Abstract:In this article, we study the existence and multiplicity of non-negative solutions of following p-fractional equation:where Ω is a bounded domain in R n , p ≥ 2, n > pα, α ∈ (0, 1), 0 < q < p − 1 < r < np n−ps − 1, λ > 0 and h, b are sign changing smooth functions. We show the existence of solutions by minimization on the suitable subset of Nehari manifold using the fibering maps. We find that there exists λ 0 such that for λ ∈ (0, λ 0 ), it has at least two solutions.
Under different assumptions on the potential functions b and c, we study the fractional equation (I − Δ) α u = λb(x)|u| p−2 u + c(x)|u| q−2 u in R N . Our existence results are based on compact embedding properties for weighted spaces.
Under different assumptions on the potential functions b and c, we study the fractional equation (I − Δ) α u = λb(x)|u| p−2 u + c(x)|u| q−2 u in R N . Our existence results are based on compact embedding properties for weighted spaces.
“…Note that the norm involves the interaction between Ω and . This type of functional setting is introduced by Servadei and Valdinoci for in and for in .…”
In this article, we study the eigenvalues of p‐fractional Hardy operator
trueright100.0ptfalse(−normalΔfalse)pαu−μfalse|ufalse|p−2u|x|pαleft=λV(x)false|ufalse|p−2u0.16em0.16emin0.16em0.16emnormalΩ,1emu=00.16em0.16emin0.16em0.16emRn∖normalΩ,where n>pα, p≥2, α∈(0,1), 0≤μ
“…In addition, these operators arise in modelling diffusion and transport in a prevalent role in physical situations such as combustion and highly heterogeneous medium. As to the concave-convex nonlinearity, this type of problems has been studied by many authors [2][3][4][5][6][7] and the references therein. If the weight functions f (x) ≡ g(x) ≡ 1, the authors in [3] have investigated the following equation:…”
This paper is devoted to study a class of fractional equations with critical exponent, concave nonlinearity and sign-changing weight functions. By means of variational methods, the multiplicity of the positive solutions to this problem is obtained.Mathematics Subject Classification. 35J20, 35S15, 47J30, 47G20.
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