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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.20013 |
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| _version_ | 1866913456766058496 |
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| author | Crespo-Blanco, Ángel Gasiński, Leszek Winkert, Patrick |
| author_facet | Crespo-Blanco, Ángel Gasiński, Leszek Winkert, Patrick |
| contents | 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*}
$Ω\subseteq \mathbb{R}^N$, $N\geq 2$, is a bounded domain with Lipschitz boundary $\partialΩ$, $1<p<N$, $p<q<p^*=\frac{Np}{N-p}$, $0 \leq μ(\cdot)\in L^\infty(Ω)$, $ψ(s) = a_0 + b_0 s^{\vartheta-1}$ for $s\in\mathbb{R}$, with $a_0 \geq 0$, $b_0>0$ and $\vartheta \geq 1$, and $f\colonΩ\times\mathbb{R}\to\mathbb{R}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_20013 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Least energy sign-changing solution for degenerate Kirchhoff double phase problems Crespo-Blanco, Ángel Gasiński, Leszek Winkert, Patrick Analysis of PDEs 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*} $Ω\subseteq \mathbb{R}^N$, $N\geq 2$, is a bounded domain with Lipschitz boundary $\partialΩ$, $1<p<N$, $p<q<p^*=\frac{Np}{N-p}$, $0 \leq μ(\cdot)\in L^\infty(Ω)$, $ψ(s) = a_0 + b_0 s^{\vartheta-1}$ for $s\in\mathbb{R}$, with $a_0 \geq 0$, $b_0>0$ and $\vartheta \geq 1$, and $f\colonΩ\times\mathbb{R}\to\mathbb{R}$ 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. |
| title | Least energy sign-changing solution for degenerate Kirchhoff double phase problems |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2310.20013 |