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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2412.10812 |
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| _version_ | 1866912156453175296 |
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| author | Agudelo, Oscar Ruf, Bernhard Velez, Carlos |
| author_facet | Agudelo, Oscar Ruf, Bernhard Velez, Carlos |
| contents | We study the {\it Hamiltonian elliptic system} \begin{eqnarray}\label{HS1-abstract} \left\{ \begin{aligned} -Δu & = λ|v|^{r-1}v +|v|^{p-1}v \qquad &\hbox{in} \ \ Ω,\\ -Δv & = μ|u|^{s-1}u +|u|^{q-1}u \qquad &\hbox{in} \ \ Ω,\\ u &>0, \ v>0 \qquad \, &\hbox{in} \ \ Ω,\\ u &=v = 0 \qquad \quad &\hbox{on} \quad \partial Ω, \end{aligned} \right. \end{eqnarray} where $Ω\subset \mathbb {R}^N$ is a smooth bounded domain, $λ$ and $ μ$ are nonnegative parameters and $r,s,p,q>0$. Our study includes the case in which the nonlinearities in \eqref{HS1-abstract} are concave near the origin and convex near infinity, and we focus on the region of non-negative {\it pairs of parameters} \red{$(λ,μ)$} that guarantee exis\-tence and multiplicity of solutions of \eqref{HS1-abstract}. \red{In particular, we show the existence of a strictly decreasing curve $λ_*(μ)$ on an interval $[0, μ]$ with $λ_*(0)> 0, λ_*(μ) = 0$ and such that the system has two solutions for $(λ,μ)$ below the curve, one solution for $(λ, μ)$ on the curve and no solution for $(λ, μ)$ above the curve. A similar statement holds reversing $λ$ and $μ$.} This work is motivated by some of the results by Ambrosseti, BRezis and Cerami from 1993. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_10812 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multiplicity results for a subcritical Hamiltonian system with concave-convex nonlinearities Agudelo, Oscar Ruf, Bernhard Velez, Carlos Analysis of PDEs 34A34, 34B08, 34B18, 35J35 We study the {\it Hamiltonian elliptic system} \begin{eqnarray}\label{HS1-abstract} \left\{ \begin{aligned} -Δu & = λ|v|^{r-1}v +|v|^{p-1}v \qquad &\hbox{in} \ \ Ω,\\ -Δv & = μ|u|^{s-1}u +|u|^{q-1}u \qquad &\hbox{in} \ \ Ω,\\ u &>0, \ v>0 \qquad \, &\hbox{in} \ \ Ω,\\ u &=v = 0 \qquad \quad &\hbox{on} \quad \partial Ω, \end{aligned} \right. \end{eqnarray} where $Ω\subset \mathbb {R}^N$ is a smooth bounded domain, $λ$ and $ μ$ are nonnegative parameters and $r,s,p,q>0$. Our study includes the case in which the nonlinearities in \eqref{HS1-abstract} are concave near the origin and convex near infinity, and we focus on the region of non-negative {\it pairs of parameters} \red{$(λ,μ)$} that guarantee exis\-tence and multiplicity of solutions of \eqref{HS1-abstract}. \red{In particular, we show the existence of a strictly decreasing curve $λ_*(μ)$ on an interval $[0, μ]$ with $λ_*(0)> 0, λ_*(μ) = 0$ and such that the system has two solutions for $(λ,μ)$ below the curve, one solution for $(λ, μ)$ on the curve and no solution for $(λ, μ)$ above the curve. A similar statement holds reversing $λ$ and $μ$.} This work is motivated by some of the results by Ambrosseti, BRezis and Cerami from 1993. |
| title | Multiplicity results for a subcritical Hamiltonian system with concave-convex nonlinearities |
| topic | Analysis of PDEs 34A34, 34B08, 34B18, 35J35 |
| url | https://arxiv.org/abs/2412.10812 |