On the Gibbons' conjecture for equations involving the $p$-Laplacian

A Curious Equation Involving the Infinity-Laplacian

We prove the uniqueness for viscosity solutions of a differential equation involving the infinity-Laplacian with a variable exponent. A version of the Harnack's inequality is derived for this minimax problem.

Multiple solutions for Dirichlet nonlinear BVPs involving fractional Laplacian

The existence of at least two solutions to superlinear integral equation in cone is proved using the Krasnosielskii Fixed Point Theorem. The result is applied to the Dirichlet BVPs with the fractional Laplacian.

Lipschitz Continuity of Solutions to a Free Boundary Problem involving the A-Laplacian

In this paper we prove local interior and boundary Lipschitz continuity of solutions of a free boundary problem involving the $A$-Laplacian. We also show that the free boundary is represented locally by graphs of a family of lower semi-continuous functions.

A concave-convex elliptic problem involving the fractional Laplacian

We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave-convex term. We characterize completely the range of parameters for which solutions of the problem exist and prove a multiplicity result.

涉及N-Laplacian和临界指数非线性项的拟线性Choquard方程

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。

A sub-super solution method for a class of nonlocal problems involving the p(x)-Laplacian operator and applications

In the present paper we study the existence of solutions for some nonlocal problems involving the p(x)-Laplacian operator. The approach is based on a new sub-supersolution method

Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros

In this paper, we use bifurcation method to investigate the existence and multiplicity of one-sign solutions of the $p$-Laplacian involving a linear/superlinear nonlinearity with zeros. To do this, we first establish a bifurcation theorem from infinity for nonlinear operator equation with homogeneous operator. To deal with the superlinear case, we establish several topological results involving superior limit.

Nonlinear problem involving the fractional p(x)-Laplacian operator by Topological degree

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.

Semilinear equations involving the fractional Laplacian on domains

Applying the method of moving planes in integral forms, we establish radial symmetry for positive solutions to a class of semilinear equations involving the fractional Laplacian in the unit ball and obtain Liouville type theorems concerning the non-existence of positive solutions in the half space. The regularity of solutions are also investigated.

A-priori bounds and multiplicity of solutions for nonlinear elliptic problems involving the fractional $p(\cdot)$-Laplacian

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.

A sub-supersolution method for a class of nonlocal system involving the p(x)-Laplacian operator and applications

In the present paper we study the existence of solutions for a class of nonlocal system involving the p(x)-Laplacian operator. The approach is based on a new sub-supersolution result.

Extremal functions for an embedding from some anisotropic space, and partial differential equation involving the "one Laplacian"

In this paper, we prove the existence of extremal functions for the best constant of embedding from anisotropic space, allowing some of the Sobolev exponents to be equal to $1$. We prove also that the extremal functions satisfy a partial differential equation involving the $1$ Laplacian.

On strongly indefinite systems involving the fractional Laplacian

In this paper we study strongly indefinite systems involving the fractional Laplacian on bounded domains. We obtain existence and non-existence results, $a priori$ estimates of Gidas-Spruck type, and the symmetric property.

Uniqueness of minimal energy solutions for a semilinear problem involving the fractional laplacian

In this paper we study a semilinear problem for the fractional laplacian that are the counterpart of the Neumann problems in the classical setting. We show uniqueness of minimal energy solutions for small domains.

Existence of a nontrival solution for Dirichlet problem involving p(x)-Laplacian

In this paper we study the nonlinear Dirichlet problem involving p(x)-Laplacian (hemivariational inequality) with nonsmooth potential. By using nonsmooth critical point theory for locally Lipschitz functionals due to Chang and the properties of variational Sobolev spaces, we establish conditions which ensure the existence of solution for our problem.

Algebraic topological techniques for elliptic problems involving fractional Laplacian

We prove the existence of infinitely many solutions to an elliptic problem by borrowing the techniques from algebraic topology. The solution(s) thus obtained will also be proved to be bounded.

Large solutions to elliptic equations involving fractional Laplacian

In this article, we study the boudary blow-up solutions for semilinear fractional equations with power absorption. Our main purpose is to obtain the existence, nonuniqueness and behavior asymptotic near the boundary.

关于一个涉及1-Laplacian的双相Dirichlet问题的解

On the solutions of a double-phase Dirichlet problem involving the 1-Laplacian

针对含1-Laplacian的非齐次Dirichlet双相问题，论文证明了弱解的存在唯一性，并通过相应极小化问题给出其变分刻画。

Signed and sign-changing solutions for a class Kirchhoff-type problem involving the fractional p-Laplacian with critical Hardy nonlinearity

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.

Positive solutions of nonlinear problems involving the square root of the Laplacian

We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type.

涉及分数阶拉普拉斯算子的半线性方程的反问题

An inverse problem for semilinear equations involving the fractional Laplacian

本文研究带零阶非线性扰动的分数阶拉普拉斯热方程与波动方程反问题，利用分数阶 Dirichlet-to-Neumann 映射、Runge 逼近和唯一延拓性质恢复非线性项。

含$p$-Laplacian算子的Choquard方程的正则性与Pohozaev恒等式

Regularity and Pohozaev identity for the Choquard equation involving the $p$-Laplacian operator

该研究针对含$p$-Laplacian算子的非线性Choquard方程，开展了弱解正则性分析并建立了Pohozaev型恒等式，丰富了相关方程的理论结果。

Segregated configurations involving the square root of the laplacian and their free boundaries

We study the local structure and the regularity of free boundaries of segregated critical configurations involving the square root of the laplacian. We develop an improvement of flatness theory and, as a consequence of this and Almgren's monotonicity formula, we obtain partial regularity (up to a small dimensional set) of the nodal set, thus extending the known resultsfor the standard diffusion to some anomalous case.

Nonnegative entire bounded solutions to some semilinear equations involving the fractional Laplacian

The main goal is to establish necessary and sufficient conditions under which the fractional semilinear elliptic equation $Δ^{\fracα{2}} u=ρ(x)\,φ(u)$ admits nonnegative nontrivial bounded solutions in the whole space $\mathbb{R}^N$.

On a quasilinear elliptic problem involving the 1-laplacian operator and a discontinuous nonlinearity

In this work, we study a quasilinear elliptic problem involving the 1-laplacian operator, with a discontinuous, superlinear and subcritical nonlinearity involving the Heaviside function $H(\cdot - β)$. Our approach is based on an analysis of the associated p-laplacian problem, followed by a thorough analysis of the asymptotic behaviour or such solutions as $p \to 1^+$. We study also the asymptotic behaviour of the solutions, as $β\to 0^+$ and we prove that it converges to a solution of the original problem, without the discontinuity in the nonlinearity.

On a class of Kirchhoff-Choquard equations involving variable-order fractional $p(\cdot)-$ Laplacian and without Ambrosetti-Rabinowitz type condition

In this article we study the existence of weak solution, existence of ground state solution using Nehari manifold and existence of infinitely many solutions using Fountain theorem and Dual fountain theorem for a class of doubly nonlocal Kirchhoff-Choquard type equations involving the variable-order fractional $p(\cdot)-$ Laplacian operator. Here the nonlinearity does not satisfy the well known Ambrosetti-Rabinowitz type condition.

Least energy sign-changing solution of fractional $p$-Laplacian problems involving singularities

In this paper we study the existence of a least energy sign-changing solution to a nonlocal elliptic PDE involving singularity by using the Nehari manifold method, the constraint variational method and Brouwer degree theory.

$p$-Laplacian problems involving critical Hardy-Sobolev exponents

We prove existence, multiplicity, and bifurcation results for $p$-Laplacian problems involving critical Hardy-Sobolev exponents. Our results are mainly for the case $λ\ge λ_1$ and extend results in the literature for $0 < λ< λ_1$. In the absence of a direct sum decomposition, we use critical point theorems based on a cohomological index and a related pseudo-index.

Symmetry of positive solutions for Lane-Emden systems involving the Logarithmic Laplacian

We study the Lane-Emden system involving the logarithmic Laplacian: $$ \begin{cases} \ \mathcal{L}_Δu(x)=v^{p}(x) ,& x\in\mathbb{R}^{n},\\ \ \mathcal{L}_Δv(x)=u^{q}(x) ,& x\in\mathbb{R}^{n}, \end{cases} $$ where $p,q>1$ and $\mathcal{L}_Δ$ denotes the Logarithmic Laplacian arising as a formal derivative $\partial_s|_{s=0}(-Δ)^s$ of fractional Laplacians at $s=0.$ By using a direct method of moving planes for the logarithmic Laplacian, we obtain the symmetry and monotonicity of the positive solutions to the Lane-Emden system. We also establish some key ingredients needed in order to apply the m…

Symmetry of Solutions to Semilinear Equations Involving the Fractional Laplacian on $\mathbb{R}^n$ and $\mathbb{R}^n_+$

Let $0<α<2$ be any real number. In this paper, we investigate the following semilinear equations involving the fractional Laplacian \begin{equation}(-\bigtriangleup)^{α/2} u(x)=f(u),\end{equation} on $\mathbb{R}^n$ and $\mathbb{R}^n_+$. Applying a direct method of moving planes for the fractional Laplacian, we prove symmetry and nonexistence of positive solutions on $\mathbb{R}^n$ and $\mathbb{R}^n_+$ under mild conditions on $f$.

Faber-Krahn inequality for Robin problem involving p-Laplacian

The eigenvalue problem for the p-Laplace operator with Robin boundary condition is considered in this paper. A Faber-Krahn type inequality is proved. More precisely, it is shown that amongst all the domains of fixed volume, the ball has the smallest first eigenvalue.

涉及对数拉普拉斯算子的狄利克雷问题的最优边界正则性与Hopf型引理

Optimal boundary regularity and a Hopf-type lemma for Dirichlet problems involving the logarithmic Laplacian

本文研究涉及对数拉普拉斯算子的狄利克雷问题，利用Kelvin变换构造障碍函数证明最优边界正则性，并由此得到非负弱解的Hopf型引理及非线性问题解的唯一性。

The Nehari manifold for fractional p-Laplacian system involving concave-convex nonlinearities and sign-changing weight functions

In this paper, we consider a fractional p-Laplacian system with both concave-convex nonlinearities and sign-changing weight functions in bounded domains. With the help of the Nehari\ manifold, we prove that the system has at least two nontrivial solutions when the pair of the parameters $(λ,μ)$ belongs to a certain subset of $\mathbb{R}^{n}$

On the bounds of the sum of eigenvalues for a Dirichlet problem involving mixed fractional Laplacians

In this paper, we show the existence of a sequence of eigenvalues for a Dirichlet problem involving two mixed fractional operators with different orders. We provide lower and upper bounds for the sum of the eigenvalues. Applications of mixed fractional operators with different orders include medicine, plasma physics, and population dynamics.

Global existence and blow-up of solutions for a parabolic equation involving the fractional $p(x)$-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.

Elliptic equations involving the $1$-Laplacian and a subcritical source term

In this paper we deal with a Dirichlet problem for an elliptic equation involving the $1$-Laplacian operator and a source term. We prove that, when the growth of the source is subcritical, there exist two bounded nontrivial solutions to our problem. Moreover, a Pohozaev type identity is proved, which holds even when the growth is supercritical. We also show explicit examples of our results.

Singular Phenomena of Solutions for Nonlinear Diffusion Equations involving $p(x)$-\hbox{Laplacian} Operator

The authors of this paper study singular phenomena(vanishing and blowing-up in finite time) of solutions to the homogeneous $\hbox{Dirichlet}$ boundary value problem of nonlinear diffusion equations involving $p(x)$-\hbox{Laplacian} operator and a nonlinear source. The authors discuss how the value of the variable exponent $p(x)$ and initial energy(data) affect the properties of solutions. At the same time, we obtain the critical extinction and blow-up exponents of solutions.

非局部狄利克雷问题中的对数 $p$-拉普拉斯算子

Nonlocal Dirichlet problems involving the Logarithmic $p$-Laplacian

本文研究了非局部狄利克雷问题中对数$p$-拉普拉斯算子的极小特征值存在性，并利用$p$-对数赛博利夫不等式证明了新存在性结果。

A priori estimates for some elliptic equations involving the $p$-Laplacian

We consider the Dirichlet problem for positive solutions of the equation $-Δ_p (u) = f(u)$ in a convex, bounded, smooth domain $Ω\subset\R^N$, with $f$ locally Lipschitz continuous. \par We provide sufficient conditions guarantying $L^{\infty} $ a priori bounds for positive solutions of some elliptic equations involving the $p$-Laplacian and extend the class of known nonlinearities for which the solutions are $L^{\infty} $ a priori bounded. As a consequence we prove the existence of positive solutions in convex bounded domains.

Uniform Holder bounds for strongly competing systems involving the square root of the laplacian

For a class of competition-diffusion nonlinear systems involving the square root of the Laplacian, including the fractional Gross-Pitaevskii system, we prove that uniform boundedness implies Holder boundedness for every exponent less than 1/2, uniformly as the interspecific competition parameter diverges. Moreover we prove that the limiting profile is Holder continuous of exponent 1/2. This system arises, for instance, in the relativistic Hartree-Fock approximation theory for mixtures of Bose-Einstein condensates in different hyperfine states.

一种涉及 $p$-拉普拉斯算子的局部与非局部耦合模型

A local and nonlocal coupling model involving the $p$-Laplacian

本文提出一种结合局部和非局部 $p$-拉普拉斯算子的耦合模型，并通过能量泛函的直接最小化证明了解的存在性和唯一性。

An obstacle control problem involving the p-Laplacian

In this paper, we consider the analogous of the obtacle problem in $H_0^1(Ω)$, on the space $W^{1,p}_0(Ω)$. We prove an existence and uniqueness of the result. In a second time, we define the optimal control problem associated. The results, here enclosed, generalize the one obtained by D.R. Adams, S. Lenhard

Symmetry and monotonicity of singular solutions to $p$-Laplacian systems involving a first order term

We consider positive singular solutions (i.e. with a non-removable singularity) of a system of PDEs driven by $p$-Laplacian operators and with the additional presence of a nonlinear first order term. By a careful use of a rather new version of the moving plane method, we prove the symmetry of the solutions. The result is already new in the scalar case.

Sign-changing solutions for the stationary Kirchhoff problems involving the fractional Laplacian in \mathcal{R}^N

In this paper, we study the existence of least energy sign-changing solutions for a Kirchhoff-type problem involving the fractional Laplacian operator. By using the constraint variational method and quantitative deformation lemma, we obtain a least energy nodal solution ub for the given problem. Moreover, we show that the energy of ub is strictly larger than twice the ground state energy. We also give a convergence property of u_b as b goes to 0, where b is regarded as a parameter.

Regularity of radial minimizers of reaction equations involving the p-Laplacian

We consider semi-stable, radially symmetric, and decreasing solutions of a reaction equation involving the p-Laplacian, where the reaction term is a locally Lipschitz function, and the domain is the unit ball. For this class of radial solutions, which includes local minimizers, we establish pointwise and Sobolev estimates which are optimal and do not depend on the specific nonlinear reaction term. Under standard assumptions we also prove the regularity of the corresponding extremal solution.

The Brezis-Nirenberg problem for the fractional $p$-Laplacian involving critical Hardy-Sobolev exponents

We obtain existence, multiplicity, and bifurcation results for the Brezis-Nirenberg problem for the fractional $p$\nobreakdash-Laplacian operator, involving critical Hardy-Sobolev exponents. Our results are mainly extend results in the literature for $α=0$. In the absence of an explicit formula for a minimizer in the fractional Hardy-Sobolev inequality $α\not= 0$, we get around this difficulty by working with certain asymptotic estimates for minimizers recently obtained in \cite{MM}.

p-Laplacian type equations involving measures

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.

On asymptotic behavior for a class of diffusion equations involving the fractional $\wp(\cdot)-$Laplacian as $\wp(\cdot)$ goes to $\infty$

In this manuscript, we will study the asymptotic behavior for a class of nonlocal diffusion equations associated with the weighted fractional $\wp(\cdot)-$Laplacian operator involving constant/variable exponent. In the case of constant exponents, under some appropriate conditions, we will study the existence of solutions and asymptotic behavior of solutions by employing the subdifferential approach and we will study the problem when $\wp$ goes to $\infty$. Already, for case the weighted fractional $\wp(\cdot)$-Laplacian operator, we will also study the asymptotic behavior of the problem soluti…

Existence of solutions for a class of $p(x)$-laplacian equations involving a concave-convex nonlinearity with critical growth in $\mathbb{R}^{N}$

We prove the existence of solutions for a class of quasilinear problems involving variable exponents and with nonlinearity having critical growth. The main tool used is the variational method, more precisely, Ekeland's Variational Principle and the Mountain Pass Theorem.

涉及对数拉普拉斯算子的非局部椭圆方程：存在性、非存在性与唯一性结果

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。

Qualitative properties of a nonlinear system involving the $p$-Laplacian operator

In this article we consider the nonlinear system involving the $p$-Laplacian $$\left\{\begin{array}{lc} |u^\prime |^{p-2} u^{\prime \prime} = u^{p-1} v^p& |v^\prime |^{p-2} v^{\prime \prime} = v^{p-1} u^p&\ {\rm on} \ \R, u\geq 0, v\geq 0& \end{array}\right.$$ for which we prove symmetry, asymptotic behavior and non degeneracy properties. This can help to a better understanding to what happens in the $N$ dimensional case, for which several authors prove a De Giorgi Type result under some additional growth and monotonicity assumptions.

同时确定涉及多项分数阶拉普拉斯算子的超扩散方程中若干标量参数与时间依赖源项的反问题

Inverse problem to determine simultaneously several scalar parameters and a time-dependent source term in a superdiffusion equation involving a multiterm fractional Laplacian

本文研究超扩散方程反问题，基于时间测量同时恢复标量系数、分数阶导数阶数、多项分数阶拉普拉斯算子参数及时间依赖源项，并利用拉普拉斯变换极点渐近性质证明了解的唯一性。

Finite many critical problems involving Fractional Laplacians in $\mathbb{R}^{N}$

In this paper, we consider the nonlocal elliptic problems in $\mathbb{R}^{N}$, which involve finite many critical exponents. By using endpoint refined Hardy--Sobolev inequality, fractional Coulomb--Sobolev space and variational method, we establish the existence of nonnegative solution. Our results generalize some results obtained by Yang and Wu [Adv. Nonlinear Stud. (2017) \cite{Yang2017}].

Regularity results for segregated configurations involving fractional Laplacian

We study the regularity of segregated profiles arising from competition - diffusion models, where the diffusion process is of nonlocal type and is driven by the fractional Laplacian of power $s \in (0,1)$. Among others, our results apply to the regularity of the densities of an optimal partition problem involving the eigenvalues of the fractional Laplacian. More precisely, we show $C^{0,α^*}$ regularity of the density, where the exponent $α^*$ is explicit and is given by \begin{equation*} α^* = \begin{cases} s & \text{for $s \in (0,1/2]$}\\ 2s-1 &\text{for $s \in (1/2,1]$}.\end{cases} \end{equ…