Critical double phase problems involving sandwich-type nonlinearities
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908528334077952 |
|---|---|
| author | Farkas, Csaba Fiscella, Alessio Ho, Ky Winkert, Patrick |
| author_facet | Farkas, Csaba Fiscella, Alessio Ho, Ky Winkert, Patrick |
| contents | In this paper we study problems with critical and sandwich-type growth represented by
\begin{align*}
-\operatorname{div}\Big(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u\Big)= λw(x)|u|^{s-2}u+θB\left(x,u\right) \quad \text{in } Ω,\quad
u= 0 \quad\text{on } \partial Ω,
\end{align*}
where $Ω\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partialΩ$, $1<p<s<q<N$, $\frac{q}{p}<1+\frac{1}{N}$, $0\leq a(\cdot)\in C^{0,1}(\overlineΩ)$, $λ$, $θ$ are real parameters, $w$ is a suitable weight and $B\colon \overlineΩ\times \mathbb{R}\to\mathbb{R}$ is given by
\begin{align*}
B(x,t) :=b_0(x)|t|^{p^*-2}t+b(x)|t|^{q^*-2}t,
\end{align*}
where $r^*:=Nr/(N-r)$ for $r\in\{p,q\}$. Here the right-hand side combines the effect of a critical term given by $B(\cdot,\cdot)$ and a sandwich-type perturbation with exponent $s \in (p,q)$. Under different values of the parameters $λ$ and $θ$, we prove the existence and multiplicity of solutions to the problem above. For this, we mainly exploit different variational methods combined with topological tools, like a new concentration-compactness principle, a suitable truncation argument and the Krasnoselskii's genus theory, by considering very mild assumptions on the data $a(\cdot)$, $b_0(\cdot)$ and $b(\cdot)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22371 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Critical double phase problems involving sandwich-type nonlinearities Farkas, Csaba Fiscella, Alessio Ho, Ky Winkert, Patrick Analysis of PDEs In this paper we study problems with critical and sandwich-type growth represented by \begin{align*} -\operatorname{div}\Big(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u\Big)= λw(x)|u|^{s-2}u+θB\left(x,u\right) \quad \text{in } Ω,\quad u= 0 \quad\text{on } \partial Ω, \end{align*} where $Ω\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partialΩ$, $1<p<s<q<N$, $\frac{q}{p}<1+\frac{1}{N}$, $0\leq a(\cdot)\in C^{0,1}(\overlineΩ)$, $λ$, $θ$ are real parameters, $w$ is a suitable weight and $B\colon \overlineΩ\times \mathbb{R}\to\mathbb{R}$ is given by \begin{align*} B(x,t) :=b_0(x)|t|^{p^*-2}t+b(x)|t|^{q^*-2}t, \end{align*} where $r^*:=Nr/(N-r)$ for $r\in\{p,q\}$. Here the right-hand side combines the effect of a critical term given by $B(\cdot,\cdot)$ and a sandwich-type perturbation with exponent $s \in (p,q)$. Under different values of the parameters $λ$ and $θ$, we prove the existence and multiplicity of solutions to the problem above. For this, we mainly exploit different variational methods combined with topological tools, like a new concentration-compactness principle, a suitable truncation argument and the Krasnoselskii's genus theory, by considering very mild assumptions on the data $a(\cdot)$, $b_0(\cdot)$ and $b(\cdot)$. |
| title | Critical double phase problems involving sandwich-type nonlinearities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.22371 |