We investigate the first eigenvalue of a highly nonlinear class of elliptic operators which includes the p –Laplace, the pseudo–p–Laplace and others. We derive the positivity of the first eigenfunction, simplicity of the first eigenvalue, Faber-Krahn and Payne-Rayner type inequalities. In another chapter we address the question of symmetry for positive solutions to more general equations. Using a Pohozaev-type inequality and isoperimetric inequalities as well as convex rearrangement methods we generalize a symmetry result of Kesavan and Pacella. Our optimal domains are level sets of a convex function H-0 . They have the so-called Wulff shape associated with H and only in special cases they are Euclidean balls.
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