Regularity results on a class of doubly nonlocal problems

Giacomoni, Jacques; Goel, Divya; Sreenadh, K.

doi:10.1016/j.jde.2019.11.009

Cited by 17 publications

(25 citation statements)

References 30 publications

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“…In the similar lines one can prove

ū \geq u

. Moreover, from [ 33, Theorem 1.2, Remark 1.5]

\underline{u}, ū \in C_{ϕ_{1}}^{+}

, where

ϕ_{1} \in C_{d^{s}}^{+}

. This implies, there exist positive constants say C 1 and C 2 such that

C_{1} d^{s} \leq \underline{u} \leq u \leq ū \leq C_{2} d^{s},

So by () and () we have

u \in C^{0} false(\overset{normalΩ}{true} false)

, hence it is a classical solution (see [ 33, Definition 2]).…”

Section: Regularity Of Solutions Of (Pλ)mentioning

confidence: 76%

Multiple positive solutions for degenerate Kirchhoff equations with singular and Choquard nonlinearity

Rawat

Sreenadh

2021

Math Methods in App Sciences

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In this paper we study the existence, multiplicity, and regularity of positive weak solutions for the following Kirchhoff–Choquard problem: M()∬ℝ2Nfalse|ufalse(xfalse)−ufalse(yfalse)false|2false|x−yfalse|N+2s0.1emdxdyfalse(−normalΔfalse)su=λuγ+()∫normalΩfalse|ufalse(yfalse)false|2μ,s∗false|x−yfalse|μ0.1emdyfalse|ufalse|2μ,s∗−2u0.51emin0.51emnormalΩ,3emu=00.51emin0.51emℝN\normalΩ, where Ω is open bounded domain of ℝN with C2 boundary, N > 2s and s ∈ (0, 1). M models Kirchhoff‐type coefficient in particular, the degenerate case where Kirchhoff coefficient M is zero at zero. (− Δ)s is fractional Laplace operator, λ > 0 is a real parameter, γ ∈ (0, 1) and 2μ,s∗=2N−μN−2s is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. We prove that each positive weak solution is bounded and satisfy Hölder regularity of order s. Furthermore, using the variational methods and truncation arguments, we prove the existence of two positive solutions.

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“…In the similar lines one can prove

ū \geq u

. Moreover, from [ 33, Theorem 1.2, Remark 1.5]

\underline{u}, ū \in C_{ϕ_{1}}^{+}

, where

ϕ_{1} \in C_{d^{s}}^{+}

. This implies, there exist positive constants say C 1 and C 2 such that

C_{1} d^{s} \leq \underline{u} \leq u \leq ū \leq C_{2} d^{s},

So by () and () we have

u \in C^{0} false(\overset{normalΩ}{true} false)

, hence it is a classical solution (see [ 33, Definition 2]).…”

Section: Regularity Of Solutions Of (Pλ)mentioning

confidence: 76%

Multiple positive solutions for degenerate Kirchhoff equations with singular and Choquard nonlinearity

Rawat

Sreenadh

2021

Math Methods in App Sciences

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“…In this section we will find two non-trivial solution of opposite sign for our problem (P λ ). It is derived with the help of H s v.s C 0 s minimizer property proved by [13]. Recall that with δ s (x) := dist(x, ∂Ω), the space C 0 s is defined as,…”

Section: Constant Sign Solutionsmentioning

confidence: 99%

“…By [13] it suffices to prove that 0 is the local minimizer for I λ on X 0 ∩ C 0 s (Ω) i.e. I λ (u) ≥ 0 for all u C 0 s < ρ 2 for some ρ 2 > 0.…”

Section: Constant Sign Solutionsmentioning

confidence: 99%

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Three solutions for a fractional elliptic problem with asymmetric critical Choquard nonlinearity

Rawat¹,

Sreenadh

2021

Preprint

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In this paper we study the existence and multiplicity of weak solutions for the following asymmetric nonlinear Choquard problem on fractional Laplacian:where Ω is open bounded domain of R N with C 2 boundary, N > 2s and s ∈ (0, 1). Here (−∆) s is the fractional Laplace operator, λ > 0 is a real parameter, q ∈ (1, 2), a > 0 and b > 0 are given constants, and 2 * µ,s = 2N −µ N −2s is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and the notation u + = max{u, 0}. We prove that the above problem has at least three nontrivial solutions using the Mountain pass Lemma and Linking theorem.

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“…In [21], symmetry and monotonicity of positive solutions are shown for the fractional Hartree equation for µ = 0 and a critical power nonlinearity, by means of the direct method of moving planes. Regularity results for a class of doubly nonlocal equations on bounded domains are obtained by Giacomoni et al in [34].…”

mentioning

confidence: 97%

On some qualitative aspects for doubly nonlocal equations

Cingolani¹,

Gallo²

2022

DCDS-S

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In this paper we investigate some qualitative properties of the solutions to the following doubly nonlocal equation<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \label{eq_abstract} (- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad \text{in } \mathbb{R}^N \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{(P)}\end{equation} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ N \geq 2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ s\in (0, 1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \alpha \in (0, N) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \mu>0 $\end{document}</tex-math></inline-formula> is fixed, <inline-formula><tex-math id="M5">\begin{document}$ (-\Delta)^s $\end{document}</tex-math></inline-formula> denotes the fractional Laplacian and <inline-formula><tex-math id="M6">\begin{document}$ I_{\alpha} $\end{document}</tex-math></inline-formula> is the Riesz potential. Here <inline-formula><tex-math id="M7">\begin{document}$ F \in C^1(\mathbb{R}) $\end{document}</tex-math></inline-formula> stands for a general nonlinearity of Berestycki-Lions type. We obtain first some regularity result for the solutions of (P). Then, by assuming <inline-formula><tex-math id="M8">\begin{document}$ F $\end{document}</tex-math></inline-formula> odd or even and positive on the half-line, we get constant sign and radial symmetry of the Pohozaev ground state solutions related to equation (P). In particular, we extend some results contained in [<xref ref-type="bibr" rid="b23">23</xref>]. Similar qualitative properties of the ground states are obtained in the limiting case <inline-formula><tex-math id="M9">\begin{document}$ s = 1 $\end{document}</tex-math></inline-formula>, generalizing some results by Moroz and Van Schaftingen in [<xref ref-type="bibr" rid="b52">52</xref>] when <inline-formula><tex-math id="M10">\begin{document}$ F $\end{document}</tex-math></inline-formula> is odd.

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Regularity results on a class of doubly nonlocal problems

Cited by 17 publications

References 30 publications

Multiple positive solutions for degenerate Kirchhoff equations with singular and Choquard nonlinearity

Multiple positive solutions for degenerate Kirchhoff equations with singular and Choquard nonlinearity

Three solutions for a fractional elliptic problem with asymmetric critical Choquard nonlinearity

On some qualitative aspects for doubly nonlocal equations

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