Global perturbative elliptic problems with critical growth in the fractional setting
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| Format: | Preprint |
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2024
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| author | Dipierro, Serena Lippi, Edoardo Proietti Valdinoci, Enrico |
| author_facet | Dipierro, Serena Lippi, Edoardo Proietti Valdinoci, Enrico |
| contents | Given $s$, $q\in(0,1)$, and a bounded and integrable function $h$ which is strictly positive in an open set, we show that there exist at least two nonnegative solutions $u$ of the critical problem $$(-Δ)^s u=\varepsilon h(x)u^q+u^{2^*_s-1},$$ as long as $\varepsilon>0$ is sufficiently small.
Also, if $h$ is nonnegative, these solutions are strictly positive.
The case $s=1$ was established in [APP00], which highlighted, in the classical case, the importance of combining perturbative techniques with variational methods: indeed, one of the two solutions branches off perturbatively in $\varepsilon$ from $u=0$, while the second solution is found by means of the Mountain Pass Theorem.
The case $s\in\left(0,\frac12\right]$ was already established, with different methods, in [DMV17] (actually, in [DMV17] it was erroneously believed that the method would have carried through all the fractional cases $s\in(0,1)$, so, in a sense, the results presented here correct and complete the ones in [DMV17]). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_19896 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global perturbative elliptic problems with critical growth in the fractional setting Dipierro, Serena Lippi, Edoardo Proietti Valdinoci, Enrico Analysis of PDEs Given $s$, $q\in(0,1)$, and a bounded and integrable function $h$ which is strictly positive in an open set, we show that there exist at least two nonnegative solutions $u$ of the critical problem $$(-Δ)^s u=\varepsilon h(x)u^q+u^{2^*_s-1},$$ as long as $\varepsilon>0$ is sufficiently small. Also, if $h$ is nonnegative, these solutions are strictly positive. The case $s=1$ was established in [APP00], which highlighted, in the classical case, the importance of combining perturbative techniques with variational methods: indeed, one of the two solutions branches off perturbatively in $\varepsilon$ from $u=0$, while the second solution is found by means of the Mountain Pass Theorem. The case $s\in\left(0,\frac12\right]$ was already established, with different methods, in [DMV17] (actually, in [DMV17] it was erroneously believed that the method would have carried through all the fractional cases $s\in(0,1)$, so, in a sense, the results presented here correct and complete the ones in [DMV17]). |
| title | Global perturbative elliptic problems with critical growth in the fractional setting |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.19896 |