Quasilinear elliptic problems via nonlinear Rayleigh quotient
Fuente:
arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909331781320704 |
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| author | Silva, Edcarlos D. Carvalho, Marcos L. M. Gasinski, Leszek Júnior, João R. Santos |
| author_facet | Silva, Edcarlos D. Carvalho, Marcos L. M. Gasinski, Leszek Júnior, João R. Santos |
| contents | It is established existence and multiplicity of solution for the following class of quasilinear elliptic problems
$$
\left\{
\begin{array}{lr}
-Δ_Φu = λa(x) |u|^{q-2}u + |u|^{p-2}u, & x\inΩ,
u = 0, & x \in \partial Ω,
\end{array}
\right.
$$
where $Ω\subset \mathbb{R}^N, N \geq 2,$ is a smooth bounded domain, $1 < q < \ell \leq m < p < \ell^*$ and $Φ: \mathbb{R} \to \mathbb{R}$ is suitable $N$-function. The main feature here is to show whether the Nehari method can be applied to find the largest positive number $λ^* > 0$ in such way that our main problem admits at least two distinct solutions for each $λ\in (0, λ^*)$. Furthermore, using some fine estimates and some extra assumptions on $Φ$, we prove the existence of at least two positive solutions for $λ= λ^*$ and $λ\in (λ^*, \overlineλ)$ where $\overlineλ > λ^*$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00861 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quasilinear elliptic problems via nonlinear Rayleigh quotient Silva, Edcarlos D. Carvalho, Marcos L. M. Gasinski, Leszek Júnior, João R. Santos Analysis of PDEs It is established existence and multiplicity of solution for the following class of quasilinear elliptic problems $$ \left\{ \begin{array}{lr} -Δ_Φu = λa(x) |u|^{q-2}u + |u|^{p-2}u, & x\inΩ, u = 0, & x \in \partial Ω, \end{array} \right. $$ where $Ω\subset \mathbb{R}^N, N \geq 2,$ is a smooth bounded domain, $1 < q < \ell \leq m < p < \ell^*$ and $Φ: \mathbb{R} \to \mathbb{R}$ is suitable $N$-function. The main feature here is to show whether the Nehari method can be applied to find the largest positive number $λ^* > 0$ in such way that our main problem admits at least two distinct solutions for each $λ\in (0, λ^*)$. Furthermore, using some fine estimates and some extra assumptions on $Φ$, we prove the existence of at least two positive solutions for $λ= λ^*$ and $λ\in (λ^*, \overlineλ)$ where $\overlineλ > λ^*$. |
| title | Quasilinear elliptic problems via nonlinear Rayleigh quotient |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.00861 |