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We shall consider the following semi-linear problem with a Neumann boundary condition
$$-\Delta u + u= a(|x|)|u|^{p-2}u- b(|x|)|u|^{q-2}u, \quad x\in B_1,$$
where $B_1$ is the unit ball in $\mathbb{R}^N$, $N\geq 2$, $a, b$ are non-negative radial functions, and
$p,q$ are distinct numbers greater than or equal to $2$. We shall assume no growth condition on $p$ and $q$. Our plan is to use a new variational principle established recently by one of the authors that allows one to deal with problem with super-critical Sobolev non-linearities.
Indeed, we first find a critical point of the Euler-Lagrange functional associated with the equation, i.e.
$$E(u):= \frac{1}{2}\int_{B_1} (|\nabla u|^2+ u^2 ) dx+ \frac{1}{q}\int_{B_1} b(|x|) u^q dx- \frac{1}{p}\int_{B_1} a(|x|)|u|^p dx,$$
over a suitable closed and convex subset of $H^1(B_1)$. Then we shall use this new variational principle to deduce that the restricted critical point of $E$ is an actual critical point.
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