We study the nodal set of solutions to equations of the form $$ (-Δ)^s u = λ_+ (u_+)^{q-1} - λ_- (u_-)^{q-1}\quad\text{in $B_1$}, $$ where $λ_+,λ_->0, q \in [1,2)$, and $u_+$ and $u_-$ are respectively the positive and negative part of $u$. This collection of nonlinearities includes the unstable two-phase membrane problem $q=1$ as well as sublinear equations for $1<q<2$. We initially prove the validity of the strong unique continuation property and the finiteness of the vanishing order, in order to implement a blow-up analysis of the nodal set. As in the local case $s=1$, we prove that the admissible vanishing orders can not exceed the critical value $k_q= 2s/(2- q)$. Moreover, we study the regularity of the nodal set and we prove a stratification result. Ultimately, for those parameters such that $k_q< 1$, we prove a remarkable difference with the local case: solutions can only vanish with order $k_q$ and the problem admits one dimensional solutions. Our approach is based on the validity of either a family of Almgren-type or a 2-parameter family of Weiss-type monotonicity formulas, according to the vanishing order of the solution.
We consider the sublinear problem \begin {equation*} \left\{\begin{array}{r c l c} -Δu & = &|u|^{q-2}u & \textrm{in }Ω, \\ u_n & = & 0 & \textrm{on }\partialΩ,\end{array}\right. \end {equation*} where $Ω\subset \real^N$ is a bounded domain, and $1 \leq q < 2$. For $q=1$, $|u|^{q-2}u$ will be identified with $\sgn(u)$. We establish a variational principle for least energy nodal solutions, and we investigate their qualitative properties. In particular, we show that they satisfy a unique continuation property (their zero set is Lebesgue-negligible). Moreover, if $Ω$ is radial, then least energy n…
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 are concerned with the nodal set of solutions to equations of the form \begin{equation*} -Δu = λ_+ \left(u^+\right)^{q-1} - λ_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $λ_+,λ_- > 0$, $q \in [1,2)$, $B_1=B_1(0)$ is the unit ball in $\mathbb{R}^N$, $N \ge 2$, and $u^+:= \max\{u,0\}$, $u^-:= \max\{-u,0\}$ are the positive and the negative part of $u$, respectively. This class includes, the \emph{unstable two-phase membrane problem} ($q=1$), as well as \emph{sublinear} equations for $1<q<2$. We prove the following main results: (a) the finiteness of the vanishing orde…
In this paper, we are interested in the least energy nodal solutions to the following nonlocal Choquard equation with a local term \begin{equation*}\left\{\begin{array}{rll} -Δu&=λ|u|^{p-2}u+μϕ(x)|u|^{q-2}u\\ -Δϕ&=|u|^q\\ u&=ϕ=0 \end{array}\right. \begin{gathered}\begin{array}{rll} &\mbox{in}\ Ω,\\ &\mbox{in}\ Ω,\\ &\mbox{on}\ \partialΩ, \end{array}\end{gathered}\end{equation*} where $ λ,μ>0, p\in [2,6), q\in (1,5)$ and $Ω\subset \mathbb{R}^3$ is a bounded domain. This problem may be seen as a nonlocal perturbation of the classical Lane-Emden equation $-Δu=λ|u|^{p-2}u$ in $Ω.$ The problem has …
In this article we deal with different forms of the unique continuation property for second order elliptic equations with nonlinear potentials of sublinear growth. Under suitable regularity assumptions, we prove the weak and the strong unique continuation property. Moreover, we also discuss the unique continuation property from measurable sets, which shows that nodal domains to these equations must have vanishing Lebesgue measure. Our methods rely on suitable Carleman estimates, for which we include the sublinear potential into the main part of the operator.
In this paper, a Sturm--Liouville problem with some nonlocal boundary conditions of the Bitsadze-Samarskii type is studied. We show that the coefficients of the problem can be uniquely determined by a dense set of nodal points. Moreover, we give an algorithm for the reconstruction of the potential function and some other coefficients in the boundary conditions.
We investigate the local properties, including the nodal set and the nodal properties of solutions to the following parabolic problem of Muckenhoupt-Neumann type: \begin{equation*} \begin{cases} \partial_t \overline{u} - y^{-a} \nabla \cdot(y^a \nabla \overline{u}) = 0 \quad &\text{ in } \mathbb{B}_1^+ \times (-1,0) \\ -\partial_y^a \overline{u} = q(x,t)u \quad &\text{ on } B_1 \times \{0\} \times (-1,0), \end{cases} \end{equation*} where $a\in(-1,1)$, is a fixed parameter $\mathbb{B}_1^+\subset \mathbb{R}^{N+1}$ is the upper unit half ball and $B_1$ is the unit ball in $\mathbb{R}^N$. Our mai…
As well known, classical catenoids in ${\mathbf{R}}^3$ possess logarithmic growth at infinity. In this note we prove that the case of nonlocal minimal surfaces is significantly different, and indeed all nonlocal catenoids must grow at least linearly. More generally, we prove that stationary sets for the nonlocal perimeter functional which grow sublinearly at infinity are necessarily half-spaces.
Our purpose of this paper is to study the isolated singularities of positive solutions to Choquard equation in the sublinear case $q \in (0,1)$ $$\displaystyle \ \ -Δu+ u =I_α[u^p] u^q\;\; {\rm in}\; \mathbb{R}^N\setminus\{0\}, % [2mm] \phantom{ } \;\; \displaystyle \lim_{|x|\to+\infty}u(x)=0, $$ where $p >0, N \geq 3, α\in (0,N)$ and $I_α[u^p](x) = \int_{\mathbb{R}^N} \frac{u^p(y)}{|x-y|^{N-α}}dy$ is the Riesz potential, which appears as a nonlocal term in the equation. We investigate the nonexistence and existence of isolated singular solutions of Choquard equation under different range of t…
In this paper, we consider the existence of nodal solutions with two bubbles to the slightly subcritical problem with the fractional Laplacian \begin{equation*} \left\{\aligned &(-Δ)^su=|u|^{p-1-\varepsilon}u\ \ \mbox{in}\ Ω&u=0\ \mbox{on}\ \partialΩ, \endaligned \right. \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb R^N$, $N>2s$, $0<s<1$, $ p=\frac{N+2s}{N-2s}$ and $\varepsilon>0$ is a small parameter, which can be seen as a nonlocal analog of the results of Bartsch, Micheletti and Pistoia (2006) \cite{Bartsch1}.
We consider the Schrödinger-Poisson-Slater (SPS) system in $\R^3$ and a nonlocal SPS type equation in balls of $\mathbb R^3$ with Dirichlet boundary conditions. We show that for every $k\in\mathbb N$ each problem considered admits a nodal radially symmetric solution which changes sign exacly $k$ times in the radial variable. Moreover when the domain is the ball of $\mathbb R^3$ we obtain the existence of radial global solutions for the associated nonlocal parabolic problem having $k+1$ nodal regions at every time.
We consider the general Choquard equations $$ -Δu + u = (I_α\ast |u|^p) |u|^{p - 2} u $$ where $I_α$ is a Riesz potential. We construct minimal action odd solutions for $p \in (\frac{N + α}{N}, \frac{N + α}{N - 2})$ and minimal action nodal solutions for $p \in (2,\frac{N + α}{N - 2})$. We introduce a new minimax principle for least action nodal solutions and we develop new concentration-compactness lemmas for sign-changing Palais--Smale sequences. The nonlinear Schrödinger equation, which is the nonlocal counterpart of the Choquard equation, does not have such solutions.
We establish a priori Lipschitz estimates for unbounded solutions of second-order Hamilton-Jacobi equations in R^N in presence of an Ornstein-Uhlenbeck drift. We generalize the results obtained by Fujita, Ishii \& Loreti (2006) in several directions. The first one is to consider more general operators. We first replace the Laplacian by a general diffusion matrix and then consider a nonlocal integro-differential operator of fractional Laplacian type. The second kind of extension is to deal with more general Hamiltonians which are merely sublinear. These results are obtained for both degenerate …
In this paper, we study the existence of least energy nodal solutions for some class of Kirchhoff type problems. Since Kirchhoff equation is a nonlocal one, the variational setting to look for sign-changing solutions is different from the local cases. By using constrained minimization on the sign-changing Nehari manifold, we prove the Kirchhoff problem has a least energy nodal solution with its energy exceeding twice the least energy. As a co-product of our approaches, we obtain the existence of least energy sign-changing solution for Choquard equations and show that the sign-changing solution…
A unified treatment of degenerate nonlocal elliptic problems
University of the National Education Commission（国家教育委员会大学）；CIEM-Conicet, Universidad Nacional de Córdoba（科尔多瓦国立大学CONICET-CEM研究所）；Universidade Federal do Pará（帕拉联邦大学）；Universidade Federal de Goiás（戈亚斯联邦大学）
This paper deals with solutions to the equation \begin{equation*} -Δu = λ_+ \left(u^+\right)^{q-1} - λ_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $λ_+,λ_- > 0$, $q \in (0,1)$, $B_1=B_1(0)$ is the unit ball in $\mathbb{R}^N$, $N \ge 2$, and $u^+:= \max\{u,0\}$, $u^-:= \max\{-u,0\}$ are the positive and the negative part of $u$, respectively. We extend to this class of \emph{singular} equations the results recently obtained in \cite{SoTe2018} for \emph{sublinear and discontinuous} equations, $1\leq q<2$, namely: (a) the finiteness of the vanishing order at every point a…
We study the equation \begin{equation} (-Δ)^{s}u+V(x)u= (I_α*|u|^{p})|u|^{p-2}u+λ(I_β*|u|^{q})|u|^{q-2}u \quad\mbox{ in } \R^{N}, \end{equation} where $I_γ(x)=|x|^{-γ}$ for any $γ\in (0,N)$, $p, q >0$, $α,β\in (0,N)$, $N\geq 3$ and $ λ\in R$. First, the existence of a groundstate solutions using minimization method on the associated Nehari manifold is obtained. Next, the existence of least energy sign-changing solutions is investigated by considering the Nehari nodal set.
We investigate the following problem $$ -{\rm div}(v(x)|\nabla u|^{m-2}\nabla u)+V(x)|u|^{m-2}u= \Big(|x|^{-θ}*\frac{|u|^{b}}{|x|^α}\Big)\frac{|u|^{b-2}}{|x|^α}u+λ\Big(|x|^{-γ}*\frac{|u|^{c}}{|x|^β}\Big)\frac{|u|^{c-2}}{|x|^β}u \quad\mbox{ in }\R^{N}, $$ where $b, c, α, β>0$, $θ,γ\in (0,N)$, $N\geq 3$, $2\leq m< \infty$ and $λ\in \R$. Here, we are concerned with the existence of groundstate solutions and least energy sign-changing solutions and that will be done by using the minimization techniques on the associated Nehari manifold and the Nehari nodal set respectively.
We consider the degenerate parabolic equation with nonlocal source given by \[ u_t=uΔu + u \int_{\mathbb{R}^n} |\nabla u|^2, \] which has been proposed as model for the evolution of the density distribution of frequencies with which different strategies are pursued in a population obeying the rules of replicator dynamics in a continuous infinite-dimensional setting. Firstly, for all positive initial data $u_0\in C^0(\mathbb{R}^n)$ satisfying $u_0\in L^p(\mathbb{R}^n)$ for some $p\in (0,1)$ as well as $\int_{\mathbb{R}^n} u_0=1$, the corresponding Cauchy problem in $\mathbb{R}^n$ is seen to pos…
We consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, whose reaction combines a sublinear term depending on a positive parameter and an asymmetric perturbation (superlinear at positive infinity, at most linear at negative infinity). By means of critical point theory and Morse theory, we prove that, for small enough values of the parameter, such problem admits at least four nontrivial solutions: two positive, one negative, and one nodal. As a tool, we prove a Brezis-Oswald type comparison result.
The paper deals with existence and multiplicity of solutions of the fractional Schrödinger--Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case \begin{equation*} (a+b[u]_{s,A}^{2θ-2})(-Δ)_A^su+V(x)u=f(x,|u|)u\,\, \quad \text{in $\mathbb{R}^N$}, \end{equation*} where $s\in (0,1)$, $N>2s$, $a\in \mathbb{R}^+_0$, $b\in \mathbb{R}^+_0$, $θ\in[1,N/(N-2s))$, $A:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a magnetic potential, $V:\mathbb{R}^N\rightarrow \mathbb{R}^+$ is an electric potential, $(-Δ)_A^s$ is the fractional magnetic o…
In this paper, we study the existence and multiplicity of weak solutions for a general class of elliptic equations (\mathscr{P}_λ) in a smooth bounded domain, driven by a nonlocal integrodifferential operator \mathscr{L}_{\mathcal{A}K} with Dirichlet boundary conditions involving variable exponents without Ambrosetti and Rabinowitz type growth conditions. Using different versions of the Mountain Pass Theorem, as well as, the Fountain Theorem and Dual Fountain Theorem with Cerami condition, we obtain the existence of weak solutions for the problem (\mathscr{P}_λ) and we show that the problem tr…
Recently, several works have been carried out in attempt to develop a theory for linear or sublinear elliptic equations involving a general class of nonlocal operators characterized by mild assumptions on the associated Green kernel. In this paper, we study the Dirichlet problem for superlinear equation (E) ${\mathbb L} u = u^p +λμ$ in a bounded domain $Ω$ with homogeneous boundary or exterior Dirichlet condition, where $p>1$ and $λ>0$. The operator ${\mathbb L}$ belongs to a class of nonlocal operators including typical types of fractional Laplacians and the datum $μ$ is taken in the optimal …
