Electron. J. Differential Equations, Vol. 2017 (2017), No. 213, pp. 1-19.

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
$$ -\Delta u + u= a(|x|)|u|^{p-2}u \quad \text{in } B_1, $$
with Neumann boundary conditions where $B_1$ is the unit ball in $\mathbb{R}^N$, a is nonnegative, radial and increasing and $p>2$. Note that for $N \ge 3$ this includes supercritical values of p. We find critical points of the functional
$$ I(u):= \frac{1}{q} \int_{B_1}a(|x|)^{1-q} |-\Delta u + u |^q \,dx - \frac{1}{p} \int_{B_1} a(|x|) |u|^p\,dx, $$
over the set of $ \{u\in H_{\rm rad}^1(B_1): 0\le u, u\text{ is increasing}\}$, 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.
$$ E(u):= \int_{B_1} \frac{| \nabla u|^2 + u^2}{2}\,dx -\int_{B_1} \frac{ a(|x|) |u|^p}{p} \,dx. $$
The novelty of using I instead of E is the hidden symmetry in I generated by $ \frac{1}{p} \int_{B_1} a(|x|) |u|^p\,dx $ 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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