Nonlinear nonhomogeneous singular problems (Q2278127)

The authors consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator with a growth of order $p-1$ near $+\infty$ and with a reaction in which there are the competing effects of a parametric singular term and a $(p-1)$-superlinear perturbation which does not satisfy the Ambrosetti-Rabinowitz condition. Employing variational tools combined with truncation and strong comparison techniques, the authors prove a bifurcation-type theorem that describes the set of positive solutions of the problem under consideration as the parameter $\lambda$ varies on the positive semiaxis. Furthermore, the authors show that for every $\lambda>0$ the problem admits a smallest positive solution $u_\lambda^*$ and they show the monotonicity and continuity properties of the map $\lambda\mapsto u_\lambda^*$. Precisely, assuming that $\Omega\subseteq \mathbb{R}^N$ is a bounded domain with a $C^2$-boundary $\partial \Omega$, the authors consider the following nonlinear, nonhomogeneous Dirichlet problem: \[ \begin{cases} -\mathrm{div}\, a(Du(z))=\lambda \vartheta (u(z))+f(z,u(z)) \text{ in }\Omega\\ u|_{\partial \Omega}=0 \end{cases},\tag{\(P_\lambda\)} \] where $\lambda>0$. In the above problem, the map $a:\mathbb{R}^N\to \mathbb{R}^N$ is strictly monotone and continuous. The hypotheses on $a$ and on the differential operator are general enough to include many well-known differential operators such as the $p$-Laplacian and the $(p,q)$-Laplacian. In the reaction of the problem, there are the competing effects of two nonlinearities: singular and superlinear. The first nonlinearity is the parametric term $\lambda \vartheta(x)$, being $\vartheta(\cdot)$ singular at $x=0$. The second term is the perturbation $f(z,x)$ which is a Carathéodory function. The authors assume that $f(z,\cdot)$ exhibits $(p-1)$-superlinear growth near $+\infty$ but they do not require that $f$ satisfies the Ambrosetti-Rabinowitz condition. It is worth pointing out that the authors employ a less restrictive condition which include in their analysis also nonlinearities with slower growth near $+\infty$. One of the main results of the paper is a bifurcation-type theorem which produces a critical parameter $\lambda^*>0$ such that for all $\lambda\in (0,\lambda^*)$ problem $(P_\lambda)$ admits at least two positive smooth solutions, for $\lambda=\lambda^*$ problem $(P_\lambda)$ admits at least one positive smooth solution, for all $\lambda>\lambda^*$ problem $(P_\lambda)$ admits no positive solutions. Moreover, for every $\lambda\in (0,\lambda^*]$ the authors show that problem $(P_\lambda)$ admits a smallest positive solution $u_\lambda^*$ and they investigate the monotonicity and continuity properties of the map $\lambda \mapsto u_\lambda^*$.