We give a simple proof of the fact that for a large class of quasilinear elliptic equations and systems the solutions that minimize the corresponding energy in the set of all solutions are radially symmetric. We require just continuous nonlinearities and no cooperative conditions for systems. Thus, in particular, our results cannot be obtained by using the moving planes method. In the case of scalar equations, we also prove that any least energy solution has a constant sign and is monotone with respect to the radial variable.
We consider singular solutions to quasilinear elliptic equations under zero Dirichlet boundary condition. Under suitable assumptions on the nonlinearity we deduce symmetry and monotonicity properties of positive solutions via an improved moving plane procedure.
In this paper, we introduce concepts of separable functions in balls and in the whole space, and develop a new method to investigate the qualitative properties of separable functions. We first study the axial symmetry and monotonicity of separable functions in unit circles by geometry analysis, and we prove the uniqueness of the symmetry axis for nontrivial separable functions. Then by using reduction dimension and convex analysis, we get the axial symmetry and monotonicity of separable functions in high dimensional spheres. Based on the above results on unit circles and spheres, we deduce the…
We consider positive singular solutions of PDEs arising from double phase functionals. Exploiting a rather new version of the moving plane method originally developed by Sciunzi, we prove symmetry and monotonicity properties of such solutions.
We establish general assumptions under which a constrained vari- ational problem involving the fractional gradient and a local nonlin- earity admits minimizers.
We extend to the case of a system involving p-Laplacians, the monotonicity and symmetry results of Damascelli and Pacella obtained in the case of a scalar p-Laplace equation with $1<p<2$. For this purpose, we use the moving hyperplanes method and we suppose that the right hand sides are increasing and locally Lipschitz continuous.
We investigate qualitative properties of positive singular solutions of some elliptic systems in bounded and unbounded domains. We deduce symmetry and monotonicity properties via the moving plane procedure. Moreover, in the unbounded case, we study some cooperative elliptic systems involving critical nonlinearities in $\mathbb{R}^n$.
We study monotonicity and 1-dimensional symmetry for positive solutions with algebraic growth of the following elliptic system: \[ \begin{cases} -Δu = -u v^2 & \text{in $\R^N$}\\ -Δv= -u^2 v & \text{in $\R^N$}, \end{cases} \] for every dimension $N \ge 2$. In particular, we prove a Gibbons-type conjecture proposed by H. Berestycki, T. C. Lin, J. Wei and C. Zhao.
We employ Clarkson's inequality to deduce that each extremal of Morrey's inequality is axially symmetric and is antisymmetric with respect to reflection about a plane orthogonal to its axis of symmetry. We also use symmetrization methods to show that each extremal is monotone in the distance from its axis of symmetry and in the direction of its axis when restricted to spheres centered at the intersection of its axis and its antisymmetry plane.
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.
In this paper, we consider radial symmetry property of positive solutions of an integral equation arising from some higher order semi-linear elliptic equations on the whole space $\mathbf{R}^n$. We do not use the usual way to get symmetric result by using moving plane method. The nice thing in our argument is that we only need a Hardy-Littlewood-Sobolev type inequality. Our main result is Theorem 1 below.
We consider nonnegative solutions to $-Δu=f(u)$ in half-planes and strips, under zero Dirichlet boundary condition. Exploiting a rotating$\&$sliding line technique, we prove symmetry and monotonicity properties of the solutions, under very general assumptions on the nonlinearity $f$. In fact we provide a unified approach that works in all the cases $f(0)<0$, $f(0)= 0$ or $f(0)> 0$. Furthermore we make the effort to deal with nonlinearities $f$ that may be not locally-Lipschitz continuous. We also provide explicite examples showing the sharpness of our assumptions on the nonlinear function $f$.
In this paper, we first establish a narrow region principle and a decay at infinity theorem to extend the direct method of moving planes for general fractional $p$-Laplacian systems. By virtue of this method, we can investigate the qualitative properties of the following Schrödinger system with fractional $p$-Laplacian \begin{equation*} \left\{\begin{array}{r@{\ \ }c@{\ \ }ll} \left(-Δ\right)_{p}^{s}u+au^{p-1}& =&f(u,v), \\[0.05cm] \left(-Δ\right)_{p}^{t}v+bv^{p-1}& =&g(u,v), \end{array}\right. \end{equation*} where $0<s,\,t<1$ and $2<p<\infty$. We obtain the radial symmetry in the unit ball o…
We advance a variational method to prove qualitative properties such as symmetries, monotonicity, upper and lower bounds, sign properties, and comparison principles for a large class of doubly-nonlinear evolutionary problems including gradient flows, some nonlocal problems, and systems of nonlinear parabolic equations. Our method is based on the so-called Weighted-Energy-Dissipation (WED) variational approach. This consists in defining a global parameter-dependent functional over entire trajectories and proving that its minimizers converge to solutions to the target problem as the parameter go…
In this paper we shall study qualitative properties of a $p$-Stokes type system, namely $$ -{\boldsymbol Δ}_p{\boldsymbol u}=-\operatorname{\bf div}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x,{\boldsymbol u})\,\, \mbox{ in $Ω$},$$ where ${\boldsymbol Δ}_p$ is the $p$-Laplacian vectorial operator. More precisely, under suitable assumptions on the domain $Ω$ and the function $\boldsymbol{ f}$, it is deduced that system solutions are symmetric and monotone. Our main results are derived from a vectorial version of the weak and strong comparison principles, which enable to procee…
We consider a nonlocal equation set in an unbounded domain with the epigraph property. We prove symmetry, monotonicity and rigidity results. In particular, we deal with halfspaces, coercive epigraphs and epigraphs that are flat at infinity.
This paper deals with solutions of semilinear elliptic equations of the type \[ \left\{\begin{array}{ll} -Δu = f(|x|, u) \qquad & \text{ in } Ω, \\ u= 0 & \text{ on } \partial Ω, \end{array} \right. \] where $Ω$ is a radially symmetric domain of the plane that can be bounded or unbounded. We consider solutions $u$ that are invariant by rotations of a certain angle $θ$ and which have a bound on their Morse index in spaces of functions invariant by these rotations. We can prove that or $u$ is radial, or, else, there exists a direction $e\in \mathcal S$ such that $u$ is symmetric with respect to …
In this manuscript we consider semilinear PDEs, with a convex nonlinearity, in a sector-like domain. Using cylindrical coordinates $(r, θ, z)$, we investigate the shape of solutions whose derivative in $θ$ vanishes at the boundary. We prove that any solution with Morse index less than two must be either independent of $θ$ or strictly monotone with respect to $θ$. In the special case of a planar domain, the result holds in a circular sector as well as in an annular, and it can also be extended to a rectangular domain. The corresponding problem in higher dimensions is also considered, as well as…
This paper is concerned about maximum principles and radial symmetry for viscosity solutions of fully nonlinear partial differential equations. We obtain the radial symmetry and monotonicity properties for nonnegative viscosity solutions of begin{equation}F (D^2 u)+u^p=0 \quad \quad \mbox{in}\ \mathbb R^n \label{abs}\end{equation} under the asymptotic decay rate $u=o(|x|^{-\frac{2}{p-1}})$ at infinity, where $p>1$ (Theorem 1, Corollary 1). As a consequence of our symmetry results, we obtain the nonexistence of any nontrivial and nonnegative solution when $F$ is the Pucci extremal operators (Co…
Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. Recently Brendle and Hung proved a sharp area bound for minimal submanifolds when the prescribed point is not the centre of the ball, which resolved a conjecture of Alexander, Hoffman and Osserman. Their proof involves asymptotic analysis of an ingeniously chosen vector field, and…
In this paper we obtain symmetry and monotonicity results for positive solutions to some $p$-Laplacian cooperative systems in bounded domains involving first order terms and under zero Dirichlet boundary condition.
In this paper, by the method of moving planes, we establish the monotonicity and symmetry properties of convex solutions for Monge-Ampere systems on bounded smooth planar domains.
We consider positive solutions to a singular semilinear elliptic equation in bounded smooth domains, with zero Dirichlet boundary conditions. We provide some weak and strong maximum principles for the H^1_0 part of the solution that allow to deduce symmetry and monotonicity properties of the solutions, via the Moving Plane Method.
In this paper, we prove even symmetry and monotonicity of certain solutions of Allen-Cahn equation in a half plane. We also show that entire solutions with {\it finite Morse index} and {\it four ends} must be evenly symmetric with respect to two orthogonal axes. A classification scheme of general entire solutions with {\it finite Morse index} is also presented using energy quantization.
The present study is concerned with the following Schrödinger-Poisson system involving critical nonlocal term with general nonlinearity: $$ \left\{ \begin{array}{ll} -Δu+V(x)u- ϕ|u|^3u= f(u), & x\in\mathbb{R}^3, -Δϕ= |u|^5, & x\in\mathbb{R}^3,\\ \end{array} \right. $$ Under certain assumptions on non-constant $V(x)$, the existence of a positive least energy solution is obtained by using some new analytical skills and Pohožaev type manifold. In particular, the Ambrosetti-Rabinowitz type condition or monotonicity assumption on the nonlinearity is not necessary.
In this paper we prove regularity results for a class of nonlinear degenerate elliptic equations of the form $\displaystyle -\operatorname{div}(A(|\nabla u|)\nabla u)+B\left( |\nabla u|\right) =f(u)$; in particular, we investigate the second order regularity of the solutions. As a consequence of these results, we obtain symmetry and monotonicity properties of positive solutions for this class of degenerate problems in convex symmetric domains via a suitable adaption of the celebrated moving plane method of Alexandrov-Serrin.
For a general class of autonomous quasi-linear elliptic equations on R^n we prove the existence of a least energy solution and show that all least energy solutions do not change sign and are radially symmetric up to a translation in R^n.
We consider the Choquard equation (also known as stationary Hartree equation or Schrödinger--Newton equation) \[ -Δu + u = (I_α\star |u|^p) |u|^{p - 2}u. \] Here $I_α$ stands for the Riesz potential of order $α\in (0,N)$, and $\frac{N - 2}{N + α} < \frac{1}{p} \le \frac{1}{2}$. We prove that least energy nodal solutions have an odd symmetry with respect to a hyperplane when $α$ is either close to $0$ or close to $N$.
In this paper, we consider nonlinear equations involving the fractional p-Laplacian $$ (-\lap)_p^s u(x)) \equiv C_{n,s,p} PV \int_{\mathbb{R}^n} \frac{|u(x)-u(y)|^{p-2}[u(x)-u(y)]}{|x-z|^{n+ps}} dz= f(x,u).$$ We prove a {\em maximum principle for anti-symmetric functions} and obtain other key ingredients for carrying on the method of moving planes, such as {\em a key boundary estimate lemma}. Then we establish radial symmetry and monotonicity for positive solutions to semilinear equations involving the fractional p-Laplacian in a unit ball and in the whole space. We believe that the methods de…
We study the pure Neumann Lane-Emden problem in a bounded domain \[ -Δu = |u|^{p-1} u \text{ in }Ω, \qquad \partial_νu=0 \text{ on }\partial Ω, \] in the subcritical, critical, and supercritical regimes. We show existence and convergence of least-energy (nodal) solutions (l.e.n.s.). In particular, we prove that l.e.n.s. converge to a l.e.n.s. of a problem with sign nonlinearity as $p\searrow 0$; to a l.e.n.s. of the critical problem as $p\nearrow 2^*$ (in particular, pure Neumann problems exhibit no blowup phenomena at the critical Sobolev exponent $2^*$); and we show that the limit as $p\to 1…
We give more precision on the regularity of the domain that is needed to have the monotonicity and symmetry results recently proved by Damascelli and Pacella, result concerning p-Laplace equations. For this purpose, we study the continuity and semicontinuity of some parameters linked with the moving hyperplane method.
We consider least energy solutions to the nonlinear equation $-Δ_g u=f(r,u)$ posed on a class of Riemannian models $(M,g)$ of dimension $n\ge 2$ which include the classical hyperbolic space $\mathbb H^n$ as well as manifolds with unbounded sectional geometry. Partial symmetry and existence of least energy solutions is proved for quite general nonlinearities $f(r,u)$, where $r$ denotes the geodesic distance from the pole of $M$.
We consider $\mathbb{S}^2$-valued maps on a domain $Ω\subset\mathbb{R}^N$ minimizing a perturbation of the Dirichlet energy with vertical penalization in $Ω$ and horizontal penalization on $\partialΩ$. We first show the global minimality of universal constant configurations in a specific range of the physical parameters using a Poincaré-type inequality. Then, we prove that any energy minimizer takes its values into a fixed meridian of the sphere $\mathbb{S}^2$, and deduce uniqueness of minimizers up to the action of the appropriate symmetry group. We also prove a comparison principle for minim…
In this paper we focus on existence and symmetry properties of solutions to the cubic Schrödinger system \[ -Δu_i +λ_i u_i = \sum_{j=1}^d β_{ij} u_j^2 u_i \quad \text{in $Ω\subset \mathbb{R}^N$},\qquad i=1,\dots d \] where $d\geq 2$, $λ_i,β_{ii}>0$, $β_{ij}=β_{ji}\in \mathbb{R}$ for $j\neq i$, $N=2,3$. The underlying domain $Ω$ is either bounded or the whole space, and $u_i\in H^1_0(Ω)$ or $u_i\in H^1_{rad}(\mathbb{R}^N)$ respectively. We establish new existence and symmetry results for least energy positive solutions in the case of mixed cooperation and competition coefficients, as well as in…
We show that solutions of the Yamabe equation on certain n-dimensional non-compact Riemannian manifolds which are bounded and L^p for p=2n/(n-2) are also L^2. This L^p-L^2-implication provides explicit constants in the surgery-monotonicity formula for the smooth Yamabe invariant in a previous article of the authors. As an application we see that the smooth Yamabe invariant of any 2-connected compact 7-dimensional manifold is at least 74.5. Similar conclusions follow in dimension 8 and in dimensions at least 11.
We consider bounded solutions of the nonlocal Allen-Cahn equation $$ (-Δ)^s u=u-u^3\qquad{\mbox{ in }}{\mathbb{R}}^3,$$ under the monotonicity condition $\partial_{x_3}u>0$ and in the genuinely nonlocal regime in which~$s\in\left(0,\frac12\right)$. Under the limit assumptions $$ \lim_{x_n\to-\infty} u(x',x_n)=-1\quad{\mbox{ and }}\quad \lim_{x_n\to+\infty} u(x',x_n)=1,$$ it has been recently shown that~$u$ is necessarily $1$D, i.e. it depends only on one Euclidean variable. The goal of this paper is to obtain a similar result without assuming such limit conditions. This type of results can be …
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…
In this paper we prove existence of least energy nodal solutions for the Hamiltonian elliptic system with Hénon-type weights \[ -Δu = |x|^β |v|^{q-1}v, \quad -Δv =|x|^α|u|^{p-1}u\quad { in } Ω, \qquad u=v=0 { on } \partial Ω, \] where $Ω$ is a bounded smooth domain in $\mathbb{R}^N$, $N\geq 1$, $α, β\geq 0$ and the nonlinearities are superlinear and subcritical, namely \[ 1> \frac{1}{p+1}+\frac{1}{q+1}> \frac{N-2}{N}. \] When $Ω$ is either a ball or an annulus centred at the origin and $N \geq 2$, we show that these solutions display the so-called foliated Schwarz symmetry. It is natural to co…
We study the nonlinear fractional reaction diffusion equation $\partial_{t}u + (-Δ)^{s} u= f(t,x,u)$, $s\in(0,1)$ in a bounded domain $Ω$ together with Dirichlet boundary conditions on $\R^N \setminus Ω$. We prove asymptotic symmetry of nonnegative globally bounded solutions in the case where the underlying data obeys some symmetry and monotonicity assumptions. More precisely, we assume that $Ω$ is symmetric with respect to reflection at a hyperplane, say ${x_1=0}$, and convex in the $x_1$-direction, and that the nonlinearity $f$ is even in $x_1$ and nonincreasing in $|x_1|$. Under rather weak…
In this paper, we consider equations involving fully nonlinear nonlocal operators $$F_α(u(x)) \equiv C_{n,α} PV \int_{\mathbb{R}^n} \frac{G(u(x)-u(z))}{|x-z|^{n+α}} dz= f(x,u).$$ We prove a maximum principle and obtain key ingredients for carrying on the method of moving planes, such as narrow region principle and decay at infinity. Then we establish radial symmetry and monotonicity for positive solutions to Dirichlet problems associated to such fully nonlinear fractional order equations in a unit ball and in the whole space, as well as non-existence of solutions on a half space. We believe th…
In a recent paper we studied an equation (called the "simple equation") introduced by one of us in 1963 for an approximate correlation function associated to the ground state of an interacting Bose gas. Solving the equation yields a relation between the density $ρ$ of the gas and the energy per particle. Our construction of solutions gave a well-defined function $ρ(e)$ for the density as a function of the energy $e$. We had conjectured that $ρ(e)$ is a strictly monotone increasing function, so that it can be inverted to yield the strictly monotone increasing function $e(ρ)$. We had also conjec…
Of concern is the a priori symmetry of traveling wave solutions for a general class of nonlocal dispersive equations \[ u_t +(u^2 +Lu)_x =0, \] where $L$ is a Fourier multiplier operator with symbol $m$. Our analysis includes both homogeneous and inhomogeneous symbols. We characterize a class of symbols $m$ guaranteeing that periodic traveling wave solutions are symmetric under a mild assumption on the wave profile. Particularly, instead of considering waves with a unique crest and trough per period or a monotone structure near troughs as classically imposed in the water wave problem, we formu…
