Generalized Polya-szegö Inequality and Applications to Some Quasi-linear Elliptic Problems

The Schwarz rearrangement of u is denoted here by u. Inequality (1.1) has numerous applications in physics. It was first used in 1945 by G. Polya and G. Szegö to prove that the capacity of a condenser diminishes or remains unchanged by applying the process of Schwarz symmetrization (see [30]). Inequality (1.1) was also the key ingredients to show that, among all bounded bodies with fixed measure, balls have the minimal capacity (see [26, Theorem 11.17]). Finally (1.1) has also played a crucial role in the solution of the famous Choquard’s conjecture (see [25]). It is heavily connected to the isoperimetric inequality and to Riesz-type rearrangement inequalities. Moreover, it turned out that (1.1) is extremely helpful in establishing the existence of ground states solutions of the nonlinear Schrödinger equation