A New Variational Principle and Supercritical Semilinear Elliptic Problems

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].