Ariel Salort, Eugenio Vecchi
In this paper we consider a Hénon-type equation driven by a nonlinear operator obtained as a combination of a local and nonlocal term. We prove existence and non-existence akin to the classical result by Ni, and a stability result as the fractional parameter $s \to 1$.
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The aim of this paper is to study a nonlocal problem with a mixed Dirichlet-Neumann exterior condition. We prove existence, nonexistence and multiplicity of positive energy solutions and describe the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.
Minimizers of functionals of mixed local and nonlocal type are locally $C^{1,β}$-regular.
Nonlocal and mixed models with Lavrentiev Gap
本文构建了用于产生非局部问题 Lavrentiev 能隙示例的通用框架，并展示其在双相型非局部及混合模型中的应用。
We establish Harnack's estimates for positive weak solutions to a mixed local and nonlocal doubly nonlinear parabolic equation. All results presented in this paper are provided together with quantitative estimates.
We give a unified proof of Hölder regularity of weak solutions for mixed local and nonlocal $p$-Laplace type parabolic equations with the full range of exponents $1<p<\infty$. Our proof is based on the expansion of positivity together with the energy estimate and De Giorgi type lemma.
Mixed local-nonlocal quasilinear problems with mixed interpolated Hardy potential
研究一类含混合插值Hardy势的混合局部-非局部拟线性问题，通过建立集中紧性原理并结合Ricceri变分原理和山路定理，得到非线性项在不同增长条件下的解的存在性。
Variational methods for nonpositive mixed local-nonlocal operators
本文运用变分方法，研究由非正定混合局部—非局部算子驱动的边值问题，并证明其弱解的存在性。
We establish a priori Lipschitz estimates for equations with mixed local and nonlocal diffusion, coercive gradient terms and unbounded right-hand side in Lebesgue spaces through an integral refinement of the Bernstein method. This relies on a nonlinear, nonlocal and variational version of the Bochner identity that involves the adjoint equation of the linearization of the initial problem.
This article proves a weak Harnack inequality with a tail term for sign changing supersolutions of a mixed local and nonlocal parabolic equation. Our argument is purely analytic. It is based on energy estimates and the Moser iteration technique. Instead of the parabolic John-Nirenberg lemma, we adopt a lemma of Bombieri to the mixed local and nonlocal parabolic case. To this end, we prove an appropriate reverse Hölder inequality and a logarithmic estimate for weak supersolutions.
On mixed local-nonlocal Sobolev-type inequalities and their connection with singular equations in the Heisenberg group
本文在Heisenberg群中建立混合局部-非局部Sobolev型不等式，证明其与奇异p-Laplace方程的解关系，并提供了一种统一框架，连接不同情形下奇异方程的存在理论。
In this article, we consider mixed local and nonlocal Sobolev $(q,p)$-inequalities with extremal in the case $0<q<1<p<\infty$. We prove that the extremal of such inequalities is unique up to a multiplicative constant that is associated with a singular elliptic problem involving the mixed local and nonlocal $p$-Laplace operator. Moreover, it is proved that the mixed Sobolev inequalities are necessary and sufficient condition for the existence of weak solutions of such singular problems. As a consequence, a relation between the singular $p$-Laplace and mixed local and nonlocal $p$-Laplace equati…
On the existence of a second positive solution to mixed local-nonlocal concave-convex critical problems
本文秉承Ambrosetti等人1994年的精神，证明了带有次线性扰动的混合局部-非局部临界半线性椭圆问题存在第二正弱解。
We consider the first Dirichlet eigenvalue problem for a mixed local/nonlocal elliptic operator and we establish a quantitative Faber-Krahn inequality. More precisely, we show that balls minimize the first eigenvalue among sets of given volume and we provide a stability result for sets that almost attain the minimum.
Mixed Local-Nonlocal Parabolic Equations in Heisenberg Group : Harnack Inequality with an Optimal Tail
本文通过正性扩展方法和聚类引理，证明了海森堡群上线性混合局部-非局部抛物方程具有最优尾部的哈纳克不等式，避免了Moser迭代和对数估计。
Multiplicity result for mixed local and nonlocal Kirchhoff problem involving critical growth
本文针对带变号权函数的临界非线性混合局部与非局部问题，利用Nehari流形方法和纤维化映射分析，证明了其存在两个非负解的多解性结果。
Radial positive solutions for mixed local and nonlocal supercritical Neumann problem
本文研究球体中混合局部与非局部超临界Neumann问题，在无需非线性增长假设下证明正非递减径向解的存在性，并给出存在非常数解的判定准则。
Harnack inequality for doubly nonlinear mixed local and nonlocal parabolic equations
本文针对双重非线性混合局部与非局部抛物型方程，利用比较原理、局部有界性估计和正性扩张方法，建立了非负弱解的Harnack不等式。
We investigate the mixed local and nonlocal parabolic $p$-Laplace equation \begin{align*} \partial_t u(x,t)-Δ_p u(x,t)+\mathcal{L}u(x,t)=0, \end{align*} where $Δ_p$ is the local $p$-Laplace operator and $\mathcal{L}$ is the nonlocal $p$-Laplace operator. Based on the combination of suitable Caccioppoli-type inequality and Logarithmic Lemma with a De Giorgi-Nash-Moser iteration, we establish the local boundedness and Hölder continuity of weak solutions for such equations.
Global and nonglobal solutions for a mixed local-nonlocal heat equation
本文针对混合局部-非局部算子控制的半线性抛物方程，建立了解的全局与非全局存在性的最优条件，并恢复了已知的Fujita指数。
Global solutions to semilinear parabolic equations driven by mixed local-nonlocal operators
研究混合局部-非局部算子驱动的半线性抛物方程柯西问题，证明Fujita现象成立且临界值与分数阶拉普拉斯算子相同。
Mixed local-nonlocal $p$-Laplace equation with variable singular nonlinearity in the Heisenberg group
本文研究了海森堡群中混合局部-非局部 $p$-拉普拉斯方程的弱解存在性、唯一性和正则性，首次在非交换设定中处理此类问题。
Positive Solutions for a Mixed Local-Nonlocal Problem with Semipositone Nonlinearity
本文利用山脊原理、比较原理和正则性原理，证明了混合局部-非局部半非负问题至少存在一个正解。
Multiplicity of positive solutions for mixed local-nonlocal singular critical problems
本文证明混合局部-非局部奇异临界椭圆问题至少存在两个正弱解，推广了奇异与临界问题的现有结果。
In this work, we study an elliptic problem involving an operator of mixed order with both local and nonlocal aspects, and in either the presence or the absence of a singular nonlinearity. We investigate existence or non-existence properties, power and exponential type Sobolev regularity results, and the boundary behavior of the weak solution, in the light of the interplay between the summability of the datum and the power exponent in singular nonlinearities.
Higher Sobolev regularity for mixed local and nonlocal equations with nonstandard growth
本研究针对可模拟两类随机过程叠加的混合局部与非局部(p,q)-Laplace方程，开展其弱解内部Sobolev正则性的系统研究。
Normalized solutions to a Choquard equation involving mixed local and nonlocal operators
本文研究了涉及混合局部与非局部扩散算子的Choquard型方程的归一化解，证明了其存在性与正则性，并建立了归一化解与归一化基态存在性之间的等价关系。
Mixed local and nonlocal semilinear elliptic equation with strongly singular and critical Choquard nonlinearity
本文研究混合局部与非局部椭圆方程，结合奇异与临界非线性，通过变分方法证明正解的存在性与多重性。
Liouville theorems for mixed local and nonlocal indefinite equations
本文研究了混合局部-非局部方程正解的定性性质，通过建立最大原理和严格单调性，结合平滑第一特征函数和合适子解，推导了混合算子(-Δ)^s - Δ的非存在性结果，并扩展到抛物情形，揭示了双重非局部性下的新定性特征。
Regularity for Mixed Local and Nonlocal Degenerate Elliptic Equations in the Heisenberg Group
本文研究了Heisenberg群中混合局部与非局部退化椭圆方程的正则性，通过推导Caccioppoli型不等式和对数估计，证明了弱解的局部有界性、Hölder连续性及Harnack不等式。
In this article, we consider the spectral problem for the mixed local and nonlocal $p$-Laplace operator. We discuss the existence and regularity of eigenfunction of the associated Dirichlet $(p,q)$-eigenvalue problem in a bounded domain $Ω\subset\mathbb{R}^N$ under the assumption that $1<p<\infty$ and $1<q<p^{*}$ where $p^{*}=\frac{N p}{N-p}$ if $1<p<N$ and $p^{*}=\infty$ if $p\geq N$.
Fujita exponent and blow-up rate for a mixed local and nonlocal heat equation
针对含混合局部-非局部扩散算子的热方程爆破问题，推导得出其Fujita指数由非局部部分决定为$p_F=1+2s/N$，并在部分情形下确定了爆破速率。
We investigate the De Giorgi-Nash-Moser theory for minimizers of mixed local and nonlocal functionals modeled after \[ v \mapsto \int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\dfrac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\,dxdy+\int_Ωa(x)|Dv|^{q}\,dx, \] where $0<s<1<p \le q$ and $a(\cdot) \ge 0$. In particular, we prove Hölder regularity and Harnack's inequality under possibly sharp assumptions on $s,p,q$ and $a(\cdot)$.
We consider mixed local and nonlocal quasilinear parabolic equations of $p$-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions, lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. We also discuss the pointwise behavior of the semicontinuous representatives. Our main results are valid for sign changing solutions. Our approach is purely analytic and is based on energy estimates and the De Giorgi theory.
Multiplicity results for mixed local-nonlocal equations with singular and critical exponential nonlinearity in R^2
本文研究R^2中带可变奇异指数与临界指数非线性项的混合局部-非局部方程，利用下解-上解技巧与变分方法证明了至少两个正弱解的存在性。
Symmetry and Approximate Symmetry for Solutions of Mixed Local-Nonlocal Singular Equations
本文研究了混合局部-非局部奇异方程解的对称性与近似对称性，通过移动平面方法建立了径向对称性，并提出了Gidas-Ni-Nirenberg定理的定量版本。
An Ahmad-Lazer-Paul-type result for indefinite mixed local-nonlocal problems
本文针对共振下不定混合局部-非局部问题，通过改造变分方法处理负特征值，证明了弱解的存在性与多重性。
We establish existence results for a class of mixed anisotropic and nonlocal $p$-Laplace equation with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To this end, we also discuss the necessary regularity properties of weak solutions of the associated non-singular problems. More precisely, we obtain local boundedness of subsolutions, Harnack inequality for solutions and weak Harnack inequality for supersolutions.
Choquard equation involving mixed local and nonlocal operators
该研究针对含混合阶局部-非局部算子与Hartree型临界非线性项的椭圆Choquard方程，通过推导对应Hardy-Littlewood-Sobolev不等式的最优常数，结合变分法与Pohožaev恒等式，得到次临界扰动问题解的存在性与不存在性结果。
On a class of mixed local and nonlocal semilinear elliptic equation with singular nonlinearity
该研究利用变分法与逼近方法，考察兼具局部与非局部 Laplace 算子的奇异半线性椭圆方程，证明参数依赖情形下弱解存在及扰动情形下多解存在。
Coexisting Automated and Human-Driven Vehicles: Well-Posedness of a Mixed Nonlocal-Local Traffic Model
提出标准车辆与信息车辆共存的宏观混合非局部-局部交通流PDE模型，并证明其生成局部Lipschitz连续半群且解唯一，确保了模型的适定性。
Mixed local and nonlocal weighted singular quasilinear elliptic problem and its associated Sobolev-type inequality
该研究针对一类混合各向异性与非局部奇异拟线性椭圆问题，采用单调逼近等方法证明弱解存在唯一性，刻画相关Sobolev型不等式最佳常数并证明其可达性，结果在p=2的混合加权拉普拉斯情形下为新成果。
On the regularity theory for mixed local and nonlocal weighted quasilinear elliptic equations
该研究针对带Muckenhoupt权的混合局部与非局部退化p-拉普拉斯方程，建立了首个系统的局部正则性理论，扩展了相关现有理论框架。
Existence of solutions for nonlinear equations with mixed local and nonlocal operators
本文研究混合局部与非局部算子驱动的非线性方程，利用临界点定理证明在特定条件下解的存在性。
On sliding methods for mixed local and nonlocal equations and Gibbons' conjecture
本文提出了一种适用于混合局部-非局部算子的精细滑动方法，解决了混合分数方程中Gibbons猜想的问题，并推导了单调性和对称性结果。
Explicit Formula Of The Critical Mass And The Energy Ground State Solution For The Mixed Local-Nonlocal Schrodinger Equation For The In-Between Critical Exponents Case
本文通过Gagliardo-Nirenberg不等式的最佳常数，推导了混合局部-非局部薛定谔方程中临界质量的显式公式，并证明了能量基态解的存在性及其作为优化器的性质。
Symmetry, existence and regularity results for a class of mixed local-nonlocal semilinear singular elliptic problem via variational characterization
本文利用变分刻画和分解性质，结合比较原理与移动平面方法，研究一类混合局部-非局部奇异椭圆问题弱解的对称性、存在性、不存在性与正则性，并用非光滑临界点理论分析跳跃问题。
Riesz potential estimates for mixed local-nonlocal problems with measure data
本文研究了具测度数据的混合局部-非局部问题的梯度正则性，通过截断1-Riesz位势建立了梯度Du的逐点界。
A Brezis-Nirenberg type result for mixed local and nonlocal operators
本文研究拉普拉斯与分数阶拉普拉斯叠加的混合阶算子的临界问题，确定其索伯列夫不等式最优常数永不可达，并建立次临界扰动问题的存在性与不存在性理论。
Mixed Local-Nonlocal Equations in the Hyperbolic Space and Sharp Sobolev Constants
本文针对双曲空间中含混合局部-非局部算子的非线性椭圆方程，采用变分法研究次临界与临界非线性情形，证明了适当条件下弱解的存在性。
We consider parabolic equations on bounded smooth open sets $\Om\subset \R^N$ ($N\ge 1$) with mixed Dirichlet type boundary-exterior conditions associated with the elliptic operator $\mathscr{L} \coloneqq - Δ+ (-Δ)^{s}$ ($0<s<1$). Firstly, we prove several well-posedness and regularity results of the associated elliptic and parabolic problems with smooth, and then with singular boundary-exterior data. Secondly, we show the existence of optimal solutions of associated optimal control problems, and we characterize the optimality conditions. This is the first time that such topics have been prese…
The best constant in the G-N inequality for the mixed local and Nonlocal Laplacian
本文研究混合局部和非局部拉普拉斯算子的G-N不等式最佳常数，提出创新方法证明最优常数的优化器为基态解，并确定最佳常数。
Some qualitative and quantitative properties of weak solutions to mixed anisotropic and nonlocal quasilinear elliptic and doubly nonlinear parabolic equations
本文研究混合各向异性与非局部p-Laplace算子及双重非线性抛物方程，建立存在唯一性、比较定理、Hardy不等式及正则性估计，并利用Picone恒等式与能量估计等方法。
In this paper we provide necessary and sufficient conditions for the existence of a unique positive weak solution for some sublinear Dirichlet problems driven by the sum of a quasilinear local and a nonlocal operator, i.e., $$\mathcal{L}_{p,s} = -Δ_p + (-Δ)^s_p.$$ Our main result is resemblant to the celebrated work by Brezis-Oswald [10]. In addition, we prove a regularity result of independent interest.
We start in this paper a systematic study of the superpositions of elliptic operators with different orders, mixing classical and fractional scenarios. For concreteness, we focus on the sum of the Laplacian and the fractional Laplacian, and we provide structural results, including existence, maximum principles (both for weak and classical solutions), interior Sobolev regularity and boundary regularity of Lipschitz type.
Klein-Gordon-Maxwell equations driven by mixed local-nonlocal operators
本文将Klein-Gordon-Maxwell型系统的经典结果推广到含实参数非正定非局部项的混合局部-非局部算子情形，针对两类势函数给出了保证作为能量泛函山路临界点的孤立驻波存在的参数范围。
A linear inverse problem with semi-nonlocal boundary conditions for a three-dimensional equation of mixed type of the second kind of the fourth order in a parallelepiped
本文研究了长方体中第二类四阶混合型三维方程带有半非局部边界条件的线性反问题，利用Fourier方法、$\varepsilon $-正则化、先验估计、逼近序列和压缩映射证明了广义解的存在性与唯一性。
We consider equations involving a combination of local and nonlocal degenerate $p$-Laplace operators. The main contribution of the paper is almost Lipschitz regularity for the homogeneous equation and Hölder continuity with an explicit Hölder exponent in the general case. For certain parameters, our results also imply Hölder continuity of the gradient. In addition, we establish existence, uniqueness and local boundedness. The approach is based on an iteration in the spirit of Moser combined with an approximation method.
Optimal boundary regularity for mixed local and nonlocal equations
本文研究了混合局部和非局部方程的最优边界正则性，通过加权霍尔德空间和固定点论证，提供精确的估计并用反例证明其最优性。
Existence Theory for a class of semilinear mixed local and nonlocal equations involving variable singularities and singular measures
本文针对含可变奇异性和奇异测度的半线性局部-非局部混合方程，通过处理可变奇异指数与测度值数据（含新的奇异测度数据），建立了其弱解的存在性。
We consider a combination of local and nonlocal $p$-Laplace equations and discuss several regularity properties of weak solutions. More precisely, we establish local boundedness of weak subsolutions, local Hölder continuity of weak solutions, Harnack inequality for weak solutions and weak Harnack inequality for weak supersolutions. We also discuss lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. Our approach is purely analytic and it is based on the De Giorgi-Nash-Moser theory, the expansion of positivity and estimates involving a tail term. The…
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