Multiplicity and asymptotic behavior of solutions to fractional (p,q)-Kirchhoff type problems with critical Sobolev-Hardy exponent

Multiplicity and asymptotic behavior of solutions to fractional (p,q)-Kirchhoff type problems with critical Sobolev-Hardy exponent

Let \(\Omega\subset\mathbb{R}^{N}\) be a bounded domain with smooth boundary and \(0\in\Omega\). For \(0

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Veröffentlicht in:Electronic journal of differential equations 2021-08, Vol.2021 (1-104), p.66
Hauptverfasser: Lin, Xiaolu, Zheng, Shenzhou
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description Let \(\Omega\subset\mathbb{R}^{N}\) be a bounded domain with smooth boundary and \(0\in\Omega\). For \(0
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For \(0<s<1\), \(1\le r<q<p\), \(0\le\alpha<ps<N\) and a positive parameter \(\lambda\), we consider the fractional \((p,q)\)-Laplacian problems involving a critical Sobolev-Hardy exponent. This model comes from a nonlocal problem of Kirchhoff type $$\displaylines{ \big(a+b[u]_{s,p}^{(\theta-1)p}\big)(-\Delta)_{p}^{s}u+(-\Delta)_{q}^{s}u =\frac{|u|^{p_{s}^{*}(\alpha)-2}u}{|x|^{\alpha}}+\lambda f(x)\frac{|u|^{r-2}u}{|x|^{c}}\quad \hbox{in }\Omega,\cr u=0\quad\text{in }\mathbb{R}^{N}\setminus\Omega, }$$ where \(a,b>0\), \(c<sr+N(1-r/p)\), \(\theta\in(1,p_{s}^{*}(\alpha)/p)\) and \(p_{s}^{*}(\alpha)\) is critical Sobolev-Hardy exponent. For a given suitable \(f(x)\), we prove that there are least two nontrivial solutions for small \(\lambda\), by way of the mountain pass theorem and Ekeland's variational principle. Furthermore, we prove that these two solutions converge to two solutions of the limiting problem as \(a\to 0^{+}\). For the limiting problem, we show the existence of infinitely many solutions, and the sequence tends to zero when \(\lambda\) belongs to a suitable range. For more information see https://ejde.math.txstate.edu/Volumes/2021/66/abstr.html]]></description><identifier>ISSN: 1072-6691</identifier><identifier>EISSN: 1072-6691</identifier><identifier>DOI: 10.58997/ejde.2021.66</identifier><language>eng</language><ispartof>Electronic journal of differential equations, 2021-08, Vol.2021 (1-104), p.66</ispartof><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c1216-97b8c36ee263853047d56b9cec05e111cf1382da94d3467c30904fec33c2c4e63</citedby><cites>FETCH-LOGICAL-c1216-97b8c36ee263853047d56b9cec05e111cf1382da94d3467c30904fec33c2c4e63</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,860,27901,27902</link.rule.ids></links><search><creatorcontrib>Lin, Xiaolu</creatorcontrib><creatorcontrib>Zheng, Shenzhou</creatorcontrib><title>Multiplicity and asymptotic behavior of solutions to fractional (p,q)-Kirchhoff type problems with critical Sobolev-Hardy exponent</title><title>Electronic journal of differential equations</title><description><![CDATA[Let \(\Omega\subset\mathbb{R}^{N}\) be a bounded domain with smooth boundary and \(0\in\Omega\). For \(0<s<1\), \(1\le r<q<p\), \(0\le\alpha<ps<N\) and a positive parameter \(\lambda\), we consider the fractional \((p,q)\)-Laplacian problems involving a critical Sobolev-Hardy exponent. This model comes from a nonlocal problem of Kirchhoff type $$\displaylines{ \big(a+b[u]_{s,p}^{(\theta-1)p}\big)(-\Delta)_{p}^{s}u+(-\Delta)_{q}^{s}u =\frac{|u|^{p_{s}^{*}(\alpha)-2}u}{|x|^{\alpha}}+\lambda f(x)\frac{|u|^{r-2}u}{|x|^{c}}\quad \hbox{in }\Omega,\cr u=0\quad\text{in }\mathbb{R}^{N}\setminus\Omega, }$$ where \(a,b>0\), \(c<sr+N(1-r/p)\), \(\theta\in(1,p_{s}^{*}(\alpha)/p)\) and \(p_{s}^{*}(\alpha)\) is critical Sobolev-Hardy exponent. For a given suitable \(f(x)\), we prove that there are least two nontrivial solutions for small \(\lambda\), by way of the mountain pass theorem and Ekeland's variational principle. Furthermore, we prove that these two solutions converge to two solutions of the limiting problem as \(a\to 0^{+}\). For the limiting problem, we show the existence of infinitely many solutions, and the sequence tends to zero when \(\lambda\) belongs to a suitable range. 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For \(0<s<1\), \(1\le r<q<p\), \(0\le\alpha<ps<N\) and a positive parameter \(\lambda\), we consider the fractional \((p,q)\)-Laplacian problems involving a critical Sobolev-Hardy exponent. This model comes from a nonlocal problem of Kirchhoff type $$\displaylines{ \big(a+b[u]_{s,p}^{(\theta-1)p}\big)(-\Delta)_{p}^{s}u+(-\Delta)_{q}^{s}u =\frac{|u|^{p_{s}^{*}(\alpha)-2}u}{|x|^{\alpha}}+\lambda f(x)\frac{|u|^{r-2}u}{|x|^{c}}\quad \hbox{in }\Omega,\cr u=0\quad\text{in }\mathbb{R}^{N}\setminus\Omega, }$$ where \(a,b>0\), \(c<sr+N(1-r/p)\), \(\theta\in(1,p_{s}^{*}(\alpha)/p)\) and \(p_{s}^{*}(\alpha)\) is critical Sobolev-Hardy exponent. For a given suitable \(f(x)\), we prove that there are least two nontrivial solutions for small \(\lambda\), by way of the mountain pass theorem and Ekeland's variational principle. Furthermore, we prove that these two solutions converge to two solutions of the limiting problem as \(a\to 0^{+}\). For the limiting problem, we show the existence of infinitely many solutions, and the sequence tends to zero when \(\lambda\) belongs to a suitable range. For more information see https://ejde.math.txstate.edu/Volumes/2021/66/abstr.html]]></abstract><doi>10.58997/ejde.2021.66</doi><oa>free_for_read</oa></addata></record>
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title Multiplicity and asymptotic behavior of solutions to fractional (p,q)-Kirchhoff type problems with critical Sobolev-Hardy exponent
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