Positive solutions of the Robin problem for semi-linear elliptic equations on annuli

Let n ≥ 3 and ωR = {x Ε Rn; R < |x| < 1}. We consider the following Robin problem: (EQUATION PRESANT) where β is a positive parameter and v is the unit outward vector normal to &partωR. Under the assumptions (F1)-(F5) in the introduction, we prove that the above problem has at most one solution when β is small enough. In addition to (F1)-(F5), if (A1) in the introduction is satisfied, then the above problem has at least k nonradial solutions when β is large enough.