Existence of positive solutions for a singular elliptic problem with critical exponent and measure data

Panda, Akasmika; Choudhuri, Debajyoti; Giri, Ratan Kr.

doi:10.1216/rmj.2021.51.973

Cited by 2 publications

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“…Recently, the study of fractional elliptic equations with singular nonlinearity attracted lot of interests by researchers in nonlinear analysis. In [5], for p = 2 and 0 < s < 1, the authors studied the existence of distributional solutions of problem (1.3) using the uniform estimates of {u n } which are solutions of the regularized problems with singular term u −α replaced by (u n + 1 n ) −α (see also [12,27,30]) for more general context. The cases, when 0 < s < 1 and p = 2, have been considered in [26] where the authors have showed the existence of multiple solutions to (1.3) using variational methods.…”

Section: Introductionmentioning

confidence: 99%

Nonlocal fractional system involving the fractional p, q-Laplacians and singular potentials

Biroud

2022

Arab. J. Math.

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In this paper, we will focus on following nonlocal quasilinear elliptic system with singular nonlinearities: $$\begin{aligned} (S) \left\{ \begin{array}{ll} (-\Delta )_{p}^{s_1} u =\dfrac{1}{v^{\alpha _1}}+v^{\beta _1}&{} \text { in }\Omega , \\ (-\Delta )_{q}^{s_2} u =\dfrac{1}{u^{\alpha _2}}+u^{\beta _2}&{} \text { in }\Omega , \\ u,v=0 &{} \text { in } ({I\!\!R}^N\setminus \Omega ) , \\ u,v>0 &{} \text { in } \Omega , \end{array} \right. \end{aligned}$$ ( S ) ( - Δ ) p s 1 u = 1 v α 1 + v β 1 in Ω , ( - Δ ) q s 2 u = 1 u α 2 + u β 2 in Ω , u , v = 0 in ( I R N \ Ω ) , u , v > 0 in Ω , where $$\Omega \subset {I\!\!R}^N$$ Ω ⊂ I R N be a smooth bounded domain, $$s_1,\,s_2\in (0,1)$$ s 1 , s 2 ∈ ( 0 , 1 ) , $$\alpha _1$$ α 1 , $$\alpha _2$$ α 2 , $$\beta _1$$ β 1 , $$\beta _2$$ β 2 are suitable positive constants, $$(-\Delta )_{p}^{s_1}$$ ( - Δ ) p s 1 and $$(-\Delta )_{q}^{s_2}$$ ( - Δ ) q s 2 are the fractional $$p-\text {Laplacian}$$ p - Laplacian and $$q-\text {Laplacian}$$ q - Laplacian operators. Using approximating arguments, Rabinowitz bifurcation Theorem, and fractional Hardy inequality, we are able to show the existence of positive solution to the above system.

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