In this paper, we study isolated singular positive solutions for the following Kirchhoff--type Laplacian problem: \begin{equation*} -\left(θ+\int_Ω |\nabla u| dx\right)Δu =u^p \quad{\rm in}\quad Ω\setminus \{0\},\qquad u=0\quad {\rm on}\quad \partial Ω, \end{equation*} where $p>1$, $θ\in \R$, $Ω$ is a bounded smooth domain containing the origin in $\R^N$ with $N\ge 2$. In the subcritical case: $1<p<N/(N-2)$ if $N\ge3$, $1<p<+\infty$ if $N=2$, we employ the Schauder fixed-point theorem to derive a sequence of positive isolated singular solutions for the above problem such that $M_θ(u)>0$. To estimate $M_θ(u)$, we make use of the rearrangement argument. Furthermore, we obtain a sequence of isolated singular solutions such that $M_θ(u)<0$, by analyzing relationship between the parameter $λ$ and the unique solution $u_λ$ of $$-Δu+λu^p=kδ_0\quad{\rm in}\quad B_1(0),\qquad u=0\quad {\rm on}\quad \partial B_1(0).$$ In the supercritical case: $N/(N-2)\le p<(N+2)/(N-2)$ with $N\ge3$, we obtain two isolated singular solutions $u_i$ with $i=1,2$ such that $M_θ(u_i)>0$ under some appropriate assumptions.
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.
We study a nonlocal parametric problem driven by the fractional Laplacian operator combined with a Kirchhoff-type coefficient and involving a critical nonlinearity term in the sense of Sobolev embeddings. Our approach is of variational and topological nature. The obtained results can be viewed as a nontrivial extension to the nonlocal setting of some recent contributions already present in the literature.
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.
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.
In this paper, we investigate the existence of solutions for a class of $p$-Laplacian fractional order Kirchhoff-type system with Riemann-Liouville fractional derivatives and a parameter $λ$. By mountain pass theorem, we obtain that system has at least one non-trivial weak solution $u_λ$ under some local superquadratic conditions for each given large parameter $λ$. We get a concrete lower bound of the parameter $λ$, and then obtain two estimates of weak solutions $u_λ$. We also obtain that $u_λ\to 0$ if $λ$ tends to $\infty$. Finally, we present an example as an application of our results.
In this article, we deal with the existence of non-negative solutions of the class of following non local problem $$ \left\{ \begin{array}{lr} \quad - M\left(\displaystyle\int_{\mathbb R^n}\int_{\mathbb R^{n}} \frac{|u(x)-u(y)|^{\frac{n}{s}}}{|x-y|^{2n}}~dxdy\right) (-Δ)^{s}_{n/s} u=\left(\displaystyle\int_Ω\frac{G(y,u)}{|x-y|^μ}~dy\right)g(x,u) \; \text{in}\; Ω,\\ \quad \quad u =0\quad\text{in} \quad \mathbb R^n \setminus Ω, \end{array} \right. $$ where $(-Δ)^{s}_{n/s}$ is the $n/s$-fractional Laplace operator, $n\geq 1$, $s\in(0,1)$ such that $n/s\geq 2$, $Ω\subset \mathbb R^n$ is a bounded …
The inf-sup condition, also called the Ladyzhenskaya--Babu\v ska--Brezzi (LBB) condition, ensures the existence, uniqueness and well-posedness of a saddle point problem, relative to a partial differential equation. Discretization by the finite element method gives the discrete problem which must satisfy the discrete inf-sup condition. But, depending on the choice of finite elements, the discrete condition may fail. This paper attempts to explain why it fails from an engineer's perspective, and reviews current methods to work around this failure. The last part recalls the mathematical bases.
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. } …
In this paper, we consider the following critical fractional Kirchhoff equation \begin{equation*} \Big(a+b{\int_{\mathbb{R}^{N}}}|(-Δ)^{\frac{s}{2}}u|^2dx\Big)(-Δ)^su=|u|^{2^*_s-2}u,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*} where $a,b>0$, $\frac{N}{4}<s<1$, $2^*_s=\frac{2N}{N-2s}$ and $(-Δ)^s$ is the fractional Laplacian. We prove the uniqueness and nondegeneracy of positive solutions to the problem, which can be used to study the singular perturbation problems concerning fractional Kirchhoff equations.
In this paper, we study the fractional Kirchhoff equation with critical nonlinearity \begin{align*} \left(a+b\int_{\mathbb R^N}|(-Δ)^{\frac{s}{2}}u|^2dx\right)(-Δ)^su+u=f(u)\ \ \mbox{in}\ \ \mathbb R^N, \end{align*} where $N>2s$ and $(-Δ)^s$ is the fractional Laplacian with $0<s<1$. By using a perturbation approach, we prove the existence of solutions to the above problem without the Ambrosetti-Rabinowitz condition when the parameter $b$ small. What's more, we obtain the asymptotic behavior of solutions as $b\to 0$.
We prove a three sphere inequality with optimal exponent at the boundary for solutions to the Kirchhoff-Love plate's equation satisfying homogeneous Dirichlet conditions. This result implies the Strong Unique Continuation Property at the Boundary (SUCPB). Our approach is based on the method of Carleman estimates, and involves the construction of an ad hoc conformal mapping preserving the structure of the operator and the employment of a suitable reflection of the solution with respect to the flatten boundary which ensures the needed regularity of the extended solution. To the authors' knowledg…
We introduce the study of isolated singularities for a semilinear equation involving the fractional Laplacian. In conformal geometry, it is equivalent to the study of singular metrics with constant fractional curvature. Our main ideas are: first, to set the problem into a natural geometric framework, and second, to perform some kind of phase portrait study for this non-local ODE.
A Gevrey class semigroup, exponential decay and Lack of analyticity for a system formed by a Kirchhoff-Love plate equation and the equation of a membrane-like electric network with indirect fractional damping
In this paper we deal with the multiplicity and concentration of positive solutions for the following fractional Schrödinger-Kirchhoff type equation \begin{equation*} M\left(\frac{1}{\varepsilon^{3-2s}} \iint_{\mathbb{R}^{6}}\frac{|u(x)- u(y)|^{2}}{|x-y|^{3+2s}} dxdy + \frac{1}{\varepsilon^{3}} \int_{\mathbb{R}^{3}} V(x)u^{2} dx\right)[\varepsilon^{2s} (-Δ)^{s}u+ V(x)u]= f(u) \, \mbox{in} \mathbb{R}^{3} \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (\frac{3}{4}, 1)$, $(-Δ)^{s}$ is the fractional Laplacian, $M$ is a Kirchhoff function, $V$ is a continuous positive potential …
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…
We consider the following $(p, q)$-Laplacian Kirchhoff type problem \begin{align*} \begin{split} &-\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{p}\, dx \right)Δ_{p}u - \left(c+d\int_{\mathbb{R}^{3}}|\nabla u|^{q}\, dx \right ) Δ_{q}u + V(x) (|u|^{p-2}u + |u|^{q-2}u)= K(x) f(u) \quad \mbox{ in } \mathbb{R}^{3}, \end{split} \end{align*} where $a, b, c, d>0$ are constants, $\frac{3}{2}< p< q<3$, $V: \mathbb{R}^{3}\rightarrow \mathbb{R}$ and $K: \mathbb{R}^{3}\rightarrow \mathbb{R}$ are positive continuous functions allowed vanishing behavior at infinity, and $f$ is a continuous function with quasicr…
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.
This paper is concerned with the following fractional $p$-Kirchhoff equation \begin{eqnarray*} \varepsilon ^{sp}M\left( {\varepsilon ^{sp - N}}\iint_{\mathbb{R}^{2N}}\frac{{{{\left| {u(x) - u(y)} \right|}^p}}}{{{{\left| {x - y} \right|}^{N + sp}}}}dxdy\right)(-Δ)_p^su + V(x){u^{p - 1}} = {u^{p_s^* - 1}}+f(u),\ \ u>0, \ \mbox{in}\ {\mathbb{R}^N}, %u \in {W^{s,p}}(\mathbb{R}^N), \end{eqnarray*} where $\varepsilon>0$ is a parameter, $M(t)=a+bt^{θ-1}$ with $a>0$, $b>0$, $θ>1$, $(-Δ)_p^s$ denotes the fractional $p$-Laplacian operator with $0<s<1$ and $1<p<\infty$, $N>sp$, $θp<p_s^*$ with $p_s^*=\fr…
This paper is concerned with the multiplicity and concentration behavior of nontrivial solutions for the following fractional Kirchhoff equation in presence of a magnetic field: \begin{equation*} \left(a\varepsilon^{2s}+b\varepsilon^{4s-3} [u]_{A/\varepsilon}^{2}\right)(-Δ)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon>0$ is a small parameter, $a, b>0$ are constants, $s\in (\frac{3}{4}, 1)$, $(-Δ)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential, $V:\mathbb{R}…
In this paper we consider the following class of fractional Kirchhoff equations with critical growth: \begin{equation*} \left\{ \begin{array}{ll} \left(\varepsilon^{2s}a+\varepsilon^{4s-3}b\int_{\mathbb{R}^{3}}|(-Δ)^{\frac{s}{2}}u|^{2}dx\right)(-Δ)^{s}u+V(x)u=f(u)+|u|^{2^{*}_{s}-2}u \quad &\mbox{ in } \mathbb{R}^{3}, \\ u\in H^{s}(\mathbb{R}^{3}), \quad u>0 &\mbox{ in } \mathbb{R}^{3}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $a, b>0$ are constants, $s\in (\frac{3}{4}, 1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-Δ)^{s}$ is the …
We prove an abstract critical point theorem based on a cohomological index theory that produces pairs of nontrivial critical points with nontrivial higher critical groups. This theorem yields pairs of nontrivial solutions that are neither local minimizers nor of mountain pass type for problems with combined nonlinearities. Applications are given to subcritical and critical $p$-Laplacian problems, Kirchhoff type nonlocal problems, and critical fractional $p$-Laplacian problems.
We study the behavior of positive solutions of p-Laplacian type elliptic equations of the form Q'(u) := -p-Laplacian(u) + V |u|^(p-2) u = 0 in Omega near an isolated singular point zeta, where 1 < p < inf, Omega is a domain in R^d with d > 1, and zeta = 0 or zeta = inf. We obtain removable singularity theorems for positive solutions near zeta. In particular, using a new three-spheres theorems for certain solutions of the above equation near zeta we prove that if V belongs to a certain Kato class near zeta and p>d (respectively, p<d), then any positive solution u of the equation Q'(u)=0 in a pu…
We study some qualitative properties (including removable singularities and superharmonicity) of non-negative solutions to $$ (-Δ)^γu=fu^p\quad\text{in }\mathbb R^n\setminusΣ$$ which are singular at $Σ$. Here $γ\in (0, \frac{n}{2})$. Among other things, we first prove that if $Σ$ is a compact set in $\mathbb R^n$ with Assouad dimension $\bf d$ (not necessarily an integer), ${\bf d}<n-2γ$, and $ u\in L_γ(\mathbb R^n)\cap L^p_{loc}({\mathbb R^n\setminusΣ})$ is a non-negative solution for some $$p>\frac{n-\bf d}{n-{\bf d}-2γ},$$ then $u\in L^p_{loc}(\mathbb R^n)$ and $u$ is a distributional solut…
In this paper, we show the existence and multiplicity of positive solutions of the following fractional Kirchhoff system\\ \begin{equation} \left\{ \begin{array}{rllll} \mc L_M(u)&=λf(x)|u|^{q-2}u+ \frac{2α}{α+β}\left|u\right|^{α-2}u|v|^β& \text{in } Ω,\\ \mc L_M(v)&=μg(x)|v|^{q-2}v+ \frac{2β}{α+β}\left|u\right|^α|v|^{β-2}v & \text{in } Ω,\\ u&=v=0 &\mbox{in } \mathbb{R}^{N}\setminus Ω, \end{array} \right. \end{equation} where $\mc L_M(u)=M\left(\displaystyle \int_Ω|(-Δ)^{\frac{s}{2}}u|^2dx\right)(-Δ)^{s} u $ is a double non-local operator due to Kirchhoff term $M(t)=a+b t$ with $a, b>0$ and f…
We investigate the existence, multiplicity and concentration of nontrivial solutions for the following fractional magnetic Kirchhoff equation with critical growth: \begin{equation*} \left(a\varepsilon^{2s}+b\varepsilon^{4s-3} [u]_{A/\varepsilon}^{2}\right)(-Δ)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u+|u|^{\2-2}u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon$ is a small positive parameter, $a, b>0$ are fixed constants, $s\in (\frac{3}{4}, 1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-Δ)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{…
In this paper, we consider the following singularly perturbed fractional Kirchhoff problem \begin{equation*} \Big(\varepsilon^{2s}a+\varepsilon^{4s-N} b{\int_{\mathbb{R}^{N}}}|(-Δ)^{\frac{s}{2}}u|^2dx\Big)(-Δ)^su+V(x)u=|u|^{p-2}u,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*} where $a,b>0$, $2s<N<4s$ with $s\in(0,1)$, $2<p<2^*_s=\frac{2N}{N-2s}$ and $(-Δ)^s$ is the fractional Laplacian. For $\varepsilon> 0$ sufficiently small and a bounded continuous function $V$, we establish a type of local Pohozǎev identity by extension technique and then we can obtain the local uniqueness of semiclassica…
In this paper, we first establish the uniqueness and non-degeneracy of positive solutions to the fractional Kirchhoff problem \begin{equation*} \Big(a+b{\int_{\mathbb{R}^{N}}}|(-Δ)^{\frac{s}{2}}u|^2dx\Big)(-Δ)^su+mu=|u|^{p-2}u,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*} where $a,b,m>0$, $0<\frac{N}{4}<s<1$, $2<p<2^*_s=\frac{2N}{N-2s}$ and $(-Δ)^s$ is the fractional Laplacian. Then, combining this non-degeneracy result and Lyapunov-Schmidt reduction method, we derive the existence of semiclassical solutions to the singularly perturbation problem \begin{equation*} \Big(\varepsilon^{2s}a+\va…
Green functions play an important role in conformal geometry. In this paper, we explain how to compute explicitly the logarithmic singularities of the Green functions of the conformal powers of the Laplacian. These operators include the Yamabe and Paneitz operators, as well as the conformal fractional powers of the Laplacian arising from scattering theory for Poincaré-Einstein metrics. The results are formulated in terms of Weyl conformal invariants arising from the ambient metric of Fefferman-Graham. As applications we obtain "Green function" characterizations of locally conformally flat mani…
The Schwarz function has played an elegant role in understanding and in generating new examples of exact solutions to the Laplacian growth (or "Hele- Shaw") problem in the plane. The guiding principle in this connection is the fact that "non-physical" singularities in the "oil domain" of the Schwarz function are stationary, and the "physical" singularities obey simple dynamics. We give an elementary proof that the same holds in any number of dimensions for the Schwarz potential, introduced by D. Khavinson and H. S. Shapiro [17] (1989). A generalization is also given for the so-called "elliptic…
We study the existence of {weak} solutions for fractional elliptic equations of the type, \begin{equation*} (-Δ)^{\frac{1}{2}} u+ V(x) u= h(u), u> 0 \;\textrm{in} \;\mathbb R, \end{equation*} %where $1<q<2,\;p>2,\;1<β\leq2\;, λ>0, K(x)>0, f$ is continuous and sign changing. where $h$ is a real valued function that behaves like $e^{u^2}$ as $u\rightarrow \infty$ and $V(x)$ is a positive, continuous unbounded function. Here $(-Δ)^{\frac{1}{2}}$ is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near $t=0$. We also study the …
We consider the problem of attainability of the best constant in the following critical fractional Hardy-Sobolev inequality: \begin{equation*} μ_{γ,s}(\R^n):= \inf\limits_{u \in H^{\fracα{2}} (\R^n)\setminus \{0\}} \frac{ \int_{\R^n} |({-}{ Δ})^{\fracα{4}}u|^2 dx - γ\int_{\R^n} \frac{|u|^2}{|x|^α}dx }{(\int_{\R^n} \frac{|u|^{2_α^*(s)}}{|x|^{s}}dx)^\frac{2}{2_α^*(s)}}, \end{equation*} where $0\leq s<α<2$, $n>α$, ${2_α^*(s)}:=\frac{2(n-s)}{n-α},$ and $γ\in \mathbb{R}$. This allows us to establish the existence of nontrivial weak solutions for the following doubly critical problem on $\R^n$, \beg…
We present and prove Polyakov-Alvarez type comparison formulas for the determinants of Friederichs extensions of Laplacians corresponding to conformally equivalent metrics on a compact Riemann surface with conical singularities. In particular, we find how the determinants depend on the orders of conical singularities. We also illustrate these general results with several examples: based on our Polyakov-Alvarez type formulas we recover known and obtain new explicit formulas for determinants of Laplacians on singular surfaces with and without boundary. In one of the examples we show that on the …
Realizations of differential operators subject to differential boundary conditions on manifolds with conical singularities are shown to have a bounded $H_{\infty}$-calculus in appropriate $L_{p}$-Sobolev spaces provided suitable conditions of parameter-ellipticity are satisfied. Applications concern the Dirichlet and Neumann Laplacian and the porous medium equation.
We study a behavior of the conformal Laplacian operator $Ł_g$ on a manifold with \emph{tame conical singularities}: when each singularity is given as a cone over a product of the standard spheres. We study the spectral properties of the operator $Ł_g$ on such manifolds. We describe the asymptotic of a general solution of the equation $Ł_g u = Q u^α$ with $1\leq α\leq \frac{n+2}{n-2}$ near each singular point. In particular, we derive the asymptotic of the Yamabe metric near such singularity.
In this paper we establish the subelliptic Picone type identities. As consequences, we obtain Hardy and Rellich type inequalities for anisotropic p-sub- Laplacians which are operators of the form $$ \mathcal{L}_p f := \sum_{i=1}^{N}X_i(|X_if|^{p_i-2}X_i f), \quad 1<p_i< \infty, $$ where $X_i, i=1,\ldots, N,$ are the generators of the first stratum of a stratified (Lie) group. Moreover, analogues of Hardy type inequalities with multiple singularities and many-particle Hardy type inequalities are obtained on stratified groups.
In this paper, we solve the fractional Lane-Emden equation in the Serrin's critical case for the fractional Laplacian by developing an innovative and self-contained approach that also applies to the classical setting ( Laplacian). We give a classification of the isolated singularities of positive solutions to the semilinear fractional elliptic equations $$(E) \qquad\qquad (-Δ)^s u = u^{\frac{N}{N-2s}}\quad {\rm in}\ \ Ω\setminus\{0\},\qquad u\geq 0\quad{\rm in}\ \ \mathbb{R}^N\setminusΩ,\qquad\qquad\quad $$ where $s\in(0,1)$, $Ω$ is a bounded domain containing the origin in $\mathbb{R}^N$ with…
We prove existence, uniqueness and optimal regularity of solutions to the stationary obstacle problem defined by the fractional Laplacian operator with drift, in the subcritical regime. We localize our problem by considering a suitable extension operator introduced by L. Caffarelli and L. Silvestre. The structure of the extension equation is different from the one considered by L. Caffarelli, S. Salsa and L. Silvestre in their study of the obstacle problem for the fractional Laplacian without drift, in that the obstacle function has less regularity, and exhibits some singularities. To take int…
We study the existence and multiplicity of solutions for a class of fractional Schrödinger-Kirchhoff type equations with the Trudinger-Moser nonlinearity. More precisely, we consider \begin{gather*} \begin{cases} M\big(\|u\|^{N/s}\big)\left[(-Δ)^s_{N/s}u+V(x)|u|^{\frac{N}{s}-1}u\right]= f(x,u) +λh(x)|u|^{p-2}u\, &{\rm in}\ \ \mathbb{R}^N,\\ \|u\|=\left(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{N/s}}{|x-y|^{2N}}dxdy+\int_{\mathbb{R}^N}V(x)|u|^{N/s}dx\right)^{s/N}, \end{cases}\end{gather*} where $M:[0,\infty]\rightarrow [0,\infty)$ is a continuous function, $s\in (0,1)$, $N\geq2$, $λ>0$ is a par…
We obtain left and right continuous embeddings for the domains of the complex powers of sectorial $\mathbb{B}$-elliptic cone differential operators. We apply this result to the heat equation on manifolds with conical singularities and provide asymptotic expansions of the unique solution close to the conical points. We further show that the decomposition of the solution in terms of asymptotics spaces, i.e. finite dimensional spaces that describe the domains of the integer powers of the Laplacian and determined by the local geometry around the singularity, is preserved under the evolution.
