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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.16624 |
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| _version_ | 1866909494939746304 |
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| author | Zhang, Weimin |
| author_facet | Zhang, Weimin |
| contents | To our knowledge, this paper is the first attempt to consider the existence issue for fractional $p$-Laplacian equation: $(-Δ)_p^s u= λf(u),\; u> 0 ~\text{in}~Ω;\; u=0\;\text{in}~ \mathbb{R}^N\setminusΩ$, where $p>1$, $s\in (0,1)$, $λ>0$ and $Ω$ is a bounded domain with $C^{1, 1}$ boundary. We first propose a notion of stable solution, then we prove that when $f$ is of class $C^1$, nondecreasing and satisfying $f(0)>0$ and $\underset{t\to \infty}{\lim}\frac{f(t)}{t^{p-1}}=\infty$, there exists an extremal parameter $λ^*\in (0, \infty)$ such that a bounded minimal solution $u_λ\in W_0^{s,p}(Ω)$ exists if $λ\in (0, λ^*)$, and no bounded solution exists if $λ>λ^*$. Moreover, no $W_0^{s,p}(Ω)$ solution exists for $λ> λ^*$ if in addition $f(t)^{\frac{1}{p-1}}$ is convex.
To handle our problems, we show a Kato-type inequality for fractional $p$-Laplacian. We show also $L^r$ estimates for the equation $(-Δ)_p^su=g$ with $g\in W_0^{s, p}(Ω)^*\cap L^q(Ω)$ for $q \geq 1$, especially for $q \le \frac{N}{sp}$. We believe that these general results have their own interests. Finally, using the stability of minimal solutions $u_λ$, under the polynomial growth or convexity assumption on $f$, we show that the extremal function $u_* =\lim_{λ\toλ^*}u_λ\in W_0^{s,p}(Ω)$ in all dimensions, and $u^*\in L^{\infty}(Ω)$ in some low dimensional cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16624 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stable solution and extremal solution for fractional $p$-Laplacian Zhang, Weimin Analysis of PDEs To our knowledge, this paper is the first attempt to consider the existence issue for fractional $p$-Laplacian equation: $(-Δ)_p^s u= λf(u),\; u> 0 ~\text{in}~Ω;\; u=0\;\text{in}~ \mathbb{R}^N\setminusΩ$, where $p>1$, $s\in (0,1)$, $λ>0$ and $Ω$ is a bounded domain with $C^{1, 1}$ boundary. We first propose a notion of stable solution, then we prove that when $f$ is of class $C^1$, nondecreasing and satisfying $f(0)>0$ and $\underset{t\to \infty}{\lim}\frac{f(t)}{t^{p-1}}=\infty$, there exists an extremal parameter $λ^*\in (0, \infty)$ such that a bounded minimal solution $u_λ\in W_0^{s,p}(Ω)$ exists if $λ\in (0, λ^*)$, and no bounded solution exists if $λ>λ^*$. Moreover, no $W_0^{s,p}(Ω)$ solution exists for $λ> λ^*$ if in addition $f(t)^{\frac{1}{p-1}}$ is convex. To handle our problems, we show a Kato-type inequality for fractional $p$-Laplacian. We show also $L^r$ estimates for the equation $(-Δ)_p^su=g$ with $g\in W_0^{s, p}(Ω)^*\cap L^q(Ω)$ for $q \geq 1$, especially for $q \le \frac{N}{sp}$. We believe that these general results have their own interests. Finally, using the stability of minimal solutions $u_λ$, under the polynomial growth or convexity assumption on $f$, we show that the extremal function $u_* =\lim_{λ\toλ^*}u_λ\in W_0^{s,p}(Ω)$ in all dimensions, and $u^*\in L^{\infty}(Ω)$ in some low dimensional cases. |
| title | Stable solution and extremal solution for fractional $p$-Laplacian |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2403.16624 |