Our objective in this thesis is to study semilinear elliptic partial differential equations when the nonlinearity may have supercritical Sobolev growth. We shall apply a new variational principle recently introduced in [25, 26] to prove the existence of solutions for such problems. We would like to emphasize that functionals that we are using in the new variational principle are different from the standard Euler-Lagrange functionals that are mostly used in the literature. The results in this thesis are twofold. Namely, we first prove the existence of a solution for the semilinear elliptic equations in the presence of a subsolution and a supersolution. Secondly, we consider the general case, and discuss the existence and smoothness of its solutions in a supercritcal case. We remark that the latter results are recently published in [9].