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 examples of domains where the associated levels do not coincide. For the case where $Ω$ is a ball or annulus and $f$ is of class $C^1$, we prove instead that the levels coincide, and that least energy nodal solutions are nonradial but axially symmetric functions. Finally, we provide stronger results for the Allen-Cahn type nonlinearities in case $Ω$ is either a ball or a square. In particular we give a complete description of the solution set for $λ\sim λ_2(Ω)$, computing the Morse index of the solutions.
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 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 adm…
We present a necessary and sufficient condition on nonnegative Radon measures $μ$ and $ν$ for the existence of a positive continuous solution of the Dirichlet problem for the sublinear elliptic equation $-Δu=μu^q+ν$ with prescribed nonnegative continuous boundary data in a general domain. Moreover, two-sided pointwise estimates of Brezis-Kamin type for positive bounded solutions and the uniqueness of a positive continuous $L^q$-solution are investigated.
Let ϕ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We prove lower bounds for the size of the nodal set {ϕ=0}.
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…
We introduce the class of quasiconvex Lipschitz domains, which covers both $C^1$ and convex domains, to the study of boundary unique continuation for elliptic operators. In particular, we prove the upper bound of the size of nodal sets for Dirichlet eigenfunctions of general elliptic equations in bounded quasiconvex Lipschitz domains. Our result is new even for Laplace operator in convex domains.
Let $Ω$ be a bounded annular $C^{1,1}$ domain in $\mathbb{R}^2$ which is left invariant under the action of the dihedral group $D_n$ of isometries of $\mathbb{R}^2$ .We show that the nodal line of a second Dirichlet eigenfunction must intersect the boundary of $Ω$, under suitable conditions on $\frac{\partial}{\partial θ}$ .
We investigate the existence of a curve $q\mapsto u_{q}$, with $q\in(0,1)$, of positive solutions for the problem $(P_{a,q})$: $-Δu=a(x)u^{q}$ in $Ω$, $u=0$ on $\partialΩ$, where $Ω$ is a bounded and smooth domain of $\mathbb{R}^{N}$ and $a:Ω\rightarrow\mathbb{R}$ is a sign-changing function (in which case the strong maximum principle does not hold). In addition, we analyze the asymptotic behavior of $u_{q}$ as $q\rightarrow0^{+}$ and $q\rightarrow1^{-}$. We also show that in some cases $u_{q}$ is the ground state solution of $(P_{a,q})$. As a byproduct, we obtain existence results for a singu…
Recent work of the authors and their collaborators has uncovered fundamental connections between the Dirichlet-to-Neumann map, the spectral flow of a certain family of self-adjoint operators, and the nodal deficiency of a Laplacian eigenfunction (or an analogous deficiency associated to a non-bipartite equipartition). Using a refined construction of the Dirichlet-to-Neumann map, we strengthen all of these results, in particular getting improved bounds on the nodal deficiency of degenerate eigenfunctions. Our framework is very general, allowing for non-bipartite partitions, non-simple eigenvalu…
We consider Dirichlet-to-Neumann operators associated to $Δ+q$ on a Lipschitz domain in a smooth manifold, where $q$ is an $L^{\infty}$ potential. We prove a Courant-type bound for the nodal count of the extensions $u_k$ of the $k$th Dirichlet-to-Neumann eigenfunctions $ϕ_k$ to the interior satisfying $(Δ+q)u_k=0$. The classical Courant nodal domain theorem is known to hold for Steklov eigenfunctions, which are the harmonic extension of the Dirichlet-to-Neumann eigenfunctions associated to $Δ$. Our result extends it to a larger family of Dirichlet-to-Neumann operators. Our proof makes use of t…
We study the nodal curves of low energy Dirichlet eigenfunctions in generalized curvilinear quadrilaterals. The techniques can be seen as a generalization of the tools developed by Grieser-Jerison in a series of works on convex planar domains and rectangles with one curved edge and a large aspect ratio. Here, we study the structure of the nodal curve in greater detail, in that we find precise bounds on its curvature, with uniform estimates up to the two points where it meets the domain at right angles, and show that many of our results hold for relatively small aspect ratios of the side length…
In this paper we consider some nodal solutions of the Hénon problem in the unit disc with Dirichlet boundary conditions and we show that they are quasiradial, that is to say they are nonradial, they have two nodal regions and their nodal line does not touch the boundary of the disc.
We consider a nonlinear Dirichlet problem driven by the $p$-Laplace differential operator with a reaction which has a subcritical growth restriction only from above. We prove two multiplicity theorems producing three nontrivial solutions, two of constant sign and the third nodal. The two multiplicity theorems differ on the geometry near the origin. In the semilinear case (that is, $p=2$), using Morse theory (critical groups), we produce a second nodal solution for a total of four nontrivial solutions. As an illustration, we show that our results incorporate and significantly extend the multipl…
Let $Ω$ be a bounded domain in $\mathbb{R}^n$ with $C^{1}$ boundary and let $u_λ$ be a Dirichlet Laplace eigenfunction in $Ω$ with eigenvalue $λ$. We show that the $(n-1)$-dimensional Hausdorff measure of the zero set of $u_λ$ does not exceed $C(Ω)\sqrtλ$. This result is new even for the case of domains with $C^\infty$-smooth boundary.
In this article we determine bounds on the maximal order of vanishing for eigenfunctions of a generalized Dirichlet-to-Neumann map (which is associated with fractional Schrödinger equations) on a compact, smooth Riemannian manifold, $(M,g)$, without boundary. Moreover, with only slight modifications these results generalize to equations with $C^1$ potentials. Here Carleman estimates are a key tool. These yield a quantitative three balls inequality which implies quantitative bulk and boundary doubling estimates and hence leads to the control of the maximal order of vanishing. Using the boundary…
In this note we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation $-Δ_p u = f(u)$ in bounded Steiner symmetric domains $Ω\subset \mathbb{R}^N$ under the zero Dirichlet boundary conditions. The nonlinearity $f$ is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet $p$-Laplacian in $Ω$. We show that the nodal set of any least energy sign-changing solution intersects the boundary of $Ω$. The proof is based on a moving polar…
We establish several results related to existence, nonexistence or bifurcation of positive solutions for a Dirichlet boundary value problem with in a smooth bounded domain. The main feature of this paper consists in the presence of a singular nonlinearity, combined with a convection term. Our approach takes into account both the sign of the potential and the decay rate around the origin of the singular nonlinearity. The proofs are based on various techniques related to the maximum principle for elliptic equations.
We establish a positivity property for a class of semilinear elliptic problems involving indefinite sublinear nonlinearities. Namely, we show that any nontrivial nonnegative solution is positive for a class of problems the strong maximum principle does not apply to. Our approach is based on a continuity argument combined with variational techniques, the sub and supersolutions method and some a priori bounds. Both Dirichlet and Neumann homogeneous boundary conditions are considered. As a byproduct, we deduce some existence and uniqueness results. Finally, as an application, we derive some posit…
We construct a multiply connected domain in $\mathbb{R}^2$ for which the second eigenfunction of the Laplacian with Robin boundary conditions has an interior nodal line. In the process, we adapt a bound of Donnelly-Fefferman type to obtain a uniform estimate on the size of the nodal sets of a sequence of solutions to a certain class of elliptic equations in the interior of a sequence of domains, which does not depend directly on any boundary behaviour. This also gives a new proof of the nodal line property of the example in the Dirichlet case.
The Dirichlet Laplacian in curved tubes of arbitrary constant cross-section rotating together with the Tang frame along a bounded curve in Euclidean spaces of arbitrary dimension is investigated in the limit when the volume of the cross-section diminishes. We show that spectral properties of the Laplacian are, in this limit, approximated well by those of the sum of the Dirichlet Laplacian in the cross-section and a one-dimensional Schrodinger operator whose potential is expressed solely in terms of the first curvature of the reference curve. In particular, we establish the convergence of eigen…
We consider a nonlinear elliptic Dirichlet equation driven by a nonlinear nonhomogeneous differential operator involving a Carathéodory reaction which is $(p-1)$-superlinear but does not satisfy the Ambrosetti-Rabinowitz condition. First we prove a three-solutions-theorem extending an earlier classical result of Wang (Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), no. 1, 43--57). Subsequently, by imposing additional conditions on the reaction $f(x,\cdot)$, we produce two more nontrivial constant sign solutions and a nodal solution for a total of five nontrivial solutions. In the special c…
The oscillation of a Laplacian eigenfunction gives a great deal of information about the manifold on which it is defined. This oscillation can be encoded in the nodal deficiency, an important geometric quantity that is notoriously hard to compute, or even estimate. Here we compare two recently obtained formulas for the nodal deficiency, one in terms of an energy functional on the space of equipartitions of the manifold, and the other in terms of a two-sided Dirichlet-to-Neumann map defined on the nodal set. We relate these two approaches by giving an explicit formula for the Hessian of the equ…
We prove new results on the existence of positive radial solutions of the elliptic equation $-Δu= λh(|x|,u)$ in an annular domain in $\mathbb{R}^{N}, N\geq 2$. Existence of positive radial solutions are determined under the conditions that the nonlinearity function $h(t,u)$ is either superlinear or sublinear growth in $u$ or satisfies some upper and lower inequalities on $h$. Our discussion is based on a fixed point theorem due to a revised version of a fixed point theorem of Gustafson and Schmitt.
We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in $\mathbb{R}^n$ - the eigenfunctions of the Dirichlet-to-Neumann map. Under the assumption that the domain $Ω$ is $C^2$, we prove a doubling property for the eigenfunction $u$. We estimate the Hausdorff $\mathcal H^{n-2}$-measure of the nodal set of $u|_{\partial Ω}$ in terms of the eigenvalue $λ$ as $λ$ grows to infinity. In case that the domain $Ω$ is analytic, we prove a polynomial bound O($λ^6$). Our arguments, which make heavy use of Almgren's frequency functions, are built on the previous wo…
We study the nodal solutions of the Lane Emden Dirichlet problem $-Δu = |u|^{p-1}u with DBC on a smooth bounded domain $Ω$ in $\IR^2$ and where $p>1$. We consider solutions $u_p$ satisfying $p \int_Ω\abs{\nabla u_p}^2\to 16πe\quad\hbox{as}p\rightarrow+\infty\qquad (*)$ and we are interested in the shape and the asymptotic behavior as $p\rightarrow+\infty$. First we prove that (*) holds for least energy nodal solutions. Then we obtain some estimates and the asymptotic profile of this kind of solutions. Finally, in some cases, we prove that $pu_p$ can be characterized as the difference of two Gr…
We consider a semilinear elliptic equation on a smooth bounded domain $\Om$ in $\R^2$, assuming that both the domain and the equation are invariant under reflections about one of the coordinate axes, say the y-axis. It is known that nonnegative solutions of the Dirichlet problem for such equations are symmetric about the axis, and, if strictly positive, they are also decreasing in $x$ for $x>0$. Our goal is to exhibit examples of equations which admit nonnegative, nonzero solutions for which the second property fails; necessarily, such solutions have a nontrivial nodal set in $\Om$. Previously…
We show the existence of simply-connected unbounded planar domains for which the second nodal line of the Dirichlet Laplacian does not touch the boundary.
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 give an example of a domain in dimension $N \geq 3$, homeomorphic to a ball and with analytic boundary, for which the second eigenvalue of the Dirichlet Laplacian has an eigenfunction with a closed nodal surface. The domain is constructed via a sequence of perturbations of the domain of S. Fournais [J. Differential Equations \textbf{173} (2001), 145-159].
Inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian have received much attention in the literature, but open problems abound. Here, we study the number of Neumann eigenvalues no greater than the first Dirichlet eigenvalue. Based on a combination of analytical and numerical results, we conjecture that this number is controlled by the isoperimetric ratio of the domain. This has applications to the nodal deficiency of eigenfunctions and is closely related to a long-standing conjecture of Yau on the Hausdorff measure of nodal sets.
We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation \[ \mathcal{L}u = σu^{q} + μ\quad \text{in} \;\; Ω, \] in the sublinear case $0<q<1$, with finite generalized energy: $\mathbb{E}_γ[u]:=\int_Ω |\nabla u|^{2} u^{γ-1}dx<\infty$, for $γ>0$. In this case $u \in L^{γ+q}(Ω, σ)\cap L^γ(Ω, μ)$, where $γ=1$ corresponds to finite energy solutions. Here $\mathcal{L} u:= -\,\text{div}(\mathcal{A}\nabla u)$ is a linear uniformly elliptic operator with bounded measurable coefficients, and $σ$, $μ$ are nonnegative functions…
In this paper we consider the Hénon problem in the unit disc with Dirichlet boundary conditions. We study the asymptotic profile of least energy and nodal least energy radial solutions and then deduce the exact computation of their Morse index for large values of the exponent p. As a consequence of this computation a multiplicity result for positive and nodal solutions is obtained.
We consider radial solutions of the slightly subcritical problem $-Δu_\varepsilon = |u_\varepsilon|^{\frac{4}{n-2}-\varepsilon}u_\varepsilon$ either on $\mathbb R^n$ ($n\geq 3$) or in a ball $B$ satisfying Dirichlet or Neumann boundary conditions. In particular, we provide sharp rates and constants describing the asymptotic behavior (as $\varepsilon\to 0$) of all local minima and maxima of $u_\varepsilon$ as well as its derivative at roots. Our proof is done by induction and uses energy estimates, blow-up/normalization techniques, a radial pointwise Pohozaev identity, and some ODE arguments. A…
In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators on manifolds with boundary. We also consider applications to curvature prescription problems on manifolds with boundary. We relate Dirichlet and Neumann eigenvalues and put the results developed here for the Escobar problem into the more general framework of boundary operators of arbitrary order.
We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for two-parametric family of partially homogeneous $(p,q)$-Laplace equations $-Δ_p u -Δ_q u=α|u|^{p-2}u+β|u|^{q-2}u$ where $p \neq q$. By virtue of the Nehari manifolds, linking theorem, and descending flow, we explicitly characterize subsets of $(α,β)$-plane which correspond to the existence of nodal solutions. In each subset the obtained solutions have prescribed signs of energy and, in some cases, exactly two nodal domains. The nonexistence of nodal solutions is also studied. Additionally, we explore se…
We study existence, regularity, and qualitative properties of solutions to the system \[ -Δu = |v|^{q-1} v\quad \text{ in }Ω,\qquad -Δv = |u|^{p-1} u\quad \text{ in }Ω,\qquad \partial_νu=\partial_νv=0\quad \text{ on }\partialΩ, \] with $Ω\subset \mathbb R^N$ bounded; in this setting, all nontrivial solutions are sign changing. Our proofs use a variational formulation in dual spaces, considering sublinear $pq< 1$ and superlinear $pq>1$ problems in the subcritical regime. In balls and annuli we show that least energy solutions (l.e.s.) are foliated Schwarz symmetric and, due to a symmetry-breaki…
Let $Ω\subset\mathbb{R}^{N}$ ($N\geq1$) be a bounded and smooth domain and $a:Ω\rightarrow\mathbb{R}$ be a sign-changing weight satisfying $\int_Ωa<0$. We prove the existence of a positive solution $u_{q}$ for the problem $(P_{a,q})$: $-Δu=a(x)u^{q}$ in $Ω$, $\frac{\partial u}{\partialν}=0$ on $\partialΩ$, if $q_{0}<q<1$, for some $q_{0}=q_{0}(a)>0$. In doing so, we improve the existence result previously established in [16]. In addition, we provide the asymptotic behavior of $u_{q}$ as $q\rightarrow1^{-}$. When $Ω$ is a ball and $a$ is radial, we give some explicit conditions on $q$ and $a$ e…
