The Brezis–Oswald Result for Quasilinear Robin Problems

In this paper we consider a nonlinear elliptic problem driven by a nonhomogeneous differential operator with Robin boundary conditions. We produce conditions on the reaction term near <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mn>0</m:mn> <m:mo>+</m:mo> </m:msup> </m:math> 0⁺ and near <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>+</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> +∞ , which imply the existence and uniqueness of a positive solution. In the particular case of equations driven by the p -Laplacian, we show that these conditions are also necessary, extending in this way the semilinear Dirichlet work of Brezis and Oswald [5].