Singular non-autonomous \((p, q)\)-equations with competing nonlinearities
Let $\Omega \subseteq \mathbb{R}^N$ be a bounded domain with a $C^2$-boundary $\partial \Omega$. In this paper the authors consider a singular Dirichlet problem of the form \N\[ \N\begin{cases}\N-\Delta_{p}^{a_1} u(z)-\Delta_{q}^{a_2} u(z)=\lambda [u(z)^{-\eta} +u(z)^{\tau-1}]+f(z,u(z)) \quad\mbox{in } \Omega, \\\Nu \Big|_{\partial \Omega} =0, \, 0< \eta <1, \, 1<\tau<q<p, \, \lambda>0, \, u > 0.\N\end{cases}\N\]\N\NGiven $1<r <\infty$ and $a \in C^{0,1}(\overline{\Omega})$ with $a(z) \geq c>0$ for all $z \in \overline{\Omega}$, by $\Delta_{r}^a$ we denote the weighted $r$-Laplace differential operator defined by \N\[\N\Delta_{r}^a u= \mbox{div }(a(z)|\nabla u|^{r-2}\nabla u) \mbox{ for all $u \in W_0^{1,r}(\Omega)$.}\N\]\N\NThe perturbation $f(z,x)$ is a Caratheodory function (that is, for all $x \in \mathbb{R}$, $z \to f(z,x)$ is measurable and for a.a. $z \in \Omega$, $x \to f(z,x)$ is continuous) which exhibits $(p-1)$-superlinear growth as $x \to +\infty$, but need not satisfy the usual in such cases Ambrosetti-Rabinowitz condition and may change sign (indefinite perturbation). In the reaction, the authors consider also the effects of two parametric terms: the singular term $u \to \lambda u^{-\eta}$ and the concave term $u \to \lambda u^{\tau-1}$. Hence, the problem in the reaction has the combined effects of singular and concave-convex nonlinearities, however the authors consider also the case when $f(z,x)$ has $(p-1)$-linear growth as $x \to +\infty$. By using appropriate auxiliary problems, hence truncation and comparison arguments, the authors obtain a bifurcation type theorem establishing the existence/non-existence/multiplicity of positive solutions. In addition, the authors study the set of positive solutions $S_\lambda$, and prove the compactness of $S_\lambda$ in $C_0^1(\overline{\Omega})$. Finally, the authors prove suitable continuity properties of the solution multifunction $\lambda \to S_\lambda$.
scientific article; zbMATH DE number 7947282
scientific article