Positive solutions for concave-convex type problems for the one-dimensional $ϕ$-Laplacian
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917685291384832 |
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| author | Kaufmann, Uriel Milne, Leandro |
| author_facet | Kaufmann, Uriel Milne, Leandro |
| contents | Let $Ω=(a,b)\subset\mathbb{R}$, $0\leq m,n\in L^{1}(Ω)$, $λ,μ>0$ be real parameters, and $ϕ:\mathbb{R}\rightarrow\mathbb{R}$ be an odd increasing homeomorphism. In this paper we consider the existence of positive solutions for problems of the form \[ \begin{cases} -ϕ\left( u^{\prime}\right) ^{\prime}=λm(x)f(u)+μn(x)g(u) & \text{ in }Ω,\\ u=0 & \text{ on }\partialΩ, \end{cases} \]
where $f,g:[0,\infty)\rightarrow\lbrack0,\infty)$ are continuous functions which are, roughly speaking, sublinear and superlinear with respect to $ϕ$, respectively. Our assumptions on $ϕ$, $m$ and $n$ are substantially weaker than the ones imposed in previous works. The approach used here combines the Guo-Krasnoselski\uı\ fixed-point theorem and the sub-supersolutions method with some estimates on related nonlinear problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_12350 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Positive solutions for concave-convex type problems for the one-dimensional $ϕ$-Laplacian Kaufmann, Uriel Milne, Leandro Classical Analysis and ODEs 34B15, 34B18 Let $Ω=(a,b)\subset\mathbb{R}$, $0\leq m,n\in L^{1}(Ω)$, $λ,μ>0$ be real parameters, and $ϕ:\mathbb{R}\rightarrow\mathbb{R}$ be an odd increasing homeomorphism. In this paper we consider the existence of positive solutions for problems of the form \[ \begin{cases} -ϕ\left( u^{\prime}\right) ^{\prime}=λm(x)f(u)+μn(x)g(u) & \text{ in }Ω,\\ u=0 & \text{ on }\partialΩ, \end{cases} \] where $f,g:[0,\infty)\rightarrow\lbrack0,\infty)$ are continuous functions which are, roughly speaking, sublinear and superlinear with respect to $ϕ$, respectively. Our assumptions on $ϕ$, $m$ and $n$ are substantially weaker than the ones imposed in previous works. The approach used here combines the Guo-Krasnoselski\uı\ fixed-point theorem and the sub-supersolutions method with some estimates on related nonlinear problems. |
| title | Positive solutions for concave-convex type problems for the one-dimensional $ϕ$-Laplacian |
| topic | Classical Analysis and ODEs 34B15, 34B18 |
| url | https://arxiv.org/abs/2302.12350 |