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Keyword: Entire Solution (2)	31 Mar 2025	Run
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articleNo Access
ON ENTIRE SOLUTIONS OF FERMAT TYPE PARTIAL DIFFERENTIAL EQUATIONS
- BAO QIN LI
International Journal of Mathematics01 Jul 2004
Preview Abstract
We shall consider Fermat type partial differential equations in Cⁿ, and give description and classification for entire solutions of the equations.
articleNo Access
Profile of solutions for nonlocal equations with critical and supercritical nonlinearities
Communications in Contemporary Mathematics28 Jan 2019
Preview Abstract
We study the fractional Laplacian problem
$(I_{𝜀}) \{\begin{matrix} {(- Δ)}^{s} u = u^{p} - 𝜀 u^{q} & i n Ω, \\ u > 0 & i n Ω, \\ u = 0 & i n ℝ^{N} \ Ω, \\ u \in H^{s} (Ω) \cap L^{q + 1} (Ω); \end{matrix}$
where $s \in (0, 1)$ , $q > p \geq \frac{N + 2 s}{N - 2 s}$ and $𝜀 > 0$ is a parameter. Here, $Ω \subseteq ℝ^{N}$ is a bounded star-shaped domain with smooth boundary and $N > 2 s$ . We establish existence of a variational positive solution $u_{𝜀}$ and characterize the asymptotic behavior of $u_{𝜀}$ as $𝜀 \to 0$ . When $p = \frac{N + 2 s}{N - 2 s}$ , we describe how the solution $u_{𝜀}$ blows up at an interior point of $Ω$ . Furthermore, we prove the local uniqueness of solution of the above problem when $Ω$ is a convex symmetric domain of $ℝ^{N}$ with $N > 4 s$ and $p = \frac{N + 2 s}{N - 2 s}$ .

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