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 $\Omega \subseteq {\mathbb{R}}^N$ be a bounded domain with Lispchitz boundary $\partial \Omega$, $\alpha \in L^{\infty }(\Omega )\backslash \{0\}$, $a(z)\ge 0$ 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 $\lambda \to 0^{+}$.