Boundary Value Problems (Jul 2021)
Nonlocal Lazer–McKenna-type problem perturbed by the Hardy’s potential and its parabolic equivalence
Abstract
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: { ( − Δ ) s u = λ u | x | 2 s + μ u γ + f in Ω , u > 0 in Ω , u = 0 in ( R N ∖ Ω ) . $$\begin{aligned} \textstyle\begin{cases} (-\Delta )^{s} u = \lambda \frac{u}{ \vert x \vert ^{2s}} + \frac{\mu }{u^{\gamma }}+f & \text{in } \Omega, \\ u>0 & \text{in } \Omega, \\ u=0 & \text{in } (\mathbb{R}^{N} \setminus \Omega ). \end{cases}\displaystyle \end{aligned}$$ Here 0 0 $\lambda >0$ , γ > 0 $\gamma >0$ , and Ω ⊂ R N $\Omega \subset \mathbb{R}^{N}$ ( N > 2 s $N > 2s$ ) is a bounded smooth domain such that 0 ∈ Ω $0 \in \Omega $ . Moreover, 0 ≤ μ , f ∈ L 1 ( Ω ) $0 \leq \mu,f \in L^{1}(\Omega )$ . For 0 Λ N , s $\lambda > \Lambda _{N,s}$ . Besides, we consider the parabolic equivalence of the above problem in the case μ ≡ 1 $\mu \equiv 1$ and some suitable f ( x , t ) $f(x,t)$ , that is, { u t + ( − Δ ) s u = λ u | x | 2 s + 1 u γ + f ( x , t ) in Ω × ( 0 , T ) , u > 0 in Ω × ( 0 , T ) , u = 0 in ( R N ∖ Ω ) × ( 0 , T ) , u ( x , 0 ) = u 0 in R N , $$\begin{aligned} \textstyle\begin{cases} u_{t}+(-\Delta )^{s} u = \lambda \frac{u}{ \vert x \vert ^{2s}} + \frac{1}{u^{\gamma }}+f(x,t) & \text{in } \Omega \times (0,T), \\ u>0 & \text{in } \Omega \times (0,T), \\ u =0 & \text{in } (\mathbb{R}^{N} \setminus \Omega ) \times (0,T), \\ u(x,0)=u_{0} & \text{in } \mathbb{R}^{N}, \end{cases}\displaystyle \end{aligned}$$ where u 0 ∈ X 0 s ( Ω ) $u_{0} \in X_{0}^{s}(\Omega )$ satisfies an appropriate cone condition. In the case 0 1 $\gamma >1$ with 2 s ( γ − 1 ) < ( γ + 1 ) $2s(\gamma -1)<(\gamma +1)$ , we show the existence of a unique solution for any 0 < λ < Λ N , s $0< \lambda < \Lambda _{N,s}$ and prove a stabilization result for certain range of λ.
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