. MR 1863405 (2002i:35065)
  • 13. S. Cingolani and J.L. Gámez, Asymmetric positive solutions for a symmetric nonlinear problem in $ {\mathbb{R}}^n$, Calc. Var. PDE, 11 (2000), 97-117. MR 1777465 (2001f:35121)
  • 14. S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Diff. Equations, 160 (2000), 118-138. MR 1734531 (2000j:35079)
  • 15. V. Coti Zelati and M.J. Esteban, Symmetry breaking and multiple solutions for a Neumann problem in an exterior domain, Proc. Royal Soc. Edinburgh, 116A (1990), 327-339. MR 1084737 (91j:35104)
  • 16. E.N. Dancer and S. Yan, Singularly perturbed elliptic problems in exterior domains, Diff. Integral Equations, 13 (2000), pp. 747-777. MR 1750049 (2001c:35022)
  • 17. M.J. Esteban, Nonsymmetric ground states of symmetric variational problems, Comm. Pure. Appl. Math., 44 (1991), 259-274. MR 1085830 (91m:35073)
  • 18. O. John and C. Stuart, Guidance properties of a cylindrical defocusing waveguide, Comm. Math. Univ. Carolinae, 35 (1994), 653-673. MR 1321236 (95m:78023)
  • 19. M.K. Kwong, Uniqueness of positive solutions of $ \Delta u-u+u^p=0$ in $ {\mathbb{R}}^n$, Arch. Rat. Mech. Anal., 105 (1989), 243-266. MR 0969899 (90d:35015)
  • 20. Y.Y. Li, Existence of multiple solutions of semilinear equations in $ {\mathbb{R}}^n$, Progr. Nonlinear Diff. Equations, 4 (1990), 134-159.
  • 21. J.H. Marburgher, Self-focusing: Theory, Prog. Quant. Electr., 4 (1975), 35-100.
  • 22. E.S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. London Math. Soc., 62 (2000), 213-227. MR 1772182 (2001k:35098)
  • 23. O. Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52. MR 1040954 (91b:35012)
  • 24. Y.R. Shen, Self-focusing: Experimental, Prog. Quant. Electr., 4 (1975), 1-34.
  • 25. C. Stuart, Guidance properties of nonlinear planar waveguides, Arch. Rat. Mech. Anal., 125 (1993), 145-200. MR 1245069 (94j:78022)
  • 26. C. Stuart, Self-trapping of an electromagnetic field and bifurcation from the essential spectrum, Arch. Rat. Mech. Anal., 113 (1990), 65-96. MR 1079182 (91j:78010)
  • 27. X. Wang and B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal., 28 (1997), 633-655. MR 1443612 (98e:81032)
  • 28. Z.-Q. Wang, Existence and symmetry of multi-bump solutions for nonlinear Schrödinger equations, J. Diff. Equations, 159 (1999), 102-137. MR 1726920 (2001h:35176)

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Additional Information

Daomin Cao
Affiliation: Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China
Email: dmcao@amt.ac.cn

Ezzat S. Noussair
Affiliation: School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia
Email: noussair@maths.unsw.edu.au

Shusen Yan
Affiliation: School of Mathematics, Statistics and Computer Science, The University of New England, Armidale, NSW 2351, Australia
Email: syan@turing.une.edu.au

DOI: https://doi.org/10.1090/S0033-569X-06-01026-5
Keywords: Nonlinear elliptic equation, asymmetric positive solutions, variational method, critical point
Received by editor(s): August 2, 2005
Published electronically: June 21, 2006
Additional Notes: The first author was supported by the Funds of Distinguished Young Scholars of NSFC and Innovation Funds of CAS in China
The second author was supported by ARC in Australia
The third author was supported by ARC in Australia
Article copyright: © Copyright 2006 Brown University

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