Positive solutions of the Robin problem for semilinear elliptic equations on annuli

Let n >= 3 and Omega(R) = {x is an element of R-n; R < vertical bar x vertical bar < 1}. We consider the following Robin problem: -Delta u = f(u), x is an element of Omega(R), u > 0, x is an element of Omega(R), partial derivative u/partial derivative v + beta u = 0, x is an element of Omega(R), where beta is a positive parameter and v is the unit outward vector normal to partial derivative Omega(R). Under the assumptions (F1)-(F5) in the introduction, we prove that the above problem has at most one solution when beta 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 beta is large enough.