Nonlocal Lazer–McKenna-type problem perturbed by the Hardy’s potential and its parabolic equivalence
Year: 2021
Type: article
Abstract: In this paper, we study the effect of Hardy potential on the existence or nonexistence of solutions to the following fractional problem involving a singular nonlinearity: $$\begin{aligned} \textstyle\begin{cases} (-\Delta )^{s} u = \lambda \frac{u}{ \vert x \vert ^{2s}} + \frac{\mu }{u^{\gamma }}+f & \text{in } \Omega, \\ u>0 & \text{in } \Omega, \\ u=0 & \text{in } (\mathbb{R}^{N} \setminus \Omega ). \end{cases}\displaystyle \end{aligned}$$ { (−Δ)su=λu|x|2s+μuγ+f in Ω, u>0 in Ω, u=0 in (RN∖Ω). Here $0 < s<1$ 0<s<1 , $\lambda >0$ λ>0 , $\gamma >0$ γ>0 , and $\Omega \subset \mathbb{R}^{N}$ Ω⊂RN ( $N > 2s$ N>2s ) is a bounded smooth domain such that $0 \in \Omega $ 0∈Ω . Moreover, $0 \leq \mu,f \in L^{1}(\Omega )$ 0≤μ,f∈L1(Ω) . For $0< \lambda \leq \Lambda _{N,s}$ 0<λ≤ΛN,s , $\Lambda _{N,s}$ ΛN,s being the best constant in the fractional Hardy inequality, we find a necessary and sufficient condition for the existence of a positive weak solution to the problem with respect to the dataμandf. Also, for a regular datum off, under suitable assumptions, we obtain some existence and uniqueness results and calculate the rate of growth of solutions. Moreover, we mention a nonexistence and a complete blowup result for the case $\lambda > \Lambda _{N,s}$ λ>ΛN,s . Besides, we consider the parabolic equivalence of the above problem in the case $\mu \equiv 1$ μ≡1 and some suitable $f(x,t)$ f(x,t) , that is, $$\begin{aligned} \textstyle\begin{cases} u_{t}+(-\Delta )^{s} u = \lambda \frac{u}{ \vert x \vert ^{2s}} + \frac{1}{u^{\gamma }}+f(x,t) & \text{in } \Omega \times (0,T), \\ u>0 & \text{in } \Omega \times (0,T), \\ u =0 & \text{in } (\mathbb{R}^{N} \setminus \Omega ) \times (0,T), \\ u(x,0)=u_{0} & \text{in } \mathbb{R}^{N}, \end{cases}\displaystyle \end{aligned}$$ { ut+(−Δ)su=λu|x|2s+1uγ+f(x,t) in Ω×(0,T), u>0 in Ω×(0,T), u=0 in (RN∖Ω)×(0,T), u(x,0)=u0 in RN, where $u_{0} \in X_{0}^{s}(\Omega )$ u0∈X0s(Ω) satisfies an appropriate cone condition. In the case $0<\gamma \leq 1$ 0<γ≤1 or $\gamma >1$ γ>1 with $2s(\gamma -1)<(\gamma +1)$ 2s
Source: Boundary Value Problems
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,570
OpenAlex ID: W3022461815