Positive solutions for double phase problems with combined nonlinearities (Q2114871)

Let \(\Omega \subseteq \mathbb{R}^N\) be a bounded domain with a Lipschitz boundary \(\partial \Omega\). The authors consider a weighted Dirichlet concave-convex problem of the form \[ \begin{cases} -\Delta_{p}^{a}u(z)- \Delta_{q}u(z)=E(z) u(z)^{\tau-1}+\lambda u(z)^{r-1} \mbox{ in } \Omega,\\ u|_{\partial \Omega}=0, \quad 1<\tau <q<p<r, \quad \lambda >0, \quad u \geq 0, \end{cases}\tag{\(P_\lambda\)} \] where \(\Delta_p^a u= \operatorname{div}(a(z)|D u|^{p-2}D u)\) for all \(u \in W^{1,p}_0(\Omega)\) is the weighted \(p\)-Laplace operator, with \(a \in C^{0,1}(\overline{\Omega})\) and \(a(z)\geq 0\) for all \(z \in \overline{\Omega}\), \(a \not \equiv 0\). For \(a=1\), we retrieve the classical \(p\)-Laplace operator. In the right-hand side, the first (concave) term is the \((p-1)\)-sublinear function \(x \to E(z)x^{\tau-1}\), \(x \geq 0\), with \(E \in L^\infty(\Omega)\setminus\{0\}\), \(E(z) \geq 0\) for a.a. \(z \in \Omega\) and the second (convex) term is the parametric \((p-1)\)-superlinear function \(x \to \lambda x^{r-1}\), \(x \geq 0\). Using the Nehari method, the authors prove that for all small values of \(\lambda >0\), the problem \((P_\lambda)\) admits at least two positive bounded solutions.