We study the existence of sign-changing solutions for a non-local version of the sinh-Poisson equation on a bounded one-dimensional interval $I$, under Dirichlet conditions in the exterior of $I$. This model is strictly related to the mathematical description of galvanic corrosion phenomena for simple electrochemical systems. By means of the finite-dimensional Lyapunov-Schmidt reduction method, we construct bubbling families of solutions developing an arbitrarily prescribed number sign-alternating peaks. With a careful analysis of the limit profile of the solutions, we also show that the number of nodal regions coincides with the number of blow-up points.
For a sinh-Poisson type problem with asymmetric exponents of interest in hydrodynamic turbulence, we establish the optimal lower bounds for the blow-up masses. We apply this result to construct solutions of mountain pass type on two-dimensional tori.
In this paper, we provide a complete blow-up picture for solution sequences to an elliptic sinh-Poisson equation with variable intensities arising in the context of the statistical mechanics description of two-dimensional turbulence, as initiated by Onsager. The vortex intensities are described in terms of a probability measure defined on the interval. Under Dirichlet boundary conditions we establish the exclusion of boundary blowup points, we show that the concentration mass does not have residual L1-terms and we determine the location of blowup points in terms of Kirchhoff's Hamiltonian. We …
We consider the sinh-Poisson equation $$(P)_λ\quad -Δu=\la\sinh u\ \hbox{in}\ Ω,\ u=0\ \hbox{on}\ \partialΩ,$$ where $Ω$ is a smooth bounded domain in $\rr^2$ and $λ$ is a small positive parameter. If $0\inΩ$ and $Ω$ is symmetric with respect to the origin, for any integer $k$ if $\la$ is small enough, we construct a family of solutions to $(P)_\la$ which blows-up at the origin whose positive mass is $4πk(k-1)$ and negative mass is $4πk(k+1).$ It gives a complete answer to an open problem formulated by Jost-Wang-Ye-Zhou in [Calc. Var. PDE (2008) 31: 263-276].
Motivated by the statistical mechanics description of stationary 2D-turbulence, for a sinh-Poisson type equation with asymmetric nonlinearity, we construct a concentrating solution sequence in the form of a tower of singular Liouville bubbles, each of which has a different degeneracy exponent. The asymmetry parameter $γ\in(0,1]$ corresponds to the ratio between the intensity of the negatively rotating vortices and the intensity of the positively rotating vortices. Our solutions correspond to a superposition of highly concentrated vortex configurations of alternating orientation; they extend in…
Given $ε\in (0,1)$ and $λ> 1$, we address the existence of solutions for the Sinh-Poisson equation with Robin boundary value condition $$ \begin{cases} Δu+ε^2 (e^{u} - e^{-u})=0 &\mbox{in }Ω\\ \frac{\partial u}{\partialν}+λu=0 &\mbox{on }\partialΩ, \end{cases} $$ where $Ω\subset\mathbb{R}^2$ is a bounded smooth domain. We prove two existence results under a suitable relation between $ε$ small and $λ$ large. When $Ω$ is symmetric with respect to an axis, we prove the existence of a family of solutions $u_{ε,λ}$ concentrating at two points with different spin, both located on the symmetry line a…
In this paper we consider the Dirichlet problem \begin{equation} \label{iniz} \begin{cases} -Δu = ρ^2 (e^{u} - e^{-u}) & \text{ in } Ω\\ u=0 & \text{ on } \partial Ω, \end{cases} \end{equation} where $ρ$ is a small parameter and $Ω$ is a $C^2$ bounded domain in $\mathbb{R}^2$. [1] proves the existence of a $m$-point blow-up solution $u_ρ$ jointly with its asymptotic behaviour. we compute the Morse index of $u_ρ$ in terms of the Morse index of the associated Hamilton function of this problem. In addition, we give an asymptotic estimate for the first $4m$ eigenvalues and eigenfunctions.
We consider a sinh-Poisson type equation with variable intensities and Dirichlet boundary condition on a pierced domain \begin{equation*} \left\{ \begin{array}{ll} Δu +ρ\left(V_1(x)e^{u}- V_2(x)e^{-τu}\right)=0 &\text{in } Ω_ε:=Ω\setminus \displaystyle \bigcup_{i=1}^m \overline{B(ξ_i,ε_i)}\\ u=0&\text{on }\partialΩ_ε, \end{array}\right. \end{equation*} where $ρ>0$, $V_1,V_2>0$ are smooth potentials in $Ω$, $τ>0$, $Ω$ is a smooth bounded domain in $\mathbb{R}^2$ and $B(ξ_i,ε_i)$ is a ball centered at $ξ_i\in Ω$ with radius $ε_i>0$, $i=1,\dots,m$. When $ρ>0$ is small enough and $m_1\in \{1,\dots…
We prove the existence of infinitely many high energy sign-changing solutions for some classes of Schrodinger-Poisson systems in bounded domains, with nonlinearities having subcritical or critical growth. Our approach is variational and relies on an application of a new sign-changing version of the symmetric mountain pass theorem.
We find a sign-changing solution for a class of Schrödinger-Poisson system in $\mathbb{R}^3$ as an existence result by minimization in a closed subset containing all the sign-changing solutions of the equation. The proof is based on variational methods in association with the deformation lemma and Miranda's theorem.
We study the existence of nonradial sign-changing solutions to the Schroedinger-Poisson system in dimension N>=3. We construct nonradial sign-changing multi-peak solutions whose peaks are displaced in suitable symmetric configurations and collapse to the same point. The proof is based on the Lyapunov-Schmidt reduction.
We investigate the existence of multiple bound state solutions, in particular sign-changing solutions. By using the method of invariant sets of descending flow, we prove that this system has infinitely many sign-changing solutions. In particular, the nonlinear term includes the power-type nonlinearity $f(u)=|u|^{p-2}u$ for the well-studied case $p\in(4,6)$, and the less-studied case $p\in(3,4)$, and for the latter case few existence results are available in the literature.
In this paper we consider the following Schrödinger-Poisson system in the whole $\mathbb R^{3}$, \begin{equation*} \left\{ \begin{array}{ll} -Δu+u+ λϕu=f(u) &\text{ in } \mathbb R^3, -Δϕ= u^2 &\text{ in } \mathbb R^3, \end{array} \right. \end{equation*} where $λ>0$ and the nonlinearity $f$ is "asymptotically cubic" at infinity. This implies that the nonlocal term $ϕu$ and the nonlinear term $f(u)$ are, in some sense, in a strict competition. We show that the system admits a least energy sign-changing and radial solution obtained by minimizing the energy functional on the so-called {nodal Nehar…
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.
In this paper, we are concerned with the existence of the least energy sign-changing solutions for the following fractional Schrödinger-Poisson system: \begin{align*} \left\{ \begin{aligned} &(-Δ)^{s} u+V(x)u+λϕ(x)u=f(x, u),\quad &\text{in}\, \ \mathbb{R}^{3},\\ &(-Δ)^{t}ϕ=u^{2},& \text{in}\,\ \mathbb{R}^{3}, \end{aligned} \right. \end{align*} where $λ\in \mathbb{R}^{+}$ is a parameter, $s, t\in (0, 1)$ and $4s+2t>3$, $(-Δ)^{s}$ stands for the fractional Laplacian. By constraint variational method and quantitative deformation lemma, we prove that the above problem has one least energy sign-cha…
In this paper, we consider the following Schrödinger-Poisson system with $p$-laplacian \begin{equation} \begin{cases} -Δ_{p}u+V(x)|u|^{p-2}u+ϕ|u|^{p-2}u=f(u)\qquad&x\in\mathbb{R}^{3},\newline -Δϕ=|u|^{p}&x\in\mathbb{R}^{3}. \end{cases}\notag \end{equation} We investigate the existence of multiple sign-changing solutions. By using the method of invariant sets of descending flow, we prove that this system has infinitely many sign-changing solutions. This system is new one coupled by Schrödinger equation of $p$-laplacian with a Poisson equation. Our results complement the study made by Zhaoli Liu…
Let $Ω$ be an open, possibly unbounded, set in Euclidean space $\R^m$ with boundary $\partialΩ,$ let $A$ be a measurable subset of $Ω$ with measure $|A|$, and let $γ\in (0,1)$. We investigate whether the solution $v_{\Om,A,γ}$ of $-Δv=γ{\bf 1}_{Ω\setminus A}-(1-γ){\bf 1}_{A}$ with $v=0$ on $\partial Ω$ changes sign. Bounds are obtained for $|A|$ in terms of geometric characteristics of $\Om$ (bottom of the spectrum of the Dirichlet Laplacian, torsion, measure, or $R$-smoothness of the boundary) such that ${\rm essinf} v_{\Om,A,γ}\ge 0$. We show that ${\rm essinf} v_{\Om,A,γ}<0$ for any measura…
We study the following Schrödinger-Poisson system (P_λ){ll} -Δu + V(x)u+λϕ(x) u =Q(x)u^{p}, x\in \mathbb{R}^3 \\ -Δϕ= u^2, \lim\limits_{|x|\to +\infty}ϕ(x)=0, u>0, where $λ\geqslant0$ is a parameter, $1 < p < +\infty$, $V(x)$ and $Q(x)$ are sign-changing or non-positive functions in $ L^{\infty}(\mathbb{R}^3)$. When $V(x)\equiv Q(x)\equiv1$, D.Ruiz \cite{RuizD-JFA} proved that ($P_λ$) with $p\in(2,5)$ has always a positive radial solution, but ($P_λ$) with $p\in(1,2]$ has solution only if $λ>0$ small enough and no any nontrivial solution if $λ\geqslant{1/4}$. By using sub-supersolution method,…
In this paper, we study the existence of localized sign-changing (or nodal) solutions for the following nonlinear Schrödinger-Poisson system \begin{equation*} \begin{cases} -\varepsilon^2 Δu+V(x)u+ϕu=K(x)f(u),&\text{in}~\mathbb{R}^3,\\ -\varepsilon^2 Δϕ=u^2,&\text{in}~ \mathbb{R}^3, \end{cases} \end{equation*} where $\varepsilon>0$ is small parameters, the linear potential $V$ and nonlinear potential $K$ are bounded and bounded away from zero. By using the penalization method together with the method of invariant sets of descending flow, we establish the existence of an infinite sequence of lo…
In this paper, we are concerned with the following Schrödinger-Poisson system with critical nonlinearity and critical nonlocal term due to the Hardy-Littlewood-Sobolev inequality \begin{equation}\begin{cases} -Δu+u+λϕ|u|^3u =|u|^4u+ |u|^{q-2}u,\ \ &\ x \in \mathbb{R}^{3},\\[2mm] -Δϕ=|u|^5, \ \ &\ x \in \mathbb{R}^{3}, \end{cases} \end{equation} where $λ\in \mathbb{R}$ is a parameter and $q\in(2,6)$. If $λ\ge (\frac{q+2}{8})^2$ and $q\in(2,6)$, the above system has no nontrivial solution. If $λ\in (λ^*,0)$ for some $λ^*<0$, we obtain a least energy radial sign-changing solution $u_λ$ to the abo…
In this paper, we study the existence of multiple positive solutions for a class of fractional Schrödinger-Poisson systems involving sign-changing potential and critical nonlinearities on an unbounded domain. With the help of Nehari manifold and Ljusternik-Schnirelmann category, we investigate how the coefficient $g(x)$ of the critical nonlinearity affects the number of positive solutions. Moreover, we present a novel relationship between the number of positive solutions and the category of the global maximum set of $g(x)$.
For an asymmetric sinh-Poisson problem arising as a mean field equation of equilibrium turbulence vortices with variable intensities of interest in hydrodynamic turbulence, we address the existence of bubbling solutions on compact Riemann surfaces. By using a Lyapunov-Schmidt reduction, we find sufficient conditions under which there exist bubbling solutions blowing up at $m$ different points of $S$: positively at $m_1$ points and negatively at $m-m_1$ points with $m\ge 1$ and $m_1\in\{0,1,...,m\}$. Several examples in different situations illustrate our results in the sphere $\mathbb S^2$ and…
In this article we study the existence of solutions to the system \begin{equation*}\left\{ \begin{array}{ll} -\left(a+b\int_Ω|\nabla u|^{2}\right)Δu +ϕu= f(x, u) &\text{in }Ω\hbox{} -Δϕ= u^{2} &\text{in }Ω\hbox{} u=ϕ=0&\text{on }\partialΩ, \hbox{} \end{array} \right. \end{equation*} where $Ω$ is a bounded smooth domain of $\mathbb{R}^N$ ($N=1,2$ or $3$), $a>0$, $b\geq0$, and $f:\overlineΩ\times \mathbb{R}\to\mathbb{R}$ is a continuous function which is $3$-superlinear. By using some variants of the mountain pass theorem established in this paper, we show the existence of three solutions: one p…
We introduce canonical principal parameters on any strongly regular minimal surface in the three dimensional sphere and prove that any such a surface is determined up to a motion by its normal curvature function satisfying the Sinh-Poisson equation. We obtain a classification theorem for bi-umbilical hypersurfaces of type number two. We prove that any minimal hypersurface of type number two with involutive distribution is generated by a minimal surface in the three-dimensional Euclidean space, or in the three dimensional sphere. Thus we prove that the theory of minimal hypersurfaces of type nu…
The new observations about the parameter-dependent Schrödinger-Poisson system
College of Science, University of Shanghai for Science and Technology（上海理工大学理学院）；College of Mathematics, Changchun Normal University（长春师范大学数学学院）；Department of Mathematics and Computer Science, University of Perugia（佩鲁贾大学数学与计算机科学系）
