Simona Dabuleanu, Vicentiu Radulescu
We prove an existence result for a class of Dirichlet boundary value problems with discontinuous nonlinearity and involving a Leray-Lions operator. The proof combines monotonicity methods for elliptic problems, variational inequality techniques and basic tools related to monotone operators.
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We give sufficient conditions for when an autonomous elliptic Leray-Lions equation in the plane has a conformal structure. This allows the Leray-Lions equation to be linearised in a special form through the hodograph transform.
We show the various existence results for degenerate $p(x)$-Laplace equations with Leray-Lions type operators. A suitable condition on degeneracy is discussed and proofs are mainly based on direct methods and critical point theories in Calculus of Variations. In particular, we investigate the various situations of the growth rates between principal operators and nonlinearities.
Evolution problems of Leray-Lions type with nonhomogeneous Neumann boundary conditions in metric random walk spaces
本文研究度量随机游走空间中带非齐次Neumann边界条件的Leray-Lions型发展问题，覆盖加权离散图$p$-Laplacian算子与$\u211d^N$非奇异核非局部算子情形。
Nonuniqueness of Leray-Hopf solutions to the forced fractional Navier-Stokes Equations in three dimensions, up to the J. L. Lions exponent
本文证明在三维受迫分数阶Navier-Stokes方程中，当指数$\alpha\in(1/2,5/4)$时存在非唯一Leray-Hopf解，从而表明J. L. Lions指数的尖锐性。
Non-Leray-Hopf solutions to 3D stochastic hyper-viscous Navier-stokes equations: beyond the Lions exponents
研究三维随机超粘性Navier-Stokes方程，证明在超越Lions指数的高粘性区域，尽管Leray-Hopf类中解唯一，但在混合Lebesgue空间中存在无穷多个非Leray-Hopf解，并建立了噪声消失极限结果。
We investigate existence and multiplicity of weak solutions for fourth-order problems involving the Leray-Lions type operators in variable exponent spaces and improve a result of Bonanno and Chinnì (2011). We use variational methods and apply a multiplicity theorem of Bonanno and Marano (2010).
Classical Sobolev approach for a critical fourth-order Leray-Lions type problem: existence and multiplicity of solutions
本文研究Navier和Dirichlet边界条件下具Sobolev临界增长的Leray-Lions型四阶椭圆问题，结合变分方法和临界点理论，在经典Sobolev空间中证明次线性扰动下解的存在性与超线性扰动下至少一个非平凡解的存在性，并推广至哈密顿系统。
We prove the existence of at least three weak solutions for the fourth-order problem with indefinite weight involving the Leray-Lions operator with nonstandard growth conditions. The proof of our main result uses variational methods and the critical theorem of Bonanno and Marano (Appl. Anal. 89 (2010), 1-10).
We study existence and regularity of weak solutions for the following PDE $$ -\dive(A(x,u)|\nabla u|^{p-2}\nabla u) = f(x,u),\;\;\mbox{in $B_1$}. $$ where $A(x,s) = A_+(x)χ_{\{s>0\}}+A_-(x)χ_{\{s\le 0\}}$ and $f(x,s) = f_+(x)χ_{\{s>0\}}+f_-(x)χ_{\{s\le 0\}}$. Under the ellipticity assumption that $\frac{1}μ\le A_{\pm} \le μ$, $A_{\pm}\in C(Ø)$ and $f_{\pm}\in L^N(Ø)$, we prove that under appropriate conditions the PDE above admits a weak solution in $W^{1,p}(B_1)$ which is also $C^{0,α}_{loc}$ for every $α\in (0,1)$ with precise estimates. Our methods relies on similar techniques as those deve…
The Leray-Lions existence theorem under general growth conditions
本文将经典 Leray-Lions 存在性定理从 $p=q$ 的自然增长推广到一般 $p,q$-增长，处理算子对 $(x,u)$ 的显式依赖，并证明相应 Dirichlet 问题弱解的存在性。
Non-uniqueness of Leray-Hopf Solutions to Forced Stochastic Hyperdissipative Navier-Stokes Equations up to Lions Index
本文证明了带线性乘性噪声和特定外力的随机分数阶Navier-Stokes方程局部强解的非唯一性（含外力扰动情形），并通过新方法得到了某些随机外力下光滑的全局Leray-Hopf解的非唯一性结果。
A priori error estimates for a Hybrid High-Order method on variable-exponent Leray-Lions problems
该研究推导变指数Leray-Lions问题的混合高阶（HHO）离散化的先验误差估计，建立局部与全局误差估计，通过数值实验验证结果，表明通量正则性假设不可省略。
A Hybrid High-Order method for variable-exponent Leray-Lions problems with convergence under minimal regularity
针对变指数Leray-Lions问题，提出基于多面体网格的混合高阶（HHO）离散方法，证明其在最小正则性下收敛，数值实验验证了方法对奇异解及指数1-6的有效性。
Ground states to a quasilinear Schrödinger equation with Berestycki-Lions type nonlinearities
本文通过拓扑流形上的临界点理论，证明了具Berestycki-Lions型非线性项的拟线性薛定谔方程在N≥3时存在基态，N≥4时存在非径向基态，N=4或N≥6时存在无穷多个非径向解，推广了经典结果。
We deal with the existence of solutions having L2 regularity for a class of non autonomous evolution equations. Associated with the equation, a general non local condition is studied. The technique we used combines a finite dimensional reduction together with the Leray-Schauder continuation principle. This approach permits to consider a wide class of nonlinear terms by allowing demicontinuity assumptions on the nonlinearity.
We deal with the existence of infinitely many solutions for a class of elliptic problems with non-symmetric nonlinearities. Our result, which is motivated by a well known conjecture formulated by A. Bahri and P.L. Lions, suggests a new approach to tackle these problems. The proof is based on a method which does not require to use techniques of deformation from the symmetry and may be applied to more general non-symmetric problems.
Quasi-linear elliptic equations with superlinear convection
本文研究了涉及Leray-Lions算子和超线性对流项的拟线性椭圆方程解的存在性与正则性，涵盖p-Laplacian方程，并推广了Boccardo等人（2024年）的部分结果。
A classical result of J.-L. Lions asserts that if a solution to the Navier-Stokes equations is such that: (i) it is in the Leray-Hopf class, and (ii) belongs to $L^4(0,T;L^4)$, then it must satisfy the energy equality in the time interval $[0,T]$. In this note we show that assumption (i) is not necessary.
We prove a radial symmetry result for bounded nonnegative solutions to the $p$-Laplacian semilinear equation $-Δ_p u=f(u)$ posed in a ball of $\mathbb R^n$ and involving discontinuous nonlinearities $f$. When $p=2$ we obtain a new result which holds in every dimension $n$ for certain positive discontinuous $f$. When $p\ge n$ we prove radial symmetry for every locally bounded nonnegative $f$. Our approach is an extension of a method of P. L. Lions for the case $p=n=2$. It leads to radial symmetry combining the isoperimetric inequality and the Pohozaev identity.
Controlling the monotonicity and growth of Leray--Lions' operators including the $p$-Laplacian plays a fundamental role in the theory of existence and regularity of solutions to second order nonlinear PDE. We collect, correct, and supply known estimates including the discussion on the constants. Moreover, we provide a comprehensive treatment of related results for operators with Orlicz growth. We pay special attention to exposition of the proofs and the use of elementary arguments.
We propose existence and multiplicity results for the system of Schrödinger equations with sign-changing nonlinearities in bounded domains or in the whole space $\mathbb{R}^N$. In the bounded domain we utilize the classical approach via the Nehari manifold, which is (under our assumptions) a differentiable manifold of class $\mathcal{C}^1$ and the Fountain theorem by Bartsch. In the space $\mathbb{R}^N$ we additionally need to assume the $\mathbb{Z}^N$-periodicity of potentials and our proofs are based on the concentration-compactness lemma by Lions and the Lusternik-Schnirelmann values.
This is a survey on problems involving equations $-\operatorname{div}{\Cal A}(x,\nabla u)=μ$, where $μ$ is a Radon measure and ${\Cal A}:\bold {R}^n\times\bold R^n\to \bold R^n$ verifies Leray-Lions type conditions. We shall discuss a potential theoretic approach when the measure is nonnegative. Existence and uniqueness, and different concepts of solutions are discussed for general signed measures.
Existence Results for the Time-incremental Elastic Contact Problem with Coulomb Friction in 2D
本文研究了二维弹性接触问题中Coulomb摩擦的存在性，通过变分不等式和Leray-Lions算子理论，证明了各向同性和各向异性弹性下的解存在性。
We prove a threshold phenomenon for the existence of solitary solutions of the dispersion management equation for positive and zero average dispersion for a large class of nonlinearities. These solutions are found as minimizers of nonlinear and nonlocal variational problems which are invariant under a large non-compact group. There exists a threshold such that minimizers exist when the power of the solitons is bigger than the threshold. Our proof of existence of minimizers is rather direct and avoids the use of Lions' concentration compactness argument. The existence of dispersion managed soli…
The concentration compactness framework for semilinear elliptic equations without compactness, set originally by P.-L.Lions for constrained minimization in the case of homogeneous nonlinearity, is extended here to the case of general nonlinearities in the standard mountain pass setting of Ambrosetti-Rabinowitz. In these setting, existence of solutions at the mountain pass level is verified under a single penalty condition analogous to that in the Lions' case. Problems on the whole space and problems with critical nonlinearity are considered. Particular attention is given to nonhomogeneous crit…
In this paper we study the following class of fractional Kirchhoff problems: \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}M(\varepsilon^{2s-N}[u]^{2}_{s})(-Δ)^{s}u + V(x) u= f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $N\geq 2$, $(-Δ)^{s}$ is the fractional Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a positive continuous function, $M: [0, \infty)\rightarrow \mathbb{R}$ is a Kirchhoff function satisfying suitable co…
Measure Data for a General Class of Nonlinear Elliptic Problems
本文研究了一类右端带有测度的非线性椭圆包含问题，其中涉及极大单调图和Leray-Lions型算子，并在各向异性Sobolev空间中探讨了该问题的合适解的概念。
Existence and summability of solutions to nonlinear X-elliptic equations with measurable coefficients
该研究针对带可测系数的非线性退化X-椭圆方程，证明了零Dirichlet边界条件下解的存在性，并得到了解的推广L^p正则性结果，涉及满足Poincaré不等式和加倍条件的向量场相关散度型算子。
A quantitative symmetry result for $p$-Laplace equations with discontinuous nonlinearities
本文研究带有不连续非线性项的 $p$-Laplace 方程正解的 Gidas-Ni-Nirenberg 型对称性定量稳定性，利用 Pólya-Szegö 原理的定量版本证明了解与其 Schwarz 对称化的偏差可由区域的等周亏格进行估计。
Existence of normalized solutions to Choquard equation with general mixed nonlinearities
该研究针对含Berestycki-Lions型一般混合非线性项的Choquard方程，探讨其满足给定$L^2$范数约束的归一化解存在性，覆盖多种临界性情形，其中拉格朗日乘子λ为未知参数。
We show existence of small solitary and periodic traveling-wave solutions in Sobolev spaces ${\mathrm{H}^s}$, ${ s > 0 }$, to a class of nonlinear, dispersive evolution equations of the form \begin{equation*} u_t + \left(Lu+ n(u)\right)_x = 0, \end{equation*} where the dispersion ${L}$ is a negative-order Fourier multiplier whose symbol is of KdV type at low frequencies and has integrable Fourier inverse ${ K }$ and the nonlinearity ${n}$ is inhomogeneous, locally Lipschitz and of superlinear growth at the origin. This generalises earlier work by Ehrnström, Groves & Wahlén on a class of equati…
The global existence of weak solutions to a class of quasilinear parabolic equations with nonlinearities depending on first order terms and integrable data in a moving domain is investigated. The class includes the $p$-Laplace equation as a special case. Weak solutions are shown to be global by obtaining appropriate estimates on the gradient as well as a suitable version of Aubin-Lions lemma in moving domains.
We prove a threshold phenomenon for the existence/non-existence of energy minimizing solitary solutions of the diffraction management equation for strictly positive and zero average diffraction. Our methods allow for a large class of nonlinearities, they are, for example, allowed to change sign, and the weakest possible condition, it only has to be locally integrable, on the local diffraction profile. The solutions are found as minimizers of a nonlinear and nonlocal variational problem which is translation invariant. There exists a critical threshold ?cr such that minimizers for this variation…
In this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on Banach space with locally monotone operators, which is a generalization of the classical result by J.L. Lions for monotone operators. In particular, we show that local monotonicity implies the pseudo-monotonicity. The main result is applied to various types of PDE such as reaction-diffusion equations, generalized Burgers equation, Navier-Stokes equation, 3D Leray-$α$ model and $p$-Laplace equation with non-monotone perturbations.
Stochastic Lagrangian Flows for SDEs with rough coefficients
本文证明了具有粗糙系数的非退化 SDE 的随机拉格朗日流及几乎处处随机流的存在性与唯一性，并应用该结果证明了 3D Navier-Stokes 方程的每个 Leray-Hopf 解在 DiPerna-Lions 意义下对应唯一的随机流。
Liouville results for $(p,q)$-Laplacian elliptic equations with source terms involving gradient nonlinearities
本文针对含梯度非线性源项的非齐次 $(p,q)$-Laplacian 椭圆方程，利用 Ishii-Lions 技术与改进的 Bernstein 方法证明了有界解的常数性与梯度先验估计，从而建立 Liouville 型定理，推广了经典结果。
We present an order-theoretical fixed point theorem for increasing multivalued operators suitable for the method of sub-supersolutions and its application to the following multivalued quasi-variational inclusion: Let $Ω\subset \mathbb R^N$ be a bounded Lipschitz domain and $W = W_0^{1,p}(Ω)$. Find $u\in W$ such that for some measurable selection $η$ of $f(\cdot,u,u)$ it holds \begin{equation*} \langle Eu,w-u\rangle + \int_Ωη(w-u) + K(w,u) - K(u,u) \geq 0\quad\text{for all }w\in W, \end{equation*} where $E\colon W \to W^\ast$ is an elliptic Leray-Lions operator of divergence form, $f\colon Ω\ti…
In this paper we prove a multiplicity result concerning the critical points of a class of functionals involving local and nonlocal nonlinearities. We apply our result to the nonlinear Schrodinger-Maxwell system and to the nonlinear elliptic Kirchhoff equation assuming on the local nonlinearity the general hypotheses introduced by Berestycki and Lions.
We carry out an analysis of the existence of solutions for a class of nonlinear partial differential equations of parabolic type. The equation is associated to a nonlocal initial condition, written in general form which includes, as particular cases, the Cauchy multipoint problem, the weighted mean value problem and the periodic problem. The dynamic is transformed into an abstract setting and by combining an approximation technique with the Leray-Schauder continuation principle, we prove global existence results. By the compactness of the semigroup generated by the linear operator, we do not a…
Weak solutions to the parabolic $p$-Laplace equation in a moving domain under a Neumann type boundary condition
本文研究了移动域中抛物体p-拉普拉斯方程的弱解存在性与唯一性，通过Galerkin方法和单调性论证，并解决了非线性梯度项的弱极限问题。
In this article, we are interested in multi-bump solutions of the singularly perturbed problem \begin{equation*} -ε^2Δv+V(x)v=f(v) \ \ \mbox{ in }\ \ \R^N. \end{equation*} Extending previous results \cite{B, DLY,W1}, we prove the existence of multi-bump solutions for an optimal class of nonlinearities $f$ satisfying the Berestycki-Lions conditions and, notably, also for more general classes of potential wells than those previously studied. We devise two novel topological arguments to deal with general classes of potential wells. Our results prove the existence of multi-bump solutions in which …
In this paper, we consider a class of quasilinear stationary Kirchhoff type potential systems with Neumann Boundary conditions, which involves a general variable exponent elliptic operator with critical growth. Under some suitable conditions on the nonlinearities, we establish existence and multiplicity of solutions for the problem by using the concentration-compactness principle of Lions for variable exponents found in [5, 7] and the Mountain Pass Theorem without the Palais-Smale condition given in [43].
We prove well-posedness for doubly nonlinear parabolic stochastic partial differential equations of the form $dX_t-\text{div}\,γ(\nabla X_t)\,dt+β(X_t)\,dt\ni B(t,X_t)\,dW_t$, where $γ$ and $β$ are the two nonlinearities, assumed to be multivalued maximal monotone operators everywhere defined on $\mathbb{R}^d$ and $\mathbb{R}$ respectively, and $W$ is a cylindrical Wiener process. Using variational techniques, suitable uniform estimates (both pathwise and in expectation) and some compactness results, well-posedness is proved under the classical Leray-Lions conditions on $γ$ and with no restric…
Existence of solutions for a model of the Earth's magnetic field
本文研究了地球内核动力学的物理真实数学模型，并证明了该模型Leray-Hopf型弱解的存在性。模型结合了液态外核的磁流体动力学方程和内核的固体力学，将外核以外的部分视为完美绝缘体，由麦克斯韦方程描述。通过Galerkin近似法证明了解的存在性，并通过定义适当的函数空间和证明Biot-Savart型结果来控制非线性项。
Mosco-convergence of convex sets and unilateral problems for differential operators with lower order terms having natural growth
本文研究障碍型凸集上变分不等式解在 Mosco 收敛下的稳定性，证明含自然增长低阶项的 Leray-Lions 型算子问题的解对约束集收敛保持稳定，推广了经典稳定性结果。
Stochastic pseudomonotone parabolic obstacle problem: well-posedness $\&$ Lewy-Stampacchia's inequalities
研究了Leray-Lions型非线性伪单调算子主导的随机抛物障碍问题，通过奇异摄动、鞅解和Gyöngy-Krylov方法证明了强解的适定性，并以此为基础建立了变分不等式和Lewy-Stampacchia不等式。
A Regularized Interface Method for Fluid-Poroelastic Structure Interaction Problems with Nonlinear Geometric Coupling
针对具有低正则性界面和非线性几何耦合的流体-多孔弹性结构相互作用问题，提出正则化界面方法，证明其弱解存在性，填补非线性移动域设置下存在性理论空白。
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