Positive solutions of logistic equations with dependence on gradient and nonhomogeneous Kirchhoff term (Q530310)

The paper is concerned with the existence of positive solutions for the following type elliptic equation: \[ -M(x,\|u\|)\Delta_p u = \lambda f(x, u,\nabla u) - g(x, u,\nabla u) \text{ in } \Omega,\;u|_{\partial \Omega} = 0, \;u \in W^{1,p}_0 (\Omega), \leqno{(P)} \] where \(\Omega \subset \mathbb{R}^N\) is a bounded smooth domain, \(\Delta_p u = \operatorname{div}(|\nabla u|^{p-2}\nabla u)\) with \(p\in (1,N)\) , \(\|\cdot \|\) is the norm of Sobolev space \(W^{1,p}_0 (\Omega)\) , \(\lambda\) is a real parameter, \(M \in C( \Omega \times \mathbb{R}^+ , \mathbb{R}^+)\), and \(f, g : \Omega \times \mathbb{R}^+ \times \mathbb{R}^N \rightarrow \mathbb{R}^+\) are Caratheodory functions. By reducing the equation (P) to a fixed point problem, and based on the fixed point index theory and the cone theoretic arguments, the authors prove that: (i) In the ``non-degenerate'' case of the function \(M\), the equation (P) has a positive solution if either \(\lambda > 0\) and \(f(x, s,t)\) is \((p - 1)\)-sublinear in \(s\), or \(\lambda >\lambda_1\) and \(f(x, s,t)\) is \((p -1)\)-linear in \(s\). (ii) In the case of ``degenerate'' of the function \(M\), the equation (P) has a positive solution for \(\lambda \in (\lambda_1, \lambda_2)\) and has no positive solution if \(\lambda > \lambda_2 + \sigma\) if \(f(x, s,t)\) is \((p - 1)\)-linear in \(s\), for some \(\sigma\). Here, \(\lambda_1, \lambda_2\) are the principal eigenvalues of some \(p-\)Laplacian problems with zero Dirichlet boundary data.