Neumann problems for nonlinear elliptic equations with lower order terms

Abstract

In the present paper we prove existence results for solutions to nonlinear elliptic Neumann problems whose prototype is $\left(\begin{array}{cc}\lambda {\left|u\right|}^{p-2}u-{\Delta }_{p}u-div\left(c\left(x\right){\left|u\right|}^{p-2}u\right)+b\left(x\right){|\nabla u|}^{p-2}\nabla u=f& \text{in}\Omega ,\\ \left({|\nabla u|}^{p-2}\nabla u+c\left(x\right){\left|u\right|}^{p-2}u\right)\cdot \underline{n}=0& \text{on}\partial \Omega \end{array}\right)$where $\Omega$ is a bounded domain of $R^N$, $N\ge 2$, with Lipschitz boundary, $1<p<N$ , $\underline{n}$ is the outer unit normal to $\partial \Omega$, $\lambda >0$, the datum $f$ belongs to the dual space of $W^{1,p}\left(\Omega \right)$ or to Lebesgue space $L^1\left(\Omega \right)$. Finally the coefficients $b$, $c$ belong to appropriate Lebesgue spaces or Lorentz spaces.

Existence results for weak solutions or renormalized solutions are proved under smallness assumptions on the coefficients $b$ and $c$.