Dirichlet problems for the p -Laplacian with a convection term

We consider the nonlinear Dirichlet boundary value problem Open image in new window in a bounded domain \(\Omega \subset \mathbb {R}^N\) with smooth boundary \(\partial \Omega \), where \(\Delta _p u\mathop {=}\limits ^{\mathrm{{def}}}\mathrm {div} (|\nabla u|^{p-2} \nabla u)\) with \(1 0\)). When \(h\not \equiv 0\) and \(-\infty< \lambda < \lambda _1\), we prove that there exists a weak solution \(u\in W_0^{1,p}(\Omega )\) to problem (P); this solution is unique provided \(\lambda < 0\) (without any further assumptions). When \(h\ge 0\), \(h\not \equiv 0\), and \(0\le \lambda < \lambda _1\), we show that the solution is positive and also unique.