Four positive solutions for a semilinear elliptic equation involving concave and convex nonlinearities

In this paper, we study the combined effect of concave and convex nonlinearities on the number of positive solutions for a semilinear elliptic equation. With the help of the Nehari manifold and the center mass function, we prove that there are at least four positive solutions for a semilinear elliptic equation in a finite strip with a hole.