\((p, q)\)-equations with singular and concave convex nonlinearities (Q2234296)

Let \(\Omega \subseteq \mathbb{R}^N\) be a bounded domain with a \(C^2\)-boundary \(\partial \Omega\). The authors study a parametric Dirichlet problem of the form \[(P_\lambda) \, \begin{cases} -\Delta_p u-\Delta_q u= \lambda[u^{-\eta}+ a(x) u^{\tau-1}] +f(x,u) \mbox{ in } \Omega,\\ u|_{\partial \Omega}=0, \quad u> 0 ,\quad \lambda>0, \quad \eta \in (0,1), \quad 1<\tau<q<p, \end{cases}\] where \(a \in L^\infty(\Omega)\) with \(a(x) \geq a_0> 0\) for a.a. \(x \in \Omega\), \(f:\Omega \times \mathbb{R} \to \mathbb{R}\) is a Carathéodory function satisfying useful properties, and \(\Delta_r u= \mathrm{div \, }(|\nabla u|^{r-2}\nabla u)\) for all \(u \in W_0^{1,r}(\Omega)\) is the classical \(r\)-Laplace operator. In this problem, \(s \to \lambda s^{-\eta}\) is a singular term, \(s \to \lambda s^{\tau-1}\) is a sublinear term, and \(s \to f(x,s)\) is a superlinear perturbation. Thus, the authors develop a variational approach to study positive solutions of \((P_\lambda)\). They prove a bifurcation-type result describing the dependence of the set of positive solutions as the parameter \(\lambda>0\) varies.