Biagio Ricceri
The aim of this paper is to provide the first application of Theorem 3 of [2] in a case where the dependence of the underlying equation from the real parameter is not of affine type.
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In this paper, we prove multiplicity of solutions for a class of quasilinear problems in $ \mathbb{R}^{N} $ involving variable exponents and nonlinearities of concave-convex type. The main tools used are variational methods, more precisely, Ekeland's variational principle and Nehari manifolds.
In this paper we prove the existence of infinitely many nontrivial solutions for the class of $(p,\, q)$ fractional elliptic equations involving concave-critical nonlinearities in bounded domains in $\mathbb{R}^N$. Further, when the nonlinearity is of convex-critical type, we establish the multiplicity of nonnegative solutions using variational methods. In particular, we show the existence of at least $cat_Ω(Ω)$ nonnegative solutions.
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.
Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
本文针对有界区域上含分数阶$p$-Laplace算子的临界Schrödinger-Kirchhoff型系统，结合Nehari流形上能量泛函性质与纤维映射分析，建立了该系统解的多解性结果。
Existence and Multiplicity of Normalized Solutions to Schrödinger Equations with General Nonlinearities in Bounded Domains
本文利用变分方法研究有界区域中具一般非线性项的薛定谔方程的归一化解，得到两个正归一化解，证明星形有界区域解的多重性及球域中非径向归一化解的存在性。
Multiplicity and Nonrelativistic limit of Bound States of Nonlinear Dirac Equations on Noncompact Metric Graphs with Localized Nonlinearities
研究非紧度量图上具有局部非线性项的狄拉克方程在L^2约束下归一化解的多重性，并证明在光速趋于无穷时解收敛到非线性薛定谔方程的解，涵盖质量次临界、临界和超临界情形。
This paper studies the multiplicity of normalized solutions to the Schrödinger equation with mixed nonlinearities \begin{equation*} \begin{cases} -Δu=λu+h(εx)|u|^{q-2}u+η|u|^{p-2}u,\quad x\in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=a^2, \end{cases} \end{equation*} where $a, ε, η>0$, $q$ is $L^2$-subcritical, $p$ is $L^2$-supercritical, $λ\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $h$ is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of $h$ when $ε$ is small enoug…
A multiplicity theorem for a class of nonlocal elliptic equations involving even nonlinearities
研究有界区域上一类非局部椭圆方程，通过特定方法在适当假设下，证明对于合适区间内的\(\lambda\)，该方程至少有三个弱解。
Well-posedness for a nonlinear Schrödinger equation with quadratic derivative nonlinearities for bounded primitive initial data
该论文研究具有二次导数非线性项的非线性薛定谔方程，证明了在有界原初数据下，$s\ge 0$ 时局部适定，$s\ge 1$ 及特殊系数情形下全局适定。
In this paper, we show the existence and multiplicity of nontrivial, non-negative solutions of the fractional $p$-Kirchhoff problem \begin{equation*} \begin{array}{rllll} M\left(\displaystyle\int_{\mathbb{R}^{2n}}\frac{|u(x)-u(y)|^p}{\left|x-y\right|^{n+ps}}dx\,dy\right)(-Δ)^{s}_p u &=λf(x)|u|^{q-2}u+ g(x)\left|u\right|^{r-2}u\, \text{in} Ω,\\ u&=0 \;\mbox{in} \mathbb{R}^{n}\setminus Ω, \end{array} \end{equation*} where $(-Δ)^{s}_p$ is the fractional $p$-Laplace operator, $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $f \in L^{\frac{r}{r-q}}(Ω)$ and $g\in L^\infty(Ω)$ are sig…
This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of $p$-$q$ type and singular nonlinearities \begin{equation*} \left\{ \begin{alignedat}{2} {} - \mathcal{L}_{p,q} u & {}= λ\frac{f(u)}{u^γ}, \ u>0 && \quad\mbox{ in } \, Ω, u & {}= 0 && \quad\mbox{ on } \partialΩ, \end{alignedat} \right. \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary, $N \geq 1$, $λ>0$ is a real parameter, $$\mathcal{L}_{p,q} u := div(|\nabla u|^{p-2} \nabla u + |\nabla u|^{q-2} …
Multiplicity and solution construction in linear elliptic problems with nonlocal nonlinearities
针对带多个非局部非线性的线性椭圆方程，结合不动点论证与渐近技术，建立解的存在性与多重性，揭示非局部项导致的新解结构。
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 $…
In this paper we study the existence of multiple nontrivial positive weak solutions to the following system of problems. \begin{align*} \begin{split} -Δ_{p}u-Δ_q u &= λf(x)|u|^{r-2}u+ν\frac{1-α}{2-α-β}h(x) |u|^{-α}|v|^{1-β}\,\,\mbox{in}\,\,Ω,\\ -Δ_{p}v-Δ_q v &= μg(x)|v|^{r-2}v+ν\frac{1-β}{2-α-β}h(x) |u|^{1-α}|v|^{-β}\,\,\mbox{in}\,\,Ω,\\ u,v&>0\,\,\mbox{in}\,\,Ω,\\ u= v &= 0\,\, \mbox{on}\,\, \partialΩ\end{split} \end{align*} where (C):~$0<α<1,\;0<β<1,$ $2-α-β<q<\frac{N(p-1)}{N-p}<p<r<p^*$, with $p^*=\frac{Np}{N-p}$. We will guarantee the existence of a solution in the Nehari manifold. Further…
Multiplicity results for critical fractional Ambrosetti-Prodi type system with nonlinearities interacting with the spectrum
本文研究涉及分数阶Laplacian算子、非线性项达临界增长且与谱相互作用的Ambrosetti-Prodi型系统，证明了解的存在性，并进一步处理了高阶特征值处的共振情形。
In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents \begin{equation*} \begin{array}{rl} (-Δ)_{p(\cdot)}^{s(\cdot)}u(x)&=λ|u(x)|^{α(x)-2}u(x)+\left(\DD\int_Ω\frac{F(y,u(y))}{|x-y|^{μ(x,y)}}dy\right)f(x,u(x)),\\ &~\hspace{6cm} x\in Ω, \\ u(x)&=0 ,\hspace{20mm} x\in Ω^c:=\mathbb R^N\setminusΩ, \end{array} \end{equation*} where $\Om\subset\mathbb R^N$ is a smooth and bounded domain, $N\geq 2$, $p,s,μ$ and $α$ are continuous functions on $\mathbb R^N\times\mathbb R^N$ and $f(x,t)$ is continuous function with…
Multiplicity of solutions for semilinear Robin problems involving sign-changing nonlinearities
研究Robin问题解的存在性和多重性，考虑局部Lipschitz函数作为非线性项，证明大λ下存在两个非负解，分析解集在γ趋近0或∞时的极限行为。
Anisotropic quasilinear elliptic systems involving nonlinearities with negative critical terms
本文研究各向异性拟线性椭圆系统，通过缩放引入参数γ，分三种情形证明解的多重性，利用抽象临界点定理和截断论证。
The present paper studies the fractional $p$-Laplacian boundary value problems with jumping nonlinearities at zero or infinity and obtain the existence of multiple solutions and sign-changing solutions by constructing the suitable pseudo-gradient vector field of the corresponding energy functional.
Existence and multiplicity of normalized solutions for $(2,q)$-Laplacian equations with generic double-behaviour nonlinearities
本文研究一类具有一般双重行为非线性的(2,q)-Laplacian方程在质量约束下的正规化解，证明了局部最小能量解和更高能量第二解的存在性。
We are concerned with a Liouville-type equation with exponential nonlinearities on a compact surface which describes the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. We provide the first multiplicity result for this class of equations by using Morse theory.
Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
本文研究带有非齐次微分算子的拟线性椭圆方程在凹-凸非线性项下解的多重性，利用Nehari流形和纤维化映射分析，证明方程至少存在一个正能量解和一个作为基态解的负能量解。
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。
Existence and multiplicity of normalized solutions for the quasi-linear Schrödinger equations with mixed nonlinearities
本文研究具混合非线性项的拟线性薛定谔方程在规定质量约束下正规化解的存在性与多重性，证明了正规化基态解和山路型正规化解的存在性，补充了已有相关结果。
Modified quasilinear equations with strongly singular and critical exponential nonlinearities
本文研究一类含 Trudinger-Moser 型临界指数增长的修正拟线性强奇异方程，建立了其解的整体多重性结果。
In this paper, by utilizing a newly established variational principle on convex sets, we provide an existence and multiplicity result for a class of semilinear elliptic problems defined on the whole $\mathbb R^N$ with nonlinearities involving sublinear and superlinear terms. We shall impose no growth restriction on the nonlinear term and consequently our problem can be super-critical by means of Sobolev spaces.
The theory of elliptic equations involving singular nonlinearities is well studied topic but the interaction of singular type nonlinearity with nonlocal nonlinearity in elliptic problems has not been investigated so far. In this article, we study the very singular and doubly nonlocal singular problem $(P_λ)$(See below). Firstly, we establish a very weak comparison principle and the optimal Sobolev regularity. Next using the critical point theory of non-smooth analysis and the geometry of the energy functional, we establish the global multiplicity of positive weak solutions.
A $p$-Laplacian elliptic problem in the presence of both strongly singular and $(p-1)$-superlinear nonlinearities is considered. We employ bifurcation theory, approximation techniques and sub-supersolution method to establish the existence of an unbounded branch of positive solutions, which is bounded in positive $λ-$direction and bifurcates from infinity at $λ=0$. As consequence of the bifurcation result, we determine intervals of existence, nonexistence and, in particular cases, global multiplicity.
Multiplicity results for a subcritical Hamiltonian system with concave-convex nonlinearities
研究具有凹-凸非线性的次临界哈密顿椭圆系统，证明存在严格递减曲线划分参数区域，使得系统在曲线下方有两个解、曲线上有一个解、上方无解，并推广到对称情形。
In this paper, we study the existence and multiplicity results of nontrivial positive solutions to a quasilinear elliptic equation in $\RN$, when $N\geq2$, as \begin{equation} \Lp u+u^{p-1}=λ\hspace{0.2mm}k(x)u^{r-1}-h(x)u^{q-1}.\nonumber \end{equation} Here, $h(x),k(x)>0$ are Lebesgue measurable functions, $1<p<q<\infty$, $p<r<\min\{p^*,q\}$ if $p<N$ while $p<r<q$ if $p\geq N$, and $λ>0$ is a parameter.
This paper studies a nonlinear Dirichlet problem for the $p$-Laplacian operator with nonlinearity consisting of power components. The problem under consideration can be though of as a perturbation of the Ambrosetti-Brezis-Cerami problem with concave-convex nonlinearity. The combined effect of power components in the perturbed nonlinearity allows to establish a higher order multiplicity of positive solutions. We study properties of the perturbed energy functional and prove the existence of three positive solutions to the problem at hand.
In this paper we establish, using variational methods combined with the Moser-Trudinger inequality, existence and multiplicity of weak solutions for a class of critical fractional elliptic equations with exponential growth without a Ambrosetti-Rabinowitz-type condition. The interaction of the nonlinearities with the spectrum of the fractional operator will used to study the existence and multiplicity of solutions. The main technical result proves that a local minimum in $C_{s}^0(\overlineΩ)$ is also a local minimum in $W^{s,p}_0$ for nonlinearities with exponential growth.
This paper is concerned with multiplicity results for parametric singular double phase problems in $\mathbb{R}^N$ via the Nehari manifold approach. It is shown that the problem under consideration has at least two nontrivial weak solutions provided the parameter is sufficiently small. The idea is to split the Nehari manifold into three disjoint parts minimizing the energy functional on two of them. The third set turns out to be the empty set for small values of the parameter.
Solutions to critical equations with a superposition of nonlocal Hartree-type nonlinearities
本文研究由Riesz势加权测度驱动的非局部椭圆方程，通过引入叠加Coulomb空间和变分方法，结合基于尺度的临界点理论，证明了径向解的存在性和多重性，并推广了Brezis-Nirenberg型问题。
Normalized Solutions to Schrödinger Equations with General Nonlinearities in Bounded Domains via a Global Bifurcation Approach
该研究采用全局分岔方法，探究有界区域中具一般非线性项的薛定谔方程指定质量正解的存在性等性质，推广了已有自治齐次非线性情形的结果，并得到相关轨道稳定性结论。
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.
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…
In this paper we establish the existence and multiplicity of nontrivial solutions to the following problem \begin{align*} \begin{split} (-Δ)^{\frac{1}{2}}u+u+(\ln|\cdot|*|u|^2)&=f(u)+μ|u|^{-γ-1}u,~\text{in}~\mathbb{R}, \end{split} \end{align*} where $μ>0$, $(*)$ is the convolution operation between two functions, $0<γ<1$, $f$ is a function with a certain type of growth. We prove the existence of a nontrivial solution at a certain mountain pass level and another ground state solution when the nonlinearity $f$ is of exponential critical growth.
In this paper, we study the existence and multiplicity results of nontrivial positive solutions to the following quasilinear elliptic equation on $\RN$, when $N\geq2$, \begin{equation} \Lp u=λ\hspace{0.2mm}K(x)u^{r-1}-V(x)u^{q-1}.\nonumber \end{equation} Here, $K(x),V(x)>0$ are suitable potentials, $1<p<r<q<\infty$, and $λ>0$ is a parameter. To study this problem, some compact embedding results regarding $\MVR\hookrightarrow\LKR$ are proved that unify and extend some recent results of the author \cite{Ha1,Ha2,Ha3}.
Fractional $p$-Laplace systems with critical Hardy nonlinearities: Existence and Multiplicity
本文研究了带有临界哈里德非线性的分数p-拉普拉斯系统，探讨了地面态解的存在性和多重性，并建立了相关的集中-紧致原理。
In this paper, we prove existence results of a one-dimensional periodic solution to equations with the fractional Laplacian of order $s\in(1/2,1)$, singular nonlinearity, and gradient term under various situations, including nonlocal contra-part of classical Lienard vector equations, as well other nonlocal versions of classical results know only in the context of second-order ODE. Our proofs are based on degree theory and Perron's method, so before that we need to establish a variety of priori estimates under different assumptions on the nonlinearities appearing in the equations. Besides, we o…
In this paper, we consider the multiplicity of solutions for a class of Kirchhoff type problems with sub-linear and critical terms on an unbounded domain. With the aid of Ekeland's variational principle and the concentration compactness principle we prove that the Kirchhoff problem has at least two solutions.
It is established existence of bound and ground state solutions for quasilinear elliptic systems driven by (ϕ1, ϕ2)-Laplacian operator. The main feature here is to consider quasilinear elliptic systems involving both nonsingular nonlinearities combined with indefinite potentials and singular cases perturbed by superlinear and subcritical couple terms. These prevent us to use arguments based on Ambrosetti-Rabinowitz condition and variational methods for differentiable functionals. By exploring the Nehari method and doing a fine analysis on the fibering map associated, we get estimates that allo…
We study a class of elliptic problems, involving a $k$-Hessian and a very fast-growing nonlinearity, on a unit ball. We prove the existence of a radial singular solution and obtain its exact asymptotic behavior in a neighborhood of the origin. Furthermore, we study the multiplicity of regular solutions and bifurcation diagrams. An essential ingredient of this study is analyzing the number of intersection points between the singular and regular solutions for rescaled problems. In the particular case of the exponential nonlinearity, we obtain the convergence of regular solutions to the singular …
Normalized solutions on large smooth domains to the Schrödinger equations with potential and combined nonlinearities: The Sobolev critical case
本文研究含位势和混合非线性项的薛定谔方程在 Sobolev 临界情形下指定质量解的存在性与多重性，克服了 Pohozaev 恒等式方法因位势项存在而失效的困难，补充了 Bartsch-Qi-Zou 的相关工作。
Let $Ω$ be a bounded domain in $\mathbb R^{N}$, $N\geq3$ with smooth boundary, $a>0, λ>0$ and $0<δ<3$ be real numbers. Define $2^*:=\displaystyle\frac{2N}{N-2}$ and the characteristic function of a set $A$ by $χ_A$. We consider the following critical problem with singular and discontinuous nonlinearity: \begin{eqnarray*} (P_\la^a)~~~~ \qquad \Biggl\{\begin{array}{rl} -Δu &= λ\left(u^{2^*-1}+ \displaystyle χ_{\{u<a\}}u^{-\de} \right), u > 0~~\text{in} ~~Ω, \\ u & = 0 ~\text{on}~ \partial Ω. \end{array} \end{eqnarray*} \noindent We study the existence and the global multiplicity of solutions to …
Quasilinear problems with mixed local-nonlocal operator and concave-critical nonlinearities: Multiplicity of positive solutions
研究混合局部-非局部算子驱动的凹-临界问题正解的存在性和多重性，证明了参数λ>0时存在正解的临界值，并扩展了相关经典结果到一般p情况。
Nonlocal elliptic PDEs with general nonlinearities
本论文研究在一般非线性假设下非局部性对物理PDE的影响，通过变分和拓扑方法分析分数阶Laplacian算子与Riesz势卷积源项的相互作用，开发新工具证明解的多重性、基态性质及集中现象，为非局部问题提供新见解。
Non-existence and multiplicity of positive solutions for Choquard equations with critical combined nonlinearities
本文研究含临界组合非线性项的Choquard方程正解的不存在性与多解性，确定了参数空间中对应方程无正基态或多正解的区域，得到了基态存在阈值与双正解条件，揭示了非局部项的特殊影响。
In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation involving a fractional Laplacian \[ (-\De)^s u = \left( \int_{\Om}\frac{|u|^{2^*_{μ,s}}}{|x-y|^μ}\mathrm{d}y \right)|u|^{2^*_{μ,s}-2}u +\la u \; \text{in } \Om,\] where $\Om $ is a bounded domain in $\mathbb R^n$ with Lipschitz boundary, $\la $ is a real parameter, $s \in (0,1)$, $n >2s$ and $2^*_{μ,s}= (2n-μ)/(n-2s)$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We obtain some existence, multiplicity, regularity and nonexistence results for solution of the above equation …
In this article, we study the following fractional Laplacian equation with critical growth and singular nonlinearity $$\quad (-Δ)^s u = λa(x) u^{-q} + u^{2^*_s-1}, \quad u>0 \; \text{in}\; Ω,\quad u = 0 \; \mbox{in}\; \mathbb{R}^n \setminusΩ,$$ where $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial Ω$, $n > 2s,\; s \in (0,1),\; λ>0,\; 0 < q \leq 1 $, $θ\leq a(x) \in L^\infty(Ω)$, for some $θ>0$ and $2^*_s=\frac{2n}{n-2s}$. We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter $λ$.
We are interested in the following Dirichlet problem $$ \left\{ \begin{array}{ll} -Δu + λu - μ\frac{u}{|x|^2} - ν\frac{u}{\mathrm{dist}\,(x,\mathbb{R}^N \setminus Ω)^2} = f(x,u) & \quad \mbox{in } Ω\\ u = 0 & \quad \mbox{on } \partial Ω, \end{array} \right. $$ on a bounded domain $Ω\subset \mathbb{R}^N$ with $0 \in Ω$. We assume that the nonlinear part is superlinear on some closed subset $K \subset Ω$ and asymptotically linear on $Ω\setminus K$. We find a solution with the energy bounded by a certain min-max level, and infinitely many solutions provided that $f$ is odd in $u$. Moreover we stu…
Existence and blow up behavior of prescribed mass solutions on large smooth domains to the Kirchhoff equation with combined nonlinearities
本文研究了大型光滑星形区域及全空间上具有混合非线性项的 Kirchhoff 方程的给定质量解，分析了其存在性、多重性与渐近行为，并指出势函数的存在导致标准 Pohozaev 恒等式方法失效。
Existence and Multiplicity of Normalized Solutions for Dirac Equations with non-autonomous nonlinearities
本文研究具有非自治非线性项的Dirac方程在L²质量约束下的正规化解问题，通过扰动方法与Lyapunov-Schmidt约化证明了解的存在性，并利用Ljusternik-Schnirelmann定理得到多重性及分支结果。
In this note, we deal with a problem of the type $$\cases {-h\left ( \int_Ω|\nabla u(x)|^2dx\right ) Δu=f(u) & in $Ω$\cr & \cr u_{|\partialΩ}=0\ .\cr}$$ As an application of a new general multiplicity result, we establish the existence of at least three solutions, two of which are global minima of the related energy functional. The only condition assumed on $f$ is that it be subcritical and superlinear: no condition on the behaviour of $f$ at $0$ is requested.
In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities: $$({-}{ Δ})^{\fracα{2}}u- γ\frac{u}{|x|^α}= λf(x) |u|^{q - 2} u + g(x) {\frac{|u|^{p-2}u}{|x|^s}} \ \text{ in } {Ω,} \quad \text{ with Dirichlet boundary condition } u = 0 \ \text{ in } \mathbb{R}^n \setminus Ω,$$ where $Ω\subset \mathbb{R}^n$ is a smooth bounded domain in $\mathbb{R}^n$ containing $0$ in its interior, and $f,g \in C(\overlineΩ)$ with $f^+,g^+ \not\equiv 0$ which may change sign in $\overlineΩ.$ We use the var…
In this work we study the following fractional critical problem $$ (P_λ)=\left\{\begin{array}{ll} (-Δ)^s u=λu^{q} + u^{2^*_{s}-1}, \quad u{>}0 & \mbox{in} Ω\\ u=0 & \mbox{in} \RR^n\setminus Ω\,, \end{array}\right. $$ where $Ω\subset \mathbb{R}^n$ is a regular bounded domain, $λ>0$, $0<s<1$ and $n>2s$. Here $(-Δ)^s$ denotes the fractional Laplace operator defined, up to a normalization factor, by $$ -(-Δ)^s u(x)={\rm P. V.} \int_{\RR^n}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy, \quad x\in \RR^n. $$ Our main results show the existence and multiplicity of solutions to problem $(P_λ)$ for differe…
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。
The paper deals with existence and multiplicity of positive solutions to nonlocal equations with critical Hrardy-Sobolev nonlinearities and external terms. We establish the profile decomposition of the Palais-Smale sequences associated with the functional and existence of at least two positive solutions to the equation.
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。
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