Craig Cowan, Abbas Moameni, Leila Salimi
We use a new variational principle to obtain a positive solution of
with Neumann boundary conditions where is the unit ball in , a is nonnegative, radial and increasing and . Note that for this includes supercritical values of p. We find critical points of the functional
over the set of , where q is the conjugate of p. We would like to emphasize the energy functional I is different from the standard Euler-Lagrange functional associated with the above equation, i.e.
The novelty of using I instead of E is the hidden symmetry in I generated by and its Fenchel dual. Additionally we are able to prove the existence of a positive nonconstant solution, in the case a(|x|)=1, relatively easy and without needing to cut off the supercritical nonlinearity. Finally, we use this new approach to prove existence results for gradient systems with supercritical nonlinearities.
Submitted April 12, 2017. Published September 13, 2017.
Math Subject Classifications: 35J15, 58E30.
Key Words: Variational principles, supercritical, Neumann boundary condition.
Show me the PDF file (320 KB), TEX file for this article.
| Craig Cowan |
University of Manitoba
Winnipeg, Manitoba, Canada
| Abbas Moameni |
School of Mathematics and Statistics
Ottawa, Ontario, Canada
| Leila Salimi |
Department of mathematics and computer sciences
Amirkabir University of Technology
Return to the EJDE web page