Mohsen Khaleghi Moghadam, Mustafa Avci
In the present paper, by using variational method, the existence of non-trivial solutions to an anisotropic discrete non-linear problem involving p(k)-Laplacian operator with Dirichlet boundary condition is investigated. The main technical tools applied here are the two local minimum theorems for differentiable functionals given by Bonanno.
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Non-existence of positive weak solutions for some nonlinear (p,q)-Laplacian systems
本文研究一类非线性（p，q）-Laplacian系统.建立了该系统正弱解的不存在性结果。
Multiple solutions for Dirichlet nonlinear BVPs involving fractional Laplacian
利用Krasnosielskii不动点定理证明了锥中超线性积分方程至少存在两个解.将结果应用于具有分数拉普拉斯算子的Dirichlet边值问题。
Regularity for Fully Nonlinear P-Laplacian Parabolic Systems: the Degenerate Case
研究一类强耦合退化抛物方程组有界弱解的Hölder正则性.
In this paper we study the existence of positive smooth solutions for a class of singular (p(x),q(x))- Laplacian systems by using sub and supersolution methods.
Existence result for differential inclusion with p(x)-Laplacian
本文研究了具有p（x）-Laplacian（半变分不等式）的非线性椭圆问题.我们证明了非平凡解的存在性。我们的方法是基于临界点理论的局部Lipschitz泛函由于张。
Quasilinear Choquard equations involving N-Laplacian and critical exponential nonlinearity
该研究聚焦于含N-Laplacian与临界指数非线性项的拟线性Choquard方程，主要探讨了这类方程正解的存在性问题。
We consider the Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. We establish local well-posedness result in Orlicz spaces. We derive the existence of global solutions for small initial data. We obtain decay estimates for large time in Lebesgue spaces.
We establish the existence and uniqueness of variational solution to the nonlinear Neumann boundary problem for the $p^{th}$-Sub-Laplacian associated to a system of Hörmander vector fields
This paper is concerned with the study of a nonlinear problems involving the fractional p(x)-Laplacian operator. By means of the Berkovits degree theory, we prove the existence of nontrivial weak solutions for this problem. The appropriate functional framework for this problems is the fractional Sobolev spaces with variable exponent.
Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros
本文利用分歧方法研究了含零点线性/超线性非线性项的$p$ -Laplacian方程单符号解的存在性和多重性.为此，我们首先建立了具有齐次算子的非线性算子方程的无穷远分支定理。为了处理超线性情形，我们建立了几个涉及上极限的拓扑结果。
Existence of Nontrivial Solutions for p-Laplacian Equations in {R}^{N}
本文考虑{R}^{N}中具有变号势和亚临界p-超线性非线性项的p-Laplacian方程.利用Degiovanni和Lancelotti在2007年提出的锥的上同调连接方法，得到了一个存在性结果。我们还给出了一维$p$ -Laplacian方程周期解的存在性的一个结果，该结果可以用同样的方法得到证明.
We obtain fundamental imbeddings for the fractional Sobolev space with variable exponent that is a generalization of well-known fractional Sobolev spaces. As an application, we obtain a-priori bounds and multiplicity of solutions to some nonlinear elliptic problems involving the fractional $p(\cdot)$-Laplacian.
Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials
对于右手边带有含时Radon测度的发展$p$ -Laplace方程的弱解，我们通过非线性抛物势得到了点态估计.
Positive solutions of nonlinear problems involving the square root of the Laplacian
本文考虑非局部算子的非线性椭圆问题：在Dirichlet边界条件为零的有界区域上Laplacian算子的平方根。对于具有幂非线性项的问题，我们建立了正解的存在性和正则性结果，以及Gidas-Spruck型的先验估计.此外，在其他结果中，我们证明了Gidas-Ni-Nirenberg型对称定理。
We consider an unsteady thermistor system with a p-Laplace type equation for the electrostatic potential.
Nonlinear Schrödinger equation for the twisted Laplacian
在一阶索伯列夫空间中建立了$\C^n$上与扭曲拉普拉斯算子相关联的非线性薛定谔方程解的局部适定性.我们的方法是基于Eschenhartz型估计，是有效的一般类的非线性，包括功率型。$n=1$的情况下，代表磁薛定谔方程在平面与磁位$A(z)=iz, z\in\C$。
We prove the existence of weak solutions to an unsteady thermistor system with p-Laplacian type equation for the electrostatic potential, where the electrical conductivity of the thermistor material is a monotone function of the electric field density.
In this paper, we study the existence of three solutions for a Kirchhoff equation involving the nonlocal fractional p-Laplacian considering Sobolev and Hardy nonlinearities at subcritical and critical growths. The proof is based on Mountain Pass Theorem, Ekeland's variational principle and constrained minimization in Nehari sets.
Gradient estimate and a Liouville theorem for a $p$-Laplacian evolution equation with a gradient nonlinearity
本文建立了一类具有快速增长梯度非线性项的$p$ -Lpalacian方程的局部梯度估计.利用这个估计，我们可以证明一个抛物型Liouville定理的古代解决方案，满足一些增长限制在无穷远。
We consider a parabolic-type PDE with a diffusion given by a fractional Laplacian operator and with a quadratic nonlinearity of the 'gradient' of the solution, convoluted with a singular term b. Our first result is the well-posedness for this problem: We show existence and uniqueness of a (local in time) mild solution. The main result is about blow-up of said solution, and in particular we find sufficient conditions on the initial datum and on the term b to ensure blow-up of the solution in finite time.
The paper deals with the existence and uniqueness of a non-trivial solution to non-homogeneous $ p ( x ) -$laplacian equations, managed by non polynomial growth operator in the framework of variable exponent Sobolev spaces on Riemannian manifolds. The mountain pass Theorem is used.
We consider a class of parametric Schrödinger equations driven by the fractional $p$-Laplacian operator and involving continuous positive potentials and nonlinearities with subcritical or critical growth. By using variational methods and Ljusternik-Schnirelmann theory, we study the existence, multiplicity and concentration of positive solutions for small values of the parameter.
A note on the existence of a unique positive entropy solution to a fractional Laplacian with singular nonlinearities
本文证明了带非线性奇异项及非负有界 Radon 测度源项的分数阶 Laplacian 问题存在唯一的正熵解。
Nonlinear perturbations of a $p(x)$-Laplacian equation with critical growth in $\mathbb{R}^N$
证明了一类非线性项具有临界增长的$p(x)$ -Laplacian方程解的存在性。在这里，我们考虑两种情况：第一种情况涉及的情况下，变量指数是周期函数。第二个涉及的情况下，变指数是非周期扰动。
Nontrival solution for nonlinear p(x)-Laplacian Dirichlet problem with the sign-changing weight
本文研究了具有非光滑位势的p（x）-Laplacian算子的非线性椭圆问题，其中权函数可以变号.利用Chang关于局部Lipschitz泛函的临界点理论，得到了问题解的存在性条件.
Propagation of low regularity for solutions of nonlinear PDEs on a Riemannian manifold with a sub-Laplacian structure
继\cite{B2}之后，我们引入了与半群相关联的仿积的概念。我们不使用傅里叶变换的参数和背景流形是加倍的，赋予一个sub-Laplacian结构。我们的主要结果是在非欧几里德框架下的非线性化定理，并应用于某些非线性偏微分方程的正则性传播。
Hydrodynamic Limit of Nonlinear Diffusions with Fractional Laplacian Operators
在包含分数拉普拉斯算子的非线性扩散模型的极限下，我们得到了一个产生于超导和超流中的“平均场”方程。对于这个方程，我们得到了唯一性、泛界和正则性结果。我们还表明，解决方案与有限的二阶矩和径向解决方案承认一个渐近大的时间限制的配置文件，这是一个特殊的自相似的解决方案：“基本涡补丁”。
We consider an optimal control problem governed by a class of boundary value problem with the spectral Dirichlet fractional Laplacian. Some sufficient condition for the existence of optimal processes is stated. The proof of the main result relies on variational structure of the problem. To show that boundary value problem with the Dirichlet fractional Laplacian has a weak solution we employ the renowned Ky Fan Theorem.
Existence of a nontrival solution for Dirichlet problem involving p(x)-Laplacian
本文研究具有非光滑位势的p（x）-Laplacian（半变分不等式）非线性狄利克雷问题.利用Chang关于局部Lipschitz泛函的非光滑临界点理论和变分索伯列夫空间的性质，建立了问题解的存在性条件.
Infinitely many non-radial sign-changing solutions for a Fractional Laplacian equation with critical nonlinearity
本文研究了具有纯临界非线性项的分数阶Laplacian问题\begin{equation*} \left\{ \begin{array}{ll} (-Δ)^{s} u=|u|^{\frac{4s}{N-2s}}u, &\mbox{in}\ \mathbb{R}^N, \\ u\in \mathcal{D}^{s,2}(\mathbb{R}^N), \end{array} \right. \end{equation*}，其中$s\in (0,1)$，$N$是正整数，$N\geq 3$，$(-Δ)^{s}$是分数阶Laplacian算子.我们将证明这个问题有无穷多个非径向变号解。
Nonlinear variable exponent Picone identity and $p(x)$-sub-Laplacian first eigenvalue for general vector fields
本文建立$p(x)$-次拉普拉斯的广义非线性变指数Picone恒等式，并据此证明其第一非平凡Dirichlet特征值的唯一性、简单性、单调性与孤立性，同时推导出变指数Hardy型不等式与Caccioppoli估计。
Existence for doubly nonlinear fractional $p$-Laplacian equations
该研究证明了含一般双重非线性（涵盖Sobolev各情形及快慢扩散）的抛物型分数阶$p$-Laplacian方程弱解的存在性，并揭示了对应算子的弱收敛方法。
We propose analogues of Green's and Picone's identities for the $p$-sub-Laplacian on stratified Lie groups. In particular, these imply a generalised Díaz-Saá inequality for the $p$-sub-Laplacian on stratified Lie groups. Using these we derive a comparison principle and uniqueness of a positive solution to nonlinear hypoelliptic equations on general stratified Lie groups extending to the setting of the general stratified Lie groups previously known results on Euclidean and Heisenberg groups.
We develop some properties of the $p-$Neumann derivative for the fractional $p-$Laplacian in bounded domains with general $p>1$. In particular, we prove the existence of a diverging sequence of eigenvalues and we introduce the evolution problem associated to such operators, studying the basic properties of solutions. Finally, we study a nonlinear problem with source in absence of the Ambrosetti-Rabinowitz condition.
Ground state sign-changing homoclinic solutions for a discrete nonlinear $p$-Laplacian equation with logarithmic nonlinearity
本文通过直接非Nehari流形方法，证明了一类带对数非线性项的离散非线性$p$-Laplacian方程存在仅变号一次的基态变号同宿解和基态同宿解，并指出变号基态能量大于基态能量的两倍。
ABP estimate and Harnack inequality for a class of degenerate fully nonlinear pseudo-$p$-Laplacian equations
本文研究了一类退化完全非线性伪p-拉普拉斯方程的ABP估计与Harnack不等式，采用滑动抛物面方法与各向异性函数相结合的方法进行证明。
We study a Dirichlet-type boundary value problem for a pseudodifferential equation driven by the fractional Laplacian, proving the existence of three nonzero solutions. When the reaction term is sublinear at infinity, we apply the second deformation theorem and spectral theory. When the reaction term is superlinear at infinity, we apply the mountain pass theorem and Morse theory.
Singular Phenomena of Solutions for Nonlinear Diffusion Equations involving $p(x)$-\hbox{Laplacian} Operator
研究了含$p(x)$ - \hbox{Laplacian}算子和非线性源的非线性扩散方程齐次$\hbox{Dirichlet}$边值问题解的奇异现象（在有限时间内消失和爆破）.讨论了变指数$p(x)$和初始能量（数据）对解的性质的影响。同时，我们得到了解的临界熄灭指数和爆破指数。
Existence of solutions for a class of integro-differential equations with the logarithmic Laplacian and transport
该研究针对L²(R)空间中含漂移项对数拉普拉斯算子的积分微分方程，利用对应线性算子的Fredholm性质，通过不动点方法建立了解的存在性。
On nonlinear potential theory, and regular boundary points, for the p-Laplacian in N space variables
我们回到一些开拓性的结果，特别是非线性位势论和非齐次边值问题的所谓的p-Laplacian算子。不幸的是，这些结果，在七十年代初获得的，被保存在阴凉处。我们相信，我们的证明仍然感兴趣，特别是由于他们的极端简单。此外，一些贡献似乎改善了在目前的文献中引用的结果。
Existence of two solutions to a parametric singular quasi-linear elliptic problem is proved. The equation is driven by the Φ-Laplacian operator and the reaction term can be non-monotone. The main tools employed are a local minimum theorem and the Mountain Pass theorem, together with the truncation technique. Global C^{1,τ} regularity of solutions is also investigated, chiefly via a priori estimates and perturbation techniques.
We study a Dirichlet problem driven by the degenerate fractional p-Laplacian and involving a nonlinear reaction, which depends on a positive parameter. The reaction is assumed to be (p-1)-sublinear near the origin and (p-1)-superlinear at infinity (including the concave-convex case). Following a variational approach based on a combination of critical point theory and suitable truncation techniques, we prove a bifurcation-type result for the existence of positive solutions.
Quasilinear parabolic problem with $p(x)$-Laplacian: existence, uniqueness of weak solutions and stabilization
讨论了拟线性抛物型方程$u_t-Δ_{p(x)}u = f(x,u)$在$ (0,T)\timesΩ$中，$u = 0$在$(0,T)\times\partialΩ$中，$u(0,x)=u_0(x)$在$Ω$中弱解的存在唯一性，其中涉及到$p(x)$ -Laplacian算子.接下来，我们讨论的整体行为的解决方案，特别是一些稳定性。
Nonlinear Schrödinger equation for the twisted Laplacian in the critical case
我们证明了非线性薛定谔方程的解的适定性，该方程与$\C^n$上的扭曲拉普拉斯算子相关联，对于一般的非线性类，包括次临界情况下的幂型$0\leq α<\frac{2}{n-1}$，参见Ratnakumar，Sohani（J. Funct. Anal. 2013年）的报告。在本文中，我们考虑临界情况$α=\frac{2}{n-1}$与$n\geq 2$。我们的方法是基于截断给定的非线性$G$，这是由Cazenave Weissler（1989）使用。我们得到的截断问题的解决方案。我们通过求极限得到原问题的解.
In this paper we study double phase problems with nonlinear boundary condition and gradient dependence. Under quite general assumptions we prove existence results for such problems where the perturbations satisfy a suitable behavior in the origin and at infinity. Our proofs make use of variational tools, truncation techniques and comparison methods. The existence of the obtained solutions depends on the first eigenvalues of the Robin and Steklov eigenvalue problems for the $p$-Laplacian.
In this paper, we consider a non-local diffusion equation involving the fractional $p(x)$-Laplacian with nonlinearities of variable exponent type. Employing the sub-differential approach we establish the existence of local solutions. By combining the potential well theory with the Nehari manifold, we obtain the existence of global solutions and finite time blow-up of solutions. Moreover, we study the asymptotic stability of global solutions as time goes to infinity in some variable exponent Lebesgue spaces.
Existence of Positive Solutions for Non Local p-Laplacian Thermistor Problems on Time Scales
利用锥上的Guo-Krasnoselskii不动点定理，证明了一类时间尺度上的非局部p-Laplacian边值问题正解的存在性.
In this paper, first we study existence results for a linearly perturbed elliptic problem driven by the fractional Laplacian. Then, we show a multiplicity result when the perturbation parameter is close to the eigenvalues. This latter result is obtained by exploiting the topological structure of the sublevels of the associated functional, which permits to apply a critical point theorem of mixed nature due to Marino and Saccon.
The main goal of this paper is the study of two kinds of nonlinear problems depending on parameters in unbounded domains. Using a nonstandard variational approach, we first prove the existence of bounded solutions for nonlinear eigenvalue problems involving the fractional Laplace operator and nonlinearities that have subcritical growth. In the second part, based on a variational principle of Ricceri [16], we study a fractional nonlinear problem with two parameters and prove the existence of multiple solutions.
Existence results for fractional p-Laplacian problems via Morse theory
我们研究一类拟线性非局部问题，包括作为一个特殊的情况下，半线性问题，涉及分数拉普拉斯算子和连续介质力学，相变现象，人口动力学和对策论的框架中产生的。在反应项的不同增长假设下，利用变分方法和拓扑方法，特别是莫尔斯理论，我们得到了不同的存在性和有限重数结果.
Nonlocal elliptic equations involving logarithmic Laplacian: Existence, non-existence and uniqueness results
本文研究涉及对数拉普拉斯算子的非局部椭圆方程的存在性、非存在性与唯一性，证明Pohozaev恒等式和Díaz-Saa型不等式，并揭示加权非局部问题的极小能量解在超线性和次线性增长下分别收敛于Brézis-Nirenberg型和逻辑型极限问题的解。
Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian
本文研究了一类具有分数阶Laplacian算子和临界指数的非线性椭圆型方程最小能量解的渐近性态和多个冒泡解的存在性。这项工作可以看作是一个非局部模拟的结果韩（安。庞加莱肛门。Non Lineaire，1991）和Rey（J. Funct.肛门，1990年）。
In this article we are interested in the following fractional $p$-Laplacian equation in $\mathbb{R}^n$ \begin{eqnarray*} &(-Δ)_{p}^αu + V(x)u^{p-2}u = f(x,u) \mbox{ in } \mathbb{R}^{n}, \end{eqnarray*} where $p\geq 2$, $0< s < 1$, $n\geq 2$ and subcritical p-superlinear nonlinearity. By using mountain pass theorem with Cerami condition we prove the existence of nontrivial solution. Furthermore, we show that this solution is radially simmetry.
Multiplicity Results for the Nonlinear Weighted Laplacian Equation on the Sierpiński Gasket
研究 Sierpiński 垫片上加权拉普拉斯方程 Dirichlet 边值问题，通过直接法、山路定理及相关多重性定理，在非线性项满足适当增长条件下给出解的存在性与多重性结果。
We study a Dirichlet type problem for an equation involving the fractional Laplacian and a reaction term subject to either subcritical or critical growth conditions, depending on a positive parameter. Applying a critical point result of Bonanno, we prove existence of one or two positive solutions as soon as the parameter lies under a (explicitly determined) threshold. As an application, we find two positive solutions for a fractional Brezis-Nirenberg problem.
We prove existence and multiplicity results for a $N$-Laplacian problem with a critical exponential nonlinearity that is a natural analog of the Brezis-Nirenberg problem for the borderline case of the Sobolev inequality. This extends results in the literature for the semilinear case $N = 2$ to all $N \ge 2$. When $N > 2$ the nonlinear operator $- Δ_N$ has no linear eigenspaces and hence this extension requires new abstract critical point theorems that are not based on linear subspaces. We prove new abstract results based on the ${\mathbb Z}_2$-cohomological index and a related pseudo-index tha…
Decay of mass for nonlinear equation with fractional Laplacian
研究了反应扩散方程$\partial_t u=-(-Δ)^{α/2}u - u^p,$ $(α\in(0,2], p>1)$在$\mathbb{R}^N$上的可积初值条件下非负解的大时间行为.我们证明了反常扩散项决定了$p>1+α/{N},$的大时间渐近性，而非线性效应在$p\leq1+α/{N}.$
We consider the nonlinear half laplacian heat equation $$ u_t+(-Δ)^{\frac{1}{2}} u-|u|^{p-1}u=0,\quad \mathbb{R}^n\times (0, T). $$ We prove that all blows-up are type I, provided that $n \leq 4$ and $ 1<p<p_{*} (n)$ where $ p_{*} (n)$ is an explicit exponent which is below $\frac{n+1}{n-1}$, the critical Sobolev exponent. Central to our proof is a Giga-Kohn type monotonicity formula for half laplacian and a Liouville type theorem for self-similar nonlinear heat equation. This is the first instance of a monotonicity formula at the level of the nonlocal equation, without invoking the extension …
Existence of Three Positive Solutions to Some $p$-Laplacian Boundary Value Problems
利用Leggett-Williams不动点定理，得到了一类时间尺度上$p$ -Laplacian边值问题至少存在三个正解的充分条件.
We obtain the global weighted $W^{1,p}$ estimates for weak solutions of nonlinear elliptic equations over Reifenberg flat domains. Where nonlinearity $A(x,z,ξ)$ is assumed to be local uniform continuous in $z$ and have small BMO semi-norm in $x$. Moreover, we derive Besov regularity for solutions of a class of special harmonic equations by making use of $W^{1,p}$ estimate. Keywords: global weighted $W^{1,p}$ estimates; quasilinear equations; Besov regularity
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