Existence of positive solutions for kirchhoff-type problem in exterior domains

We consider the following Kirchhoff-type problem in an unbounded exterior domain Omega subset of R-3: { -(a+b integral(Omega) |del u|(2) dx) Delta u+lambda u = f(u), x is an element of Omega, (*) u = 0, x is an element of partial derivative Omega, where a >0, b >= 0, and lambda>0 are constants, partial derivative Omega not equal theta, R-3\Omega is bounded, u is an element of H-0(1)(Omega), and f is an element of C-1(R;R) is subcritical and superlinear near infinity. Under some mild conditions, we prove that if -Delta u + lambda u = f(u); x is an element of R-3 has only finite number of positive solutions in H-1(R-3) and the diameter of the hole R-3\Omega is small enough, then the problem (*) admits a positive solution. Same conclusion holds true if Omega is fixed and lambda>0 is small. To our best knowledge, there is no similar result published in the literature concerning the existence of positive solutions to the above Kirchhoff equation in exterior domains.