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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 |
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| 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: |
| Zusammenfassung: | <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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| Umfang: | 1351-1382 |
| ISSN: |
2191-950X
2191-9496 |
| DOI: | 10.1515/anona-2020-0056 |