Electron. J. Differential Equations,
Vol. 2017 (2017), No. 213, pp. 119.
Existence of solutions to supercritical Neumann problems via a new
variational principle
Craig Cowan, Abbas Moameni, Leila Salimi
Abstract:
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 EulerLagrange 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.
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Craig Cowan
University of Manitoba
Winnipeg, Manitoba, Canada
email: Craig.Cowan@umanitoba.ca

Abbas Moameni
School of Mathematics and Statistics
Carleton University
Ottawa, Ontario, Canada
email: momeni@math.carleton.ca

Leila Salimi
Department of mathematics and computer sciences
Amirkabir University of Technology
Tehran, Iran
email: l_salimi@aut.ac.ir

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