We shall consider the following semilinear problem with a Neumann boundary condition
−∆u + u = a(x)u^{p−2}u− b(x)u^{q−2}u, x ∈ B_{1}, 

where B_{1} is the unit ball in \mathbbR^{N}, N ≥ 2, a, b are nonnegative 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 supercritical Sobolev nonlinearities.
Indeed, we first find a critical point of the EulerLagrange functional associated with the equation, i.e.
E(u): = 
1
2

 ⌠ ⌡

B_{1}

(∇u^{2}+ u^{2} ) dx+ 
1
q

 ⌠ ⌡

B_{1}

b(x) u^{q} dx− 
1
p

 ⌠ ⌡

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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