Skip to main content Accesibility Help
×
×
Home

On the Solvability of a Neumann Boundary Value Problem at Resonance

  • Chung-Cheng Kuo (a1)
Abstract

We study the existence of solutions of the semilinear equations (1) in which the non-linearity g may grow superlinearly in u in one of directions u → ∞ and u → −∞, and (2) −Δu + g(x, u) = h, in which the nonlinear term g may grow superlinearly in u as |u| → ∞. The purpose of this paper is to obtain solvability theorems for (1) and (2) when the Landesman-Lazer condition does not hold. More precisely, we require that h may satisfy are arbitrarily nonnegative constants, . The proofs are based upon degree theoretic arguments.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      On the Solvability of a Neumann Boundary Value Problem at Resonance
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      On the Solvability of a Neumann Boundary Value Problem at Resonance
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      On the Solvability of a Neumann Boundary Value Problem at Resonance
      Available formats
      ×
Copyright
References
Hide All
1. Agmon, S., Douglis, A. and Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Comm. Pure Appl. Math 12 (1959), 623727.
2. Berestycki, H. and de Figueiredo, D. G., Double resonance in semilinear elliptic equations. Comm. Partial Differential Equations 6 (1981), 91120.
3. Gupta, C. P., Perturbations of second order elliptic problems by unbounded nonlinearities. Nonlinear Anal. 6 (1982), 919933.
4. Hirano, N., Nonlinear perturbations of second order linear elliptic boundary value problems. Houston J. Math. 14 (1988), 105114.
5. Iannacci, R., Nkashama, M. N. and Ward, J. R. Jr., Nonlinear second order elliptic partial differential equations at resonance. Trans. Amer.Math. Soc. 311 (1989), 711726.
6. Kannan, R. and Ortega, R., Landesman-Lazer conditions for problems with “one-side unbounded” nonlinearities. Nonlinear Anal. 9 (1985), 13131317.
7. Kuo, C.-C., On the solvability of a nonlinear second order elliptic equation at resonance. Proc.Amer.Math. Soc. 24 (1996), 8387.
8. Landesman, E. M. and Lazer, A. C., Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. Math. Mech. 19 (1970), 609623.
9. Lloyd, N. G., Degree theory. Cambridge University Press, New York, 1978.
10. McKenna, P. J. and Rauch, J., Strongly nonlinear perturbations of nonnegative boundary value problems with kernel. J. Differential Equations 28 (1978), 253265.
11. Robinson, S. B. and Landesman, E. M., A general approach to solvability conditions for semilinear elliptic boundary value problems at resonance. Differential Integral Equations 8 (1995), 15551569.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed