High multiplicity of positive solutions in a superlinear problem of Moore-Nehari type
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arXiv
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| Natura: | Preprint |
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2024
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| author | Cubillos, Pablo López-Gómez, Julián Tellini, Andrea |
| author_facet | Cubillos, Pablo López-Gómez, Julián Tellini, Andrea |
| contents | In this paper we consider a superlinear one-dimensional elliptic boundary value problem that generalizes the one studied by Moore and Nehari in [43]. Specifically, we deal with piecewise-constant weight functions in front of the nonlinearity with an arbitrary number $κ\geq 1$ of vanishing regions. We study, from an analytic and numerical point of view, the number of positive solutions, depending on the value of a parameter $λ$ and on $κ$.
Our main results are twofold. On the one hand, we study analytically the behavior of the solutions, as $λ\downarrow-\infty$, in the regions where the weight vanishes. Our result leads us to conjecture the existence of $2^{κ+1}-1$ solutions for sufficiently negative $λ$. On the other hand, we support such a conjecture with the results of numerical simulations which also shed light on the structure of the global bifurcation diagrams in $λ$ and the profiles of positive solutions.
Finally, we give additional numerical results suggesting that the same high multiplicity result holds true for a much larger class of weights, also arbitrarily close to situations where there is uniqueness of positive solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_19084 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | High multiplicity of positive solutions in a superlinear problem of Moore-Nehari type Cubillos, Pablo López-Gómez, Julián Tellini, Andrea Analysis of PDEs Numerical Analysis Classical Analysis and ODEs 35J25, 34B08, 35J60, 65N06, 65P30 In this paper we consider a superlinear one-dimensional elliptic boundary value problem that generalizes the one studied by Moore and Nehari in [43]. Specifically, we deal with piecewise-constant weight functions in front of the nonlinearity with an arbitrary number $κ\geq 1$ of vanishing regions. We study, from an analytic and numerical point of view, the number of positive solutions, depending on the value of a parameter $λ$ and on $κ$. Our main results are twofold. On the one hand, we study analytically the behavior of the solutions, as $λ\downarrow-\infty$, in the regions where the weight vanishes. Our result leads us to conjecture the existence of $2^{κ+1}-1$ solutions for sufficiently negative $λ$. On the other hand, we support such a conjecture with the results of numerical simulations which also shed light on the structure of the global bifurcation diagrams in $λ$ and the profiles of positive solutions. Finally, we give additional numerical results suggesting that the same high multiplicity result holds true for a much larger class of weights, also arbitrarily close to situations where there is uniqueness of positive solutions. |
| title | High multiplicity of positive solutions in a superlinear problem of Moore-Nehari type |
| topic | Analysis of PDEs Numerical Analysis Classical Analysis and ODEs 35J25, 34B08, 35J60, 65N06, 65P30 |
| url | https://arxiv.org/abs/2402.19084 |