On the existence and nonexistence of global solutions for certain semilinear exterior problems with nontrivial Robin boundary conditions
We consider three types of semilinear equations (elliptic, parabolic and hyperbolic) posed in the N-dimensional exterior domain RN D, where N ≥ 2 and D is the closed unit ball in RN. A nontrivial Robin boundary condition is imposed on the boundary of D. Using a test function approach with judicious choices of the test functions, we show that the considered problems share a common critical behavior. We discuss separately the cases N = 2 and N ≥ 3. Moreover, in the case N ≥ 3, the dependence of the critical exponent on initial data is discussed. To the best our knowledge, the study of the critical behavior in an exterior domain with a nontrivial Robin boundary condition has never been studied in the literature.