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} &-Δ_p^αu(z)-Δ_q u(z)=λ\vert u(z)\vert^{τ-2}u(z)+f(z, u(z)), \quad \quad \hbox{in }Ω,\\ &u|_{\partial Ω}=0, \end{aligned} \right. \end{align*} where $Ω\subseteq \mathbb{R}^N$ be a bounded domain with Lispchitz boundary $\partialΩ$, $α\in L^{\infty}(Ω)\backslash \{0\}$, $a(z)\geq 0$ for a.e. $z \in Ω$, $ 1<τ< q 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 $λ>0$, the problem has at least three nontrivial bounded solutions (positive, negative, nodal), which are ordered and asymptotically vanish as $λ\rightarrow 0^{+}$.