Critical $(p,q)$-fractional problems involving a sandwich type nonlinearity
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929659123335168 |
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| author | Bhakta, Mousomi Fiscella, Alessio Gupta, Shilpa |
| author_facet | Bhakta, Mousomi Fiscella, Alessio Gupta, Shilpa |
| contents | In this paper, we deal with the following $(p,q)$-fractional problem $$ (-Δ)^{s_{1}}_{p}u +(-Δ)^{s_{2}}_{q}u=λP(x)|u|^{k-2}u+θ|u|^{p_{s_{1}}^{*}-2}u \, \mbox{ in }\, Ω,\qquad u=0\, \mbox{ in }\, \mathbb{R}^{N} \setminus Ω, $$ where $Ω\subseteq\mathbb{R}^{N}$ is a general open set, $0<s_{2}<s_{1}<1$, $1<q<k<p<N/s_{1}$, parameter $λ,\ θ>0$, $P$ is a nontrivial nonnegative weight, while $p_{s_{1}}^{*}=Np/(N-ps_{1})$ is the critical exponent. We prove that there exists a decreasing sequence $\{θ_j\}_j$ such that for any $j\in\mathbb N$ and with $θ\in(0,θ_j)$, there exist $λ_*$, $λ^*>0$ such that above problem admits at least $j$ distinct weak solutions with negative energy for any $λ\in (λ_*,λ^*)$. On the other hand, we show there exists $\overlineλ>0$ such that for any $λ>\overlineλ$, there exists $θ^*=θ^*(λ)>0$ such that the above problem admits a nonnegative weak solution with negative energy for any $θ\in(0,θ^*)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13986 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Critical $(p,q)$-fractional problems involving a sandwich type nonlinearity Bhakta, Mousomi Fiscella, Alessio Gupta, Shilpa Analysis of PDEs 35J62, 35J70, 35R11, 35J20, 49J35 In this paper, we deal with the following $(p,q)$-fractional problem $$ (-Δ)^{s_{1}}_{p}u +(-Δ)^{s_{2}}_{q}u=λP(x)|u|^{k-2}u+θ|u|^{p_{s_{1}}^{*}-2}u \, \mbox{ in }\, Ω,\qquad u=0\, \mbox{ in }\, \mathbb{R}^{N} \setminus Ω, $$ where $Ω\subseteq\mathbb{R}^{N}$ is a general open set, $0<s_{2}<s_{1}<1$, $1<q<k<p<N/s_{1}$, parameter $λ,\ θ>0$, $P$ is a nontrivial nonnegative weight, while $p_{s_{1}}^{*}=Np/(N-ps_{1})$ is the critical exponent. We prove that there exists a decreasing sequence $\{θ_j\}_j$ such that for any $j\in\mathbb N$ and with $θ\in(0,θ_j)$, there exist $λ_*$, $λ^*>0$ such that above problem admits at least $j$ distinct weak solutions with negative energy for any $λ\in (λ_*,λ^*)$. On the other hand, we show there exists $\overlineλ>0$ such that for any $λ>\overlineλ$, there exists $θ^*=θ^*(λ)>0$ such that the above problem admits a nonnegative weak solution with negative energy for any $θ\in(0,θ^*)$. |
| title | Critical $(p,q)$-fractional problems involving a sandwich type nonlinearity |
| topic | Analysis of PDEs 35J62, 35J70, 35R11, 35J20, 49J35 |
| url | https://arxiv.org/abs/2409.13986 |