Solution to nonlinear gradient dependent systems with a balance law.
Using an approximation method, we show the existence of solutions for some noncooperative elliptic systems defined on an unbounded domain.
We study symmetry properties of least energy positive or nodal solutions of semilinear elliptic problems with Dirichlet or Neumann boundary conditions. The proof is based on symmetrizations in the spirit of Bartsch, Weth and Willem (J. Anal. Math., 2005) together with a strong maximum principle for quasi-continuous functions of Ancona and an intermediate value property for such functions.