Nodal solutions for the double phase problems

Abstract

We consider a parametric nonautonomous $(p, q)$-equation with unbalanced growth as follows \begin{align*} \left\{ \begin{aligned} &-\Delta_p^\alpha u(z)-\Delta_q u(z)=\lambda \vert u(z)\vert^{\tau-2}u(z)+f(z, u(z)), \quad \quad \hbox{in }\Omega,\\ &u|_{\partial \Omega}=0, \end{aligned} \right. \end{align*} where $\Omega \subseteq \mathbb{R}^N$ be a bounded domain with Lispchitz boundary $\partial\Omega$, $\alpha \in L^{\infty}(\Omega)\backslash \{0\}$, $a(z)\geq 0$ for a.e. $z \in \Omega$, $ 1<\tau< q<p<N$ and $\lambda>0$. In the reaction there is a parametric concave term and a perturbation $f(z, x)$. Under the minimal conditions on $f(z, 0)$, which essentially restrict its growth near zero, by employing variational tools, truncation and comparison techniques, as well as critical groups, we prove that for all small values of the parameter $\lambda>0$, the problem has at least three nontrivial bounded solutions (positive, negative, nodal), which are ordered and asymptotically vanish as $\lambda \rightarrow 0^{+}$.