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A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems
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| Zeitschriftentitel: | Advances in Nonlinear Analysis |
|---|---|
| Personen und Körperschaften: | |
| In: | Advances in Nonlinear Analysis, 9, 2020, 1, S. 1351-1382 |
| Format: | E-Article |
| Sprache: | Unbestimmt |
| veröffentlicht: |
Walter de Gruyter GmbH
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| Schlagwörter: |
| author_facet |
López-Martínez, Salvador López-Martínez, Salvador |
|---|---|
| author |
López-Martínez, Salvador |
| spellingShingle |
López-Martínez, Salvador Advances in Nonlinear Analysis A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems Analysis |
| author_sort |
lópez-martínez, salvador |
| spelling |
López-Martínez, Salvador 2191-950X 2191-9496 Walter de Gruyter GmbH Analysis http://dx.doi.org/10.1515/anona-2020-0056 <jats:title>Abstract</jats:title> <jats:p>In this paper we deal with the elliptic problem</jats:p> <jats:p><jats:disp-formula id="j_anona-2020-0056_eq_001"><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_anona-2020-0056_eq_001.png" position="float" orientation="portrait" /> <jats:tex-math>$$\begin{array}{} \begin{cases} \displaystyle-{\it\Delta} u=\lambda u+\mu(x)\frac{|\nabla u|^q}{u^\alpha}+f(x)\quad &\text{ in }{\it\Omega}, \\ u \gt 0 \quad &\text{ in }{\it\Omega}, \\ u=0\quad &\text{ on }\partial{\it\Omega}, \end{cases} \end{array} $$</jats:tex-math></jats:alternatives></jats:disp-formula></jats:p> <jats:p>where <jats:italic>Ω</jats:italic> ⊂ ℝ<jats:sup><jats:italic>N</jats:italic></jats:sup> is a bounded smooth domain, 0 ≨ <jats:italic>μ</jats:italic> ∈ <jats:italic>L</jats:italic><jats:sup>∞</jats:sup>(<jats:italic>Ω</jats:italic>), 0 ≨ <jats:italic>f</jats:italic> ∈ <jats:italic>L</jats:italic><jats:sup><jats:italic>p</jats:italic><jats:sub>0</jats:sub></jats:sup>(<jats:italic>Ω</jats:italic>) for some <jats:italic>p</jats:italic><jats:sub>0</jats:sub> > <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_anona-2020-0056_eq_002.png" /> <jats:tex-math>$\begin{array}{} \frac{N}{2} \end{array}$</jats:tex-math></jats:alternatives></jats:inline-formula>, 1 < <jats:italic>q</jats:italic> < 2, <jats:italic>α</jats:italic> ∈ [0 1] and <jats:italic>λ</jats:italic> ∈ ℝ. We establish existence and multiplicity results for <jats:italic>λ</jats:italic> > 0 and <jats:italic>α</jats:italic> < <jats:italic>q</jats:italic> – 1, including the non-singular case <jats:italic>α</jats:italic> = 0. In contrast, we also derive existence and uniqueness results for <jats:italic>λ</jats:italic> > 0 and <jats:italic>q</jats:italic> – 1 < <jats:italic>α</jats:italic> ≤ 1. We thus complement the results in [1, 2], which are concerned with <jats:italic>α</jats:italic> = <jats:italic>q</jats:italic> – 1, and show that the value <jats:italic>α</jats:italic> = <jats:italic>q</jats:italic> – 1 plays the role of a break point for the multiplicity/uniqueness of solution.</jats:p> A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems Advances in Nonlinear Analysis |
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10.1515/anona-2020-0056 |
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Online Free |
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Mathematik |
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ElectronicArticle |
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DE-Gla1 DE-Zi4 DE-15 DE-Pl11 DE-Rs1 DE-105 DE-14 DE-Ch1 DE-L229 DE-D275 DE-Bn3 DE-Brt1 DE-Zwi2 DE-D161 |
| imprint |
Walter de Gruyter GmbH, 2020 |
| imprint_str_mv |
Walter de Gruyter GmbH, 2020 |
| issn |
2191-950X 2191-9496 |
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2191-950X 2191-9496 |
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Walter de Gruyter GmbH (CrossRef) |
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lopezmartinez2020asingularityasabreakpointforthemultiplicityofsolutionstoquasilinearellipticproblems |
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2020 |
| publisher |
Walter de Gruyter GmbH |
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ai |
| record_format |
ai |
| series |
Advances in Nonlinear Analysis |
| source_id |
49 |
| title |
A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems |
| title_unstemmed |
A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems |
| title_full |
A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems |
| title_fullStr |
A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems |
| title_full_unstemmed |
A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems |
| title_short |
A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems |
| title_sort |
a singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems |
| topic |
Analysis |
| url |
http://dx.doi.org/10.1515/anona-2020-0056 |
| publishDate |
2020 |
| physical |
1351-1382 |
| description |
<jats:title>Abstract</jats:title>
<jats:p>In this paper we deal with the elliptic problem</jats:p>
<jats:p><jats:disp-formula id="j_anona-2020-0056_eq_001"><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_anona-2020-0056_eq_001.png" position="float" orientation="portrait" />
<jats:tex-math>$$\begin{array}{}
\begin{cases}
\displaystyle-{\it\Delta} u=\lambda u+\mu(x)\frac{|\nabla u|^q}{u^\alpha}+f(x)\quad &\text{ in }{\it\Omega},
\\
u \gt 0 \quad &\text{ in }{\it\Omega},
\\
u=0\quad &\text{ on }\partial{\it\Omega},
\end{cases}
\end{array}
$$</jats:tex-math></jats:alternatives></jats:disp-formula></jats:p>
<jats:p>where <jats:italic>Ω</jats:italic> ⊂ ℝ<jats:sup><jats:italic>N</jats:italic></jats:sup> is a bounded smooth domain, 0 ≨ <jats:italic>μ</jats:italic> ∈ <jats:italic>L</jats:italic><jats:sup>∞</jats:sup>(<jats:italic>Ω</jats:italic>), 0 ≨ <jats:italic>f</jats:italic> ∈ <jats:italic>L</jats:italic><jats:sup><jats:italic>p</jats:italic><jats:sub>0</jats:sub></jats:sup>(<jats:italic>Ω</jats:italic>) for some <jats:italic>p</jats:italic><jats:sub>0</jats:sub> > <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_anona-2020-0056_eq_002.png" />
<jats:tex-math>$\begin{array}{}
\frac{N}{2}
\end{array}$</jats:tex-math></jats:alternatives></jats:inline-formula>, 1 < <jats:italic>q</jats:italic> < 2, <jats:italic>α</jats:italic> ∈ [0 1] and <jats:italic>λ</jats:italic> ∈ ℝ. We establish existence and multiplicity results for <jats:italic>λ</jats:italic> > 0 and <jats:italic>α</jats:italic> < <jats:italic>q</jats:italic> – 1, including the non-singular case <jats:italic>α</jats:italic> = 0. In contrast, we also derive existence and uniqueness results for <jats:italic>λ</jats:italic> > 0 and <jats:italic>q</jats:italic> – 1 < <jats:italic>α</jats:italic> ≤ 1. We thus complement the results in [1, 2], which are concerned with <jats:italic>α</jats:italic> = <jats:italic>q</jats:italic> – 1, and show that the value <jats:italic>α</jats:italic> = <jats:italic>q</jats:italic> – 1 plays the role of a break point for the multiplicity/uniqueness of solution.</jats:p> |
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| author | López-Martínez, Salvador |
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| description | <jats:title>Abstract</jats:title> <jats:p>In this paper we deal with the elliptic problem</jats:p> <jats:p><jats:disp-formula id="j_anona-2020-0056_eq_001"><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_anona-2020-0056_eq_001.png" position="float" orientation="portrait" /> <jats:tex-math>$$\begin{array}{} \begin{cases} \displaystyle-{\it\Delta} u=\lambda u+\mu(x)\frac{|\nabla u|^q}{u^\alpha}+f(x)\quad &\text{ in }{\it\Omega}, \\ u \gt 0 \quad &\text{ in }{\it\Omega}, \\ u=0\quad &\text{ on }\partial{\it\Omega}, \end{cases} \end{array} $$</jats:tex-math></jats:alternatives></jats:disp-formula></jats:p> <jats:p>where <jats:italic>Ω</jats:italic> ⊂ ℝ<jats:sup><jats:italic>N</jats:italic></jats:sup> is a bounded smooth domain, 0 ≨ <jats:italic>μ</jats:italic> ∈ <jats:italic>L</jats:italic><jats:sup>∞</jats:sup>(<jats:italic>Ω</jats:italic>), 0 ≨ <jats:italic>f</jats:italic> ∈ <jats:italic>L</jats:italic><jats:sup><jats:italic>p</jats:italic><jats:sub>0</jats:sub></jats:sup>(<jats:italic>Ω</jats:italic>) for some <jats:italic>p</jats:italic><jats:sub>0</jats:sub> > <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_anona-2020-0056_eq_002.png" /> <jats:tex-math>$\begin{array}{} \frac{N}{2} \end{array}$</jats:tex-math></jats:alternatives></jats:inline-formula>, 1 < <jats:italic>q</jats:italic> < 2, <jats:italic>α</jats:italic> ∈ [0 1] and <jats:italic>λ</jats:italic> ∈ ℝ. We establish existence and multiplicity results for <jats:italic>λ</jats:italic> > 0 and <jats:italic>α</jats:italic> < <jats:italic>q</jats:italic> – 1, including the non-singular case <jats:italic>α</jats:italic> = 0. In contrast, we also derive existence and uniqueness results for <jats:italic>λ</jats:italic> > 0 and <jats:italic>q</jats:italic> – 1 < <jats:italic>α</jats:italic> ≤ 1. We thus complement the results in [1, 2], which are concerned with <jats:italic>α</jats:italic> = <jats:italic>q</jats:italic> – 1, and show that the value <jats:italic>α</jats:italic> = <jats:italic>q</jats:italic> – 1 plays the role of a break point for the multiplicity/uniqueness of solution.</jats:p> |
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| spelling | López-Martínez, Salvador 2191-950X 2191-9496 Walter de Gruyter GmbH Analysis http://dx.doi.org/10.1515/anona-2020-0056 <jats:title>Abstract</jats:title> <jats:p>In this paper we deal with the elliptic problem</jats:p> <jats:p><jats:disp-formula id="j_anona-2020-0056_eq_001"><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_anona-2020-0056_eq_001.png" position="float" orientation="portrait" /> <jats:tex-math>$$\begin{array}{} \begin{cases} \displaystyle-{\it\Delta} u=\lambda u+\mu(x)\frac{|\nabla u|^q}{u^\alpha}+f(x)\quad &\text{ in }{\it\Omega}, \\ u \gt 0 \quad &\text{ in }{\it\Omega}, \\ u=0\quad &\text{ on }\partial{\it\Omega}, \end{cases} \end{array} $$</jats:tex-math></jats:alternatives></jats:disp-formula></jats:p> <jats:p>where <jats:italic>Ω</jats:italic> ⊂ ℝ<jats:sup><jats:italic>N</jats:italic></jats:sup> is a bounded smooth domain, 0 ≨ <jats:italic>μ</jats:italic> ∈ <jats:italic>L</jats:italic><jats:sup>∞</jats:sup>(<jats:italic>Ω</jats:italic>), 0 ≨ <jats:italic>f</jats:italic> ∈ <jats:italic>L</jats:italic><jats:sup><jats:italic>p</jats:italic><jats:sub>0</jats:sub></jats:sup>(<jats:italic>Ω</jats:italic>) for some <jats:italic>p</jats:italic><jats:sub>0</jats:sub> > <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_anona-2020-0056_eq_002.png" /> <jats:tex-math>$\begin{array}{} \frac{N}{2} \end{array}$</jats:tex-math></jats:alternatives></jats:inline-formula>, 1 < <jats:italic>q</jats:italic> < 2, <jats:italic>α</jats:italic> ∈ [0 1] and <jats:italic>λ</jats:italic> ∈ ℝ. We establish existence and multiplicity results for <jats:italic>λ</jats:italic> > 0 and <jats:italic>α</jats:italic> < <jats:italic>q</jats:italic> – 1, including the non-singular case <jats:italic>α</jats:italic> = 0. In contrast, we also derive existence and uniqueness results for <jats:italic>λ</jats:italic> > 0 and <jats:italic>q</jats:italic> – 1 < <jats:italic>α</jats:italic> ≤ 1. We thus complement the results in [1, 2], which are concerned with <jats:italic>α</jats:italic> = <jats:italic>q</jats:italic> – 1, and show that the value <jats:italic>α</jats:italic> = <jats:italic>q</jats:italic> – 1 plays the role of a break point for the multiplicity/uniqueness of solution.</jats:p> A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems Advances in Nonlinear Analysis |
| spellingShingle | López-Martínez, Salvador, Advances in Nonlinear Analysis, A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems, Analysis |
| title | A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems |
| title_full | A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems |
| title_fullStr | A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems |
| title_full_unstemmed | A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems |
| title_short | A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems |
| title_sort | a singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems |
| title_unstemmed | A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems |
| topic | Analysis |
| url | http://dx.doi.org/10.1515/anona-2020-0056 |