A supercritical elliptic equation in the annulus (Q2691951)

Summary: By a combination of variational and topological techniques in the presence of invariant cones, we detect a new type of positive axially symmetric solutions of the Dirichlet problem for the elliptic equation \[ -\Delta u+u=a(x)|u|^{p-2}u \] in an annulus \(A\subset\mathbb{R}^N\) \((N\geq 3)\). Here \(p>2\) is allowed to be supercritical and \(a(x)\) is an axially symmetric but possibly nonradial function with additional symmetry and monotonicity properties, which are shared by the solution \(u\) we construct. In the case where \(a\) equals a positive constant, we detect conditions, only depending on the exponent \(p\) and on the inner radius of the annulus, that ensure that the solution is nonradial.