Positive solutions for convective double phase problems (Q6616859)

In this paper the authors study a double phase problem with a parametric concave term and a convective perturbation of the form\N\begin{align*}\N\begin{cases} -\Delta_p^au(z)-\Delta_qu(z)=\lambda u(z)^{\tau-1}+f(z,u(z),Du(z)) &\text{in }\Omega,\\\Nu\big|_{\partial\Omega}=0, \ u>0, \ \lambda >0, \ 1<\tau<q<p<N, \end{cases}\N\end{align*}\Nwhere \(a \in L^\infty(\Omega)\), \(a(z)\geq 0\) for a.a.\,\(z\in\Omega\), \(\Delta_p^a\) denotes the weighted p-Laplace differential operator while \(f\colon\Omega\times\mathbb{R}\times\mathbb{R}^N\to\mathbb{R}\) is a Carathéodory function which is defined only locally. Using truncation and comparison techniques and the theory of nonlinear operators of monotone type, it is shown that for all \(\lambda>0\) sufficiently small, the problem above has a bounded positive solution.