1,042 research outputs found

    Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian

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    We study the regularity of stable solutions to the problem {(−Δ)su=f(u)inB1 ,u≡0inRn∖B1 , \left\{ \begin{array}{rcll} (-\Delta)^s u &=& f(u) & \text{in} \quad B_1\,, u &\equiv&0 & \text{in} \quad \mathbb R^n\setminus B_1\,, \end{array} \right. where s∈(0,1)s\in(0,1). Our main result establishes an L∞L^\infty bound for stable and radially decreasing HsH^s solutions to this problem in dimensions 2≤n<2(s+2+2(s+1))2 \leq n < 2(s+2+\sqrt{2(s+1)}). In particular, this estimate holds for all s∈(0,1)s\in(0,1) in dimensions 2≤n≤62 \leq n\leq 6. It applies to all nonlinearities f∈C2f\in C^2. For such parameters ss and nn, our result leads to the regularity of the extremal solution when ff is replaced by λf\lambda f with λ>0\lambda > 0. This is a widely studied question for s=1s=1, which is still largely open in the nonradial case both for s=1s=1 and s<1s<1

    Boundedness of stable solutions to semilinear elliptic equations: a survey

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    This article is a survey on boundedness results for stable solutions to semilinear elliptic problems. For these solutions, we present the currently known L∞L^{\infty} estimates that hold for all nonlinearities. Such estimates are known to hold up to dimension 4. They are expected to be true also in dimensions 5 to 9, but this is still an open problem which has only been established in the radial case

    Boundedness of stable solutions to semilinear elliptic equations: a survey

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    This article is a survey on boundedness results for stable solutions to semilinear elliptic problems. For these solutions, we present the currently known L∞L^{\infty} estimates that hold for all nonlinearities. Such estimates are known to hold up to dimension 4. They are expected to be true also in dimensions 5 to 9, but this is still an open problem which has only been established in the radial case

    Regularity of radial extremal solutions for some non local semilinear equations

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    We investigate stable solutions of elliptic equations of the type \begin{equation*} \left \{ \begin{aligned} (-\Delta)^s u&=\lambda f(u) \qquad {\mbox{ in B1⊂RnB_1 \subset \R^{n}}} \\ u&= 0 \qquad{\mbox{ on ∂B1\partial B_1,}}\end{aligned}\right . \end{equation*} where n≥2n\ge2, s∈(0,1)s \in (0,1), λ≥0\lambda \geq 0 and ff is any smooth positive superlinear function. The operator (−Δ)s(-\Delta)^s stands for the fractional Laplacian, a pseudo-differential operator of order 2s2s. According to the value of λ\lambda, we study the existence and regularity of weak solutions uu

    Mass and Asymptotics associated to Fractional Hardy-Schr\"odinger Operators in Critical Regimes

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    We consider linear and non-linear boundary value problems associated to the fractional Hardy-Schr\"odinger operator Lγ,α:=(−Δ)α2−γ∣x∣α L_{\gamma,\alpha}: = ({-}{ \Delta})^{\frac{\alpha}{2}}- \frac{\gamma}{|x|^{\alpha}} on domains of Rn\mathbb{R}^n containing the singularity 00, where 0<α<20<\alpha<2 and 0≤γ<γH(α) 0 \le \gamma < \gamma_H(\alpha), the latter being the best constant in the fractional Hardy inequality on Rn\mathbb{R}^n. We tackle the existence of least-energy solutions for the borderline boundary value problem (Lγ,α−λI)u=u2α⋆(s)−1∣x∣s(L_{\gamma,\alpha}-\lambda I)u= {\frac{u^{2^\star_\alpha(s)-1}}{|x|^s}} on Ω\Omega, where 0≤s<α<n0\leq s <\alpha <n and 2α⋆(s)=2(n−s)n−α 2^\star_\alpha(s)={\frac{2(n-s)}{n-{\alpha}}} is the critical fractional Sobolev exponent. We show that if γ\gamma is below a certain threshold γcrit\gamma_{crit}, then such solutions exist for all 0<λ<λ1(Lγ,α)0<\lambda <\lambda_1(L_{\gamma,\alpha}), the latter being the first eigenvalue of Lγ,αL_{\gamma,\alpha}. On the other hand, for γcrit<γ<γH(α)\gamma_{crit}<\gamma <\gamma_H(\alpha), we prove existence of such solutions only for those λ\lambda in (0,λ1(Lγ,α))(0, \lambda_1(L_{\gamma,\alpha})) for which the domain Ω\Omega has a positive {\it fractional Hardy-Schr\"odinger mass} mγ,λ(Ω)m_{\gamma, \lambda}(\Omega). This latter notion is introduced by way of an invariant of the linear equation (Lγ,α−λI)u=0(L_{\gamma,\alpha}-\lambda I)u=0 on Ω\Omega
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