Anderson L. A. de Araujo
We present a result of existence of infinitely many solutions for the Dirichlet problem involving the p-Laplacian in annular domains, when $p\leq N$, contouring the failure of compactness of $W^{1,p}(Ω)$ in $C^0(\barΩ)$ applying a variable change.
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It is proved that the minimal Dirichlet eigenvalue of the Laplacian in an annulus is a monotonically decreasing function of the displacement of the center of the smaller disc. The maximal value of the minimal eigenvalue is attained when the annulus is formed by two concentric discs.
In this paper, we build infinitely many non-radial sign-changing solutions to the critical problem: \begin{equation*} \left\{\begin{array}{rlll} -Δu&=|u|^{\frac{4}{N-2}}u, &\hbox{ in }Ω,\\ u&=0, &\hbox{ on }\partialΩ. \end{array}\right. \eqno(P) \end{equation*} on the annulus $Ω:=\{x\in \mathbb{R}^N: a<|x|<b\}$, $N\geq 3.$ In particular, for any integer $k$ large enough, we build a non-radial solution which look like the unique positive solution $u_0$ to $(P)$ crowned by $k$ negative bubbles arranged on a regular polygon with radius $r_0$ such that $r_0^{\frac{N-2}{2}}u_0(r_0)=:\displaystyle\m…
In this work, the following fractional Laplacian problem with pure critical nonlinearity is considered \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*} where $s\in (0,1)$, $N$ is a positive integer with $N\geq 3$, $(-Δ)^{s}$ is the fractional Laplacian operator. We will prove that this problem has infinitely many non-radial sign-changing solutions.
We consider the fully nonlinear problem \begin{equation*} \begin{cases} -F(x,D^2u)=|u|^{p-1}u & \text{in $Ω$}\\ u=0 & \text{on $\partialΩ$} \end{cases} \end{equation*} where $F$ is uniformly elliptic, $p>1$ and $Ω$ is either an annulus or a ball in $\Rn$, $n\geq2$. \\ We prove the following results: \begin{itemize} \item[i)] existence of a positive/negative radial solution for every exponent $p>1$, if $Ω$ is an annulus; \item[ii)] existence of infinitely many sign changing radial solutions for every $p>1$, characterized by the number of nodal regions, if $Ω$ is an annulus; \item[iii)] existenc…
Infinitely many solutions for three quasilinear Laplacian systems on weighted graphs
该研究利用无紧性条件的抽象临界点定理，证明了加权图上三类带参数的拟线性拉普拉斯方程组在参数处于确定范围时存在无穷多范数无界的非平凡解。
In this paper, we consider the following Schrödinger-Poisson system with $p$-laplacian \begin{equation} \begin{cases} -Δ_{p}u+V(x)|u|^{p-2}u+ϕ|u|^{p-2}u=f(u)\qquad&x\in\mathbb{R}^{3},\newline -Δϕ=|u|^{p}&x\in\mathbb{R}^{3}. \end{cases}\notag \end{equation} We investigate the existence of multiple sign-changing solutions. By using the method of invariant sets of descending flow, we prove that this system has infinitely many sign-changing solutions. This system is new one coupled by Schrödinger equation of $p$-laplacian with a Poisson equation. Our results complement the study made by Zhaoli Liu…
Infinitely many solutions for two generalized poly-Laplacian systems on weighted graphs
本研究利用基于山路定理和拓扑度理论的变分方法，证明了加权有限图及带Dirichlet边值的加权局部有限图上的两类广义多拉普拉斯系统，均存在能量分别趋于正负无穷的两列无穷多解。
Let $1<p<N$, $p^{*}=Np/(N-p)$, $0<s<p$, $p^{*}(s)=(N-s)p/(N-p)$, and $\Om\in C^{1}$ be a bounded domain in $\R^{N}$ with $0\in\bar{\Om}.$ In this paper, we study the following problem \[ \begin{cases} -Δ_{p}u=μ|u|^{p^{*}-2}u+\frac{|u|^{p^{*}(s)-2}u}{|x|^{s}}+a(x)|u|^{p-2}u, & \text{in }\Om,\\ u=0, & \text{on }\pa\Om, \end{cases} \] where $μ\ge0$ is a constant, $\De_{p}$ is the $p$-Laplacian operator and $a\in C^{1}(\bar{\Om})$. By an approximation argument, we prove that if $N>p^{2}+p,a(0)>0$ and $Ω$ satisfies some geometry conditions if $0\in\partialΩ$, say, all the principle curvatures of $\…
Infinitely many solutions for $(p,q)$-Schrödinger-Poisson system with concave and convex nonlinearities
本文研究由p-Laplacian Schrödinger方程与q-Laplacian Poisson方程耦合、含凹凸非线性项和不定权函数的拟线性系统，证明了其存在无穷多个解。
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.
We study the dependence of least nontrivial critical levels of the energy functional corresponding to the zero Dirichlet problem $-Δ_p u = f(u)$ in a bounded domain $Ω\subset \mathbb{R}^N$ upon domain perturbations. Assuming that the nonlinearity $f$ is superlinear and subcritical, we establish Hadamard-type formulas for such critical levels. As an application, we show that among all (generally eccentric) spherical annuli $Ω$ least nontrivial critical levels attain maximum if and only if $Ω$ is concentric. As a consequence of this fact, we prove the nonradiality of least energy nodal solutions…
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.
We prove the existence of infinitely many solutions to a class of non-symmetric Dirichlet problems with exponential nonlinearities. Here the domain $Ω\subset\subset \mathbb{R}^{2l}$ where $2l$ is the order of the equation. Considered are the problem with no symmetry requirements on the domain, the radial problem on an annulus, and the radial problem on a ball with a Hardy potential term.
Let $1<p<+\infty$ and let $Ω\subset\mathbb R^N$ be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the type \[ -Δ_p u = f(u), \quad u>0 \mbox{ in } Ω, \quad \partial_νu = 0 \mbox{ on } \partialΩ. \] We suppose that $f(0)=f(1)=0$ and that $f$ is negative between the two zeros and positive after. In case $Ω$ is a ball, we also require that $f$ grows less than the Sobolev-critical power at infinity. We prove a priori bounds of radial solutions, focusing in particular…
In this paper, we are concern with the multiplicity of solutions for a p-Laplacian problem. A weaker super-quadratic assumptions is required on the nonlinearity. Under the weaker condition we give a new proof for the infinite solutions having a prescribed number of nodes to the problem. It turns out that the weaker condition on nonlinearity suffices to guarantee the infinitely many solutions. At the same time, a global characterization of the critical values of the nodal radial solutions are given.
We provide fundamental properties of the first eigenpair for fractional $p$-Laplacian eigenvalue problems under singular weights, which is related to Hardy type inequality, and also show that the second eigenvalue is well-defined. We obtain a-priori bounds and the continuity of solutions to problems with such singular weights with some additional assumptions. Moreover, applying the above results, we show a global bifurcation emenating from the first eigenvalue, the Fredholm alternative for non-resonant problems, and obtain the existence of infinitely many solutions for some nonlinear problems …
The Neumann condition for the superposition of fractional Laplacians
本文提出了一种新的函数设定，用于研究分数拉普拉斯算子叠加的Neumann条件，并探讨了相关数学性质及热方程的分析。
For $1<p<\infty$, we consider the following problem $$ -Δ_p u=f(u),\quad u>0\text{ in }Ω,\quad\partial_νu=0\text{ on }\partialΩ, $$ where $Ω\subset\mathbb R^N$ is either a ball or an annulus. The nonlinearity $f$ is possibly supercritical in the sense of Sobolev embeddings; in particular our assumptions allow to include the prototype nonlinearity $f(s)=-s^{p-1}+s^{q-1}$ for every $q>p$. We use the shooting method to get existence and multiplicity of non-constant radial solutions. With the same technique, we also detect the oscillatory behavior of the solutions around the constant solution $u\e…
Multiple solutions for elliptic equations driven by higher order fractional Laplacian
本文研究由高阶分数阶拉普拉斯算子驱动、带齐次狄利克雷边界条件的椭圆方程，在超线性、凹-凸及对称等条件下，利用山路定理、Ekeland 变分原理和 Fountain 定理证明其存在两个非平凡弱解以及无穷多个正能量和负能量弱解。
We use a variant of the fountain Theorem to prove the existence of infinitely many weak solutions for the following fractional p-Laplace equation (-Δ)^{s}_{p}u+V(x)|u|^{p-2}u=f(x,u) in R^N, where $s \in (0,1)$,$ p \geq 2$,$ N \geq 2$, $(-Δ)^{s}_{p}$ is the fractional $p$-Laplace operator, the nonlinearity f is $p$-superlinear at infinity and the potential V(x) is allowed to be sign-changing.
This paper is devoted to the following class of nonlinear fractional Schrödinger equations: \begin{equation*} (-Δ)^{s} u + V(x)u = f(x,u) + λg(x,u), \quad \text{in}\: \mathbb{R}^N, \end{equation*} where $s\in (0,1)$, $N>2s$, $(-Δ)^{s}$ stands for the fractional Laplacian, $λ\in \mathbb{R}$ is a parameter, $V\in C(\mathbb{R}^N,R)$, $f(x,u)$ is superlinear and $g(x,u)$ is sublinear with respect to $u$, respectively. We prove the existence of infinitely many high energy solutions of the aforementioned equation by means of the Fountain theorem. Some recent results are extended and sharply improved…
We investigate the existence of infinitely many radially symmetric solutions to the following problem $$(-Δ_p)^s u=g(u) \ \ \textrm{ in } \ \ \mathbb{R}^N, \ \ u\in W^{s,p}(\mathbb{R}^N),$$ where $s\in (0,1)$, $2 \leq p < \infty$, $sp \leq N $, $2 \leq N \in \mathbb{N}$ and $(-Δ_p)^s$ is the fractional $p$-Laplacian operator. We treat both of cases $sp=N$ and $sp<N.$ The nonlinearity $g$ is a function of Berestycki-Lions type with critical exponential growth if $sp=N$ and critical polynomial growth if $sp<N$. We also prove the existence of a ground state solution for the same problem.
In the present work we investigate the existence and multiplicity of nontrivial solutions for the Choquard Logarithmic equation $(-Δ)^{\frac{1}{2}} u + au + λ(\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ in } \mathbb{R}$, for $ a>0 $, $ λ>0 $ and a nonlinearity $f$ with exponential critical growth. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under exponential critical and subcritical growth. Morever, when $ f $ has subcritical growth we guarantee the existence of infinitely many solutions, via genus theory.
We study the bifurcation of solutions of semilinear elliptic boundary value problems of the form \begin{align*} \begin{aligned} -Δu &= f_λ(|x|,u,|\nabla u|) &&\text{in }Ω, u &= 0 &&\text{on }\partialΩ, \end{aligned} \end{align*} on an annulus $Ω\subset\mathbb{R}^N$, with a concave-convex nonlinearity, a special case being the nonlinearity first considered by Ambrosetti, Brezis and Cerami: $f_λ(|x|,u,|\nabla u|)=λ|u|^{q-2}u + |u|^{p-2}u$ with $1<q<2<p$. Although the trivial solution $u_0\equiv0$ is nondegenerate if $λ=0$ we prove that $(λ_0,u_0)=(0,0)$ is a bifurcation point. In fact, the bifur…
We study an eigenvalue problem for the infinity-Laplacian on bounded domains. We prove the existence of the principal eigenvalue and a corresponding positive eigenfunction. The work also contains existence results when the parameter, in the equation, is less than the first eigenvalue. A comparison principle applicable to these problems is also proven. Some additional results are shown, in particular, that on star- shaped domains and on C^2 domains higher eigenfunctions change sign. When the domain is a ball, we prove that the first eigenfunction has one sign, radial principal eigenfunction exi…
This paper concerns the existence of normalized solutions to a class of $(2,q)$-Laplacian equations in all the possible cases according to the value of $p$ with respect to the critical exponent $2(1+2/N)$. In the $L^2$-subcritical case, we study a global minimization problem and obtain a ground state solution. While in the $L^2$-critical case, we prove several nonexistence results, extended also in the $L^q$-critical case. At last, we derive a ground state and infinitely many radial solutions in the $L^2$-supercritical case. Compared with the classical Schrödinger equation, the $(2,q)$-Laplaci…
In this paper, we consider the following problem involving fractional Laplacian operator: \begin{equation}\label{eq:0.1} (-Δ)^α u= |u|^{2^*_α-2-\varepsilon}u + λu\,\, {\rm in}\,\, Ω,\quad u=0 \,\, {\rm on}\, \, \partialΩ, \end{equation} where $Ω$ is a smooth bounded domain in $\mathbb{R}^N$, $\varepsilon\in [0, 2^*_α-2)$, $0<α<1,\, 2^*_α= \frac {2N}{N-2α}$. We show that for any sequence of solutions $u_n$ of \eqref{eq:0.1} corresponding to $\varepsilon_n\in [0, 2^*_α-2)$, satisfying $\|u_n\|_{H}\le C$ in the Sobolev space $H$ defined in \eqref{eq:1.1a}, $u_n$ converges strongly in $H$ provided…
Entire solutions to a quasilinear purely critical competitive system
本文证明了$p$-拉普拉斯纯临界竞争系统存在非负完全非平凡解，并证明了纯临界方程有无穷多个结点解。
Existence and nonexistence of infinitely many solutions to elliptic problems with oscillating nonlinearities
研究了在有界域内具有Dirichlet边界条件的椭圆问题-Δ_p u = λf(u)中，非负解无限多个存在的精确条件，基于非线性项f的正零点序列收敛性质及参数λ的范围，利用变分方法和ODE论证得出存在与不存在结果。
Infinitely many solutions for $p$-fractional Choquard type equations involving general nonlocal nonlinearities with critical growth via the concentration compactness method
本文通过对称山路引理和非局部集中紧性方法，证明了含临界增长的分数阶Choquard型方程存在无穷多解。
Existence and multiplicity of solutions involving the $p(x)$-Laplacian equations: On the effect of two nonlocal terms
研究一类含双非局部项的$p(x)$-基尔霍夫问题，利用变分方法证明两个非平凡解及无穷多解的存在性，推广了已有结果。
Existence of infinitely many solutions for the fractional Schrödinger- Maxwell equations
该研究运用变分方法与临界点理论，在对非线性项 f 增加适当假设后，证明了 ℝ³ 中一类含分数阶 Laplacian 的 Schrödinger-Maxwell 系统存在无穷多解。
We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularities \begin{align} (-Δ)_{p(\cdot)}^{s} u&=\fracλ{|u|^{γ(x)-1}u}+f(x,u)~\text{in}~Ω,\nonumber u&=0~\text{in}~\mathbb{R}^N\setminusΩ,\nonumber \end{align} where $Ω\subset\mathbb{R}^N,\, N\geq2$ is a smooth, bounded domain, $λ>0$, $s\in (0,1)$, $γ(x)\in(0,1)$ for all $x\in\barΩ$, $N>sp(x,y)$ for all $(x,y)\in\barΩ\times\barΩ$ and $(-Δ)_{p(\cdot)}^{s}$ is the fractional $p(\cdot)$-Laplacian operator with variable exponent. The nonlinear function $f$ sa…
In this paper, we obtain nonexistence results of positive solutions, and also the existence of an unbounded sequence of solutions that changing sign for some critical problems involving conformally invariant operators on the standard unit sphere, and the fractional Laplacian operator in the Euclidean space. Our arguments are based on a reduction of the initial problem in the Euclidean space to an equivalent problem on the standard unit sphere and vice versa, what together to blow up arguments, a variant of Pohozaev's type identity, a refinement of regularity results for this type operators, an…
An overdetermined eigenvalue problem and the Critical Catenoid conjecture
该研究针对二维球面上的一类超定特征值问题，通过边界梯度局部常数与无穷多最大值点条件分类正解为旋转对称解，并由此刻画了临界悬链面的唯一性特征。
In the present work we obtain the existence and multiplicity of nontrivial solutions for the Choquard logarithmic equation $(-Δ)_{p}^{s}u + |u|^{p-2}u + (\ln|\cdot|\ast |u|^{p})|u|^{p-2}u = f(u) \textrm{ \ in \ } \mathbb{R}^N $ , where $ N=sp $, $ s\in (0, 1) $, $ p>2 $, $ a>0 $, $ λ>0 $ and $f: \mathbb{R}\rightarrow \mathbb{R} $ a continuous nonlinearity with exponential critical and subcritical growth. We guarantee the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under critical and subcritical growth. Morever, when $ f $ has subcritical…
In this paper, we deal with the following elliptic type problem $$ \begin{cases} (-Δ)_{q(.)}^{s(.)}u + λVu = α\left\vert u\right\vert^{p(.)-2}u+β\left\vert u\right\vert^{k(.)-2}u & \text{ in }Ω, \\[7pt] u =0 & \text{ in }\mathbb{R}^{n}\backslash Ω, \end{cases} $$ where $q(.):\overlineΩ\times \overlineΩ\rightarrow \mathbb{R}$ is a measurable function and $s(.):\mathbb{R}^n\times \mathbb{R}^n\rightarrow (0,1)$ is a continuous function, $n>q(x,y)s(x,y)$ for all $(x,y)\in Ω\times Ω$, $(-Δ)_{q(.)}^{s(.)}$ is the variable-order fractional Laplace operator, and $V$ is a positive continuous potential.…
Gradient constraints, Born-Infeld, and maximal surfaces via superposition of infinitely many $p$-Laplacians
研究无穷多个 p-Laplacian 叠加的 Dirichlet 问题，通过饱和通量刻画梯度约束下的可解性，并应用于 Born-Infeld 模型，获得电荷密度等周界和弱场展开收敛速率。
We study the question of the existence of infinitely many weak solutions for nonlocal equations of fractional Laplacian type with homogeneous Dirichlet boundary data, in presence of a superlinear term. Starting from the well-known Ambrosetti-Rabinowitz condition, we consider different growth assumptions on the nonlinearity, all of superlinear type. We obtain three different existence results in this setting by using the Fountain Theorem, which extend some classical results for semilinear Laplacian equations to the nonlocal fractional setting.
Some nonlinear problems for the superposition of fractional operators with Neumann boundary conditions
本文研究了受Neumann边界条件约束的分数算子叠加非线性问题，通过引入新的泛函分析工具和特征值分析，将存在理论分为山脊方法和链接技术两条分支。
An existence theory for nonlinear superposition operators of mixed fractional order
本文提出了一种混合分数阶非线性叠加算子的存在性理论，解决了临界类型非线性问题多个解的存在性问题。
Superlinear problems involving nonlinear superposition operators of mixed fractional order
本文研究了一类涉及混合分数阶非线性叠加算子的超线性椭圆问题，通过变分方法和Fountain定理证明了无限多个弱解的存在性。
This paper is devoted to study the existence and multiplicity solutions for the nonlinear Schrödinger-Poisson systems involving fractional Laplacian operator: \begin{equation}\label{eq*} \left\{ \aligned &(-Δ)^{s} u+V(x)u+ ϕu=f(x,u), \quad &\text{in }\mathbb{R}^3, &(-Δ)^{t} ϕ=u^2, \quad &\text{in }\mathbb{R}^3, \endaligned \right. \end{equation} where $(-Δ)^α$ stands for the fractional Laplacian of order $α\in (0\,,\,1)$. Under certain assumptions on $V$ and $f$, we obtain infinitely many high energy solutions for \eqref{eq*} without assuming the Ambrosetti-Rabinowitz condition by using the fo…
Existence and Multiplicity of Solutions for Fractional $p$-Laplacian Equation Involving Critical Concave-convex Nonlinearities
本文研究含临界凹凸非线性项的分数阶 $p$-Laplacian Dirichlet 问题，利用变分方法建立正解存在性的参数二分性、小参数下两个正解及无穷多解。
Infinitely many solutions for a class of fractional Schrodinger equations coupled with neutral scalar field
本文研究了与中性标量场耦合的分数阶薛定谔方程，在外部势的适当假设下，借助改进的Fountain定理处理强不定变分问题，证明了该系统存在无穷多非平凡解。
Normalized solutions for $p$-Laplacian equation with critical Sobolev exponent and mixed nonlinearities
本文研究带临界Sobolev指数与混合非线性项的$p$-Laplacian方程的规范化解，运用多种变分方法得到不同参数条件下解的存在性、多重性、不存在性及渐近行为结果。
We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space $\mathbb{R}^N$. We assume that the nonlinear term satisfies the locally super-$(m_1,m_2)$ condition, that is, $\lim_{|(u,v)|\rightarrow+\infty}\frac{F(x,u,v)}{|u|^{m_1}+|v|^{m_2}}=+\infty \mbox{ for a.e. } x \in G$ where $G$ is a domain in $\mathbb{R}^N$, which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, $\lim_{|(u,v)|\rightarrow+\infty}\frac{F(x,u,v)}{|u|^{m_1}+|v|^{m_2}}=+\infty \mbox{ for a.e. } …
Green function and Poisson kernel associated to root systems for annular regions
本文研究环形区域上的 Dunkl Laplacian，求解其 Dirichlet 问题，构造并用球谐函数表示 Green 函数，进而给出 Poisson-Jensen 公式及半线性问题正连续解的应用。
We prove the hydrodynamic limit for the symmetric exclusion process with long jumps given by a mean zero probability transition rate with infinite variance and in contact with infinitely many reservoirs with density $α$ at the left of the system and $β$ at the right of the system. The strength of the reservoirs is ruled by $κ$N --$θ$ > 0. Here N is the size of the system, $κ$ > 0 and $θ$ $\in$. Our results are valid for $θ$ $\le$ 0. For $θ$ = 0, we obtain a collection of fractional reaction-diffusion equations indexed by the parameter $κ$ and with Dirichlet boundary conditions. Their solutions…
Mixed double phase equations with local and nonlocal operators
研究结合局部和非局部算子的混合双相问题，通过变分和拓扑方法建立弱解的存在性和多重性结果。
Elliptic problems involving a diffusion operator of vectorial $p\&q$-Laplacian type
该研究针对含向量p&q-拉普拉斯型扩散算子的各向异性椭圆狄利克雷问题，建立比较原理，分析不同反应指数下解的存在性、非存在性等结果，结合多种变分方法完成论证。
Normalized solutions to a class of $(2, q)$-Laplacian equationsin the strongly sublinear regime
本文针对强次线性非线性项的 $(2,q)$-Laplacian 规范化问题，通过逼近问题与最低能量解收敛分析证明最低能量解存在，并在适当函数子空间中证明无穷多解。
Existence and multiplicity of solutions for critical Kirchhoff-Choquard equations involving the fractional $p$-Laplacian on the Heisenberg group
本文研究Heisenberg群上含分数阶$p$-Laplacian的临界Kirchhoff-Choquard方程，分别借助Krasnoselskii亏格定理与分数阶集中紧性原理，证明参数$\mu$充分大时有无穷多解、充分小时有$m$对解。
We consider nonlinear second order elliptic problems of the type \[ -Δu=f(u) \text{ in } Ω, \qquad u=0 \text{ on } \partial Ω, \] where $Ω$ is an open $C^{1,1}$-domain in $\mathbb{R}^N$, $N\geq 2$, under some general assumptions on the nonlinearity that include the case of a sublinear pure power $f(s)=|s|^{p-1}s$ with $0<p<1$ and of Allen-Cahn type $f(s)=λ(s-|s|^{p-1}s)$ with $p>1$ and $λ>λ_2(Ω)$ (the second Dirichlet eigenvalue of the Laplacian). We prove the existence of a least energy nodal (i.e. sign changing) solution, and of a nodal solution of mountain-pass type. We then give explicit e…
We investigate the existence of least energy solutions and infinitely many solutions for the following nonlinear fractional equation (-Δ)^{s} u = g(u) \mbox{ in } \mathbb{R}^{N}, where $s\in (0,1)$, $N\geq 2$, $(-Δ)^{s}$ is the fractional Laplacian and $g: \mathbb{R} \rightarrow \mathbb{R}$ is an odd $\mathcal{C}^{1, α}$ function satisfying Berestycki-Lions type assumptions. The proof is based on the symmetric mountain pass approach developed by Hirata, Ikoma and Tanaka in \cite{HIT}. Moreover, by combining the mountain pass approach and an approximation argument, we also prove the existence o…
In the present work we briefly explain how to adapt techniques already used in fractional and $p$-fractional Laplacian cases to obtain the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution, for the critical case, and the existence of infinitely many solutions, for the subcritical case, to the Choquard Logarithmic equation, $-Δ_N u + a(x)|u|^{N-2}u + λ(\ln|\cdot|\ast |u|^{N})|u|^{N-2}u = f(u) \textrm{ in } \mathbb{R}^N $, where $ a:\mathbb{R}^N \rightarrow \mathbb{R} $, $ λ>0 $, $ N \geq 3 $ and $f: \mathbb{R} \rightarrow [0, \infty) $ is conti…
On the density and multiplicity of solutions to the fractional Nirenberg problem
本文利用变分粘合、气泡行为精细分析、延拓公式和爆破分析等方法，证明了分数阶Nirenberg问题无穷多个多峰解的存在性、共形度量光滑曲率函数在C⁰拓扑中的稠密性，以及渐近周期情形下分数阶Laplacian方程的无穷多解。
We study existence of minimizers of the least gradient problem \[\inf_{v \in BV_g} \int_Ωφ(x, Dv),\] where $BV_g=\{v \in BV(Ω): \int_{\partial Ω}gv=1\}$, $φ(x,p): Ω\times \R^n \rightarrow \R$ is a convex, continuous, and homogeneous function of degree $1$ with respect to the $p$ variable, and $g$ satisfies the comparability condition $\int_{\partial Ω} g dS=0$. We prove that for every $0\not \equiv g \in L^{\infty}(\partial Ω)$ there are infinitely many minimizers in $BV(Ω)$. Moreover there exists a divergence free vector field $T\in (L^{\infty}(Ω))^n$ that determines the structure of level se…
Fractional $Q$-curvature on the sphere and optimal partitions
本文通过变分方法，利用对称性和新的Hölder正则性结果，证明了球面上与分数阶Q-曲率相关的最优划分的存在性，并建立了非局部竞争系统无穷多个解的存在性及其渐近行为。
In this paper we prove the existence of infinitely many nontrivial solutions of the following equations driven by a nonlocal integro-differential operator $L_K$ with concave-convex nonlinearities and homogeneous Dirichlet boundary conditions \begin{eqnarray*} \mathcal{L}_{K} u + μ\, |u|^{q-1}u + λ\,|u|^{p-1}u &=& 0 \quad\text{in}\quad Ω, \\[2mm] u&=&0 \quad\mbox{in}\quad\mathbb{R}^N\setminusΩ, \end{eqnarray*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^N$, $N>2s$, $s\in(0, 1)$, $0<q<1<p\leq \frac{N+2s}{N-2s}$. Moreover, when $L_K$ reduces to the fractional laplacian operator $-(-Δ)^s $…
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