Nonlocal Lazer–McKenna-type problem perturbed by the Hardy’s potential and its parabolic equivalence

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 ) s u = λ u | x | 2 s + μ u γ + f in Ω , u > 0 in Ω , u = 0 in ( R N ∖ Ω ) . Here $0 < s<1$ 0 < s < 1 , $\lambda >0$ λ > 0 , $\gamma >0$ γ > 0 , and $\Omega \subset \mathbb{R}^{N}$ Ω ⊂ R N ($N > 2s$ N > 2 s ) is a bounded smooth domain such that $0 \in \Omega $ 0 ∈ Ω . Moreover, $0 \leq \mu,f \in L^{1}(\Omega )$ 0 ≤ μ , f ∈ L 1 ( Ω ) . For $0< \lambda \leq \Lambda _{N,s}$ 0 < λ ≤ Λ N , s , $\Lambda _{N,s}$ Λ N , 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 data μ and f. Also, for a regular datum of f, 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 > Λ N , 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, $$\begin{aligned} \textstyle\begin{cases} u_{t}+(-\Delta )^{s} u = \lambda \frac{u}{ \vert x \vert ^{2s}} + \frac{1}{u^{\gamma }}+f(x,t) & \text{in } \Omega \times (0,T), \\ u>0 & \text{in } \Omega \times (0,T), \\ u =0 & \text{in } (\mathbb{R}^{N} \setminus \Omega ) \times (0,T), \\ u(x,0)=u_{0} & \text{in } \mathbb{R}^{N}, \end{cases}\displaystyle \end{aligned}$$ { 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 , where $u_{0} \in X_{0}^{s}(\Omega )$ u 0 ∈ X 0 s ( Ω ) satisfies an appropriate cone condition. In the case $0<\gamma \leq 1$ 0 < γ ≤ 1 or $\gamma >1$ γ > 1 with $2s(\gamma -1)<(\gamma +1)$ 2 s ( γ − 1 ) < ( γ + 1 ) , we show the existence of a unique solution for any $0< \lambda < \Lambda _{N,s}$ 0 < λ < Λ N , s and prove a stabilization result for certain range of λ.