Least energy sign-changing solution for degenerate Kirchhoff double phase problems

Abstract

In this paper we study the following nonlocal Dirichlet equation of double phase type \begin{align*} -ψ\left [ \int_Ω\left ( \frac{|\nabla u |^p}{p} + μ(x) \frac{|\nabla u|^q}{q}\right)\,\mathrm{d} x\right] \mathcal{G}(u) = f(x,u)\quad \text{in } Ω, \quad u = 0\quad \text{on } \partialΩ, \end{align*} where $\mathcal{G}$ is the double phase operator given by \begin{align*} \mathcal{G}(u)=\operatorname{div} \left(|\nabla u|^{p-2}\nabla u + μ(x) |\nabla u|^{q-2}\nabla u \right)\quad u\in W^{1,\mathcal{H}}_0(Ω), \end{align*} $\Omega \subseteq {\mathbb{R}}^N$, $N\ge 2$, is a bounded domain with Lipschitz boundary $\partial \Omega$, $1<p<N$, $p<q<p^{\ast }=\frac{Np}{N-p}$, $0\le \mu (\cdot )\in L^{\infty }(\Omega )$, $\psi (s)=a_0+b_0{s}^{\vartheta -1}$ for $s\in \mathbb{R}$, with $a_0\ge 0$, $b_0>0$ and $\vartheta \ge 1$, and $f:\Omega \times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function that grows superlinearly and subcritically. We prove the existence of two constant sign solutions (one is positive, the other one negative) and of a sign-changing solution which turns out to be a least energy sign-changing solution of the problem above. Our proofs are based on variational tools in combination with the quantitative deformation lemma and the Poincaré-Miranda existence theorem.