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We build blowing-up solutions for a supercritical perturbation of the Yamabe problem on manifolds with umbilic boundary, provided the dimension of the manifold is n ≥ 8 and that the Weyl tensor W g is not vanishing on ∂M.

Let (M,g) a compact Riemannian $n$-dimensional manifold with umbilic boundary. It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have the boundary of M as a constant mean curvature hypersurface. In this paper we prove that these metrics are a compact set in the case of low dimensional manifolds, that is n=6,7,8, provided that the Weyl tensor is always not vanishing on the boundary., arXiv admin note: text overlap with arXiv:1903.10990, arXiv:1912.11482

Let ( M , g ) a compact Riemannian n -dimensional manifold with umbilic boundary. It is well known that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have ∂ M as a constant mean curvature hypersurface. In this paper we prove that these metrics are a compact set, provided n = 8 and the Weyl tensor of the boundary is always different from zero, or if n > 8 and the Weyl tensor of M is always different from zero on the boundary.

Let M,g a compact Riemannian n-dimensional manifold. It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have the boundary of M as a constant mean curvature hypersurface. Also, under certain hypothesis, it is known that these metrics are a compact set. In this paper we prove that, both in the case of umbilic and non-umbilic boundary, if we linearly perturb the mean curvature term with a negative smooth function, the set of solutions of Yamabe problem is still a compact set., arXiv admin note: substantial text overlap with arXiv:1903.10990

We consider the nonlinear Schr\"odinger equation with focusing power-type nonlinearity on compact graphs with at least one terminal edge, i.e. an edge ending with a vertex of degree 1. On the one hand, we introduce the associated action functional and we provide a profile description of positive low action solutions at large frequencies, showing that they concentrate on one terminal edge, where they coincide with suitable rescaling of the unique solution to the corresponding problem on the real line. On the other hand, a Ljapunov-Schmidt reduction procedure is performed to construct one-peaked and multipeaked positive solutions with sufficiently large frequency, exploiting the presence of one or more terminal edges., Comment: 25 pages, 1 figure

We study the semiclassical limit to a singularly perturbed nonlinear Klein–Gordon–Maxwell–Proca system, with Neumann boundary conditions, on a Riemannian manifold 𝔐 {\mathfrak{M}} with boundary. We exhibit examples of manifolds, of arbitrary dimension, on which these systems have a solution which concentrates at a closed submanifold of the boundary of 𝔐 {\mathfrak{M}} , forming a positive layer, as the singular perturbation parameter goes to zero. Our results allow supercritical nonlinearities and apply, in particular, to bounded domains in ℝ N {\mathbb{R}^{N}} . Similar results are obtained for the more classical electrostatic Klein–Gordon–Maxwell system with appropriate boundary conditions.

We build blowing-up solutions for linear perturbation of the Yamabe problem on manifolds with umbilic boundary, provided the Weyl tensor is nonzero everywhere on the boundary and the dimension of the manifold is n ≥ 11 .

We show that the number of solutions of a double singularly perturbed Schrodinger Maxwell system on a smooth bounded domain Ω ⊂ R 3 depends on the topological properties of the domain. In particular if Ω is non contractible we obtain cat ( Ω ) + 1 positive solutions. The result is obtained via Lusternik–Schnirelmann category theory.

Given a 3-dimensional Riemannian manifold ( M , g ) , we investigate the existence of positive solutions of the Klein–Gordon–Maxwell system { − e 2 Δ g u + a u = u p − 1 + ω 2 ( q v − 1 ) 2 u in M , − Δ g v + ( 1 + q 2 u 2 ) v = q u 2 in M and Schrodinger–Maxwell system { − e 2 Δ g u + u + ω u v = u p − 1 in M , − Δ g v + v = q u 2 in M when p ∈ ( 4 , 6 ) . We prove that the number of one peak solutions depends on the topological properties of the manifold M, by means of the Lusternik Schnirelmann category.

Let $M$ be a connected compact $C^\infty$ manifold of dimension $n\ge4$ without boundary. Let $ \mathcal{M}^k$ be the set of all $C^k$ Riemannian metrics on $M$. Any $g\in\mathcal{M}^k$ determines the Weyl tensor $$ \mathcal W^g\colon M\to \mathbb R^{4n},\qquad \mathcal W^g(\xi):=(W^g_{ijkl}(\xi))_{i,j,k,l=1,\dots,n}.$$ We prove that the set $$\mathcal{A}:=\big\{g\in \mathcal{M}^k : |\mathcal W^g(\xi)|+|D \mathcal W^g(\xi)|+|D^2 \mathcal W^g(\xi)|> 0\ \hbox{for any}\ \xi\in M\big\}$$ is an open dense subset of $\mathcal{M}^k$.

Given a 3-dimensional Riemannian manifold $(M,g)$, we investigate the existence of positive solutions of the Klein-Gordon-Maxwell system $$ \left\{ \begin{array}{cc} -\varepsilon^{2}\Delta_{g}u+au=u^{p-1}+\omega^{2}(qv-1)^{2}u & \text{in }M\\ -\Delta_{g}v+(1+q^{2}u^{2})v=qu^{2} & \text{in }M \end{array}\right. $$ and Schrodinger-Maxwell system $$ \left\{ \begin{array}{cc} -\varepsilon^{2}\Delta_{g}u+u+\omega uv=u^{p-1} & \text{in }M\\ -\Delta_{g}v+v=qu^{2} & \text{in }M \end{array}\right. $$ when $p\in(2,6). $ We prove that if $\varepsilon$ is small enough, any stable critical point $\xi_0$ of the scalar curvature of $g$ generates a positive solution $(u_\varepsilon,v_\varepsilon)$ to both the systems such that $u_\varepsilon$ concentrates at $\xi_0$ as $\varepsilon$ goes to zero.

For a bounded domain $\Omega\subset\mathbb{R}^n$ let $H_\Omega:\Omega\times\Omega\to\mathbb{R}$ be the regular part of the Dirichlet Green function for the Laplace operator. Given a fixed arbitrary ${\mathcal C}^2$ function $f:{\mathcal D}\to\mathbb{R}$, defined on an open subset ${\mathcal D}\subset\mathbb{R}^{nN}$, and fixed coefficients $\lambda_1,\dots,\lambda_N\in\mathbb{R}\setminus\{0\}$ we consider the function $f_\Omega:{\mathcal D}\cap\Omega^N\to\mathbb{R}$ defined as \[ f_\Omega(x_1,\dots,x_N) = f(x_1,\dots,x_N) - \sum_{j,k=1}^N \lambda_j\lambda_k H_\Omega(x_j,x_k). \] We prove that $f_\Omega$ is a Morse function for most domains $\Omega$ of class ${\mathcal C}^{m+2,\alpha}$, any $m\ge0$, $0, Comment: 14 pages

Given a symmetric Riemannian manifold $(M,g)$, we show some results of genericity for non degenerate sign changing solutions of singularly perturbed nonlinear elliptic problems with respect to the parameters: the positive number $\varepsilon$ and the symmetric metric $g$. Using these results we obtain a lower bound on the number of non degenerate solutions which change sign exactly once.

We study the behavior of the soliton solutions of the equation i((\partial{\psi})/(\partialt))=-(1/(2m)){\Delta}{\psi}+(1/2)W_{{\epsilon}}'({\psi})+V(x){\psi} where W_{{\epsilon}}' is a suitable nonlinear term which is singular for {\epsilon}=0. We use the "strong" nonlinearity to obtain results on existence, shape, stability and dynamics of the soliton. The main result of this paper (Theorem 1) shows that for {\epsilon}\to0 the orbit of our soliton approaches the orbit of a classical particle in a potential V(x)., Comment: 29 pages

We show that the number of solutions of a nonlinear elliptic problem on a Riemannian manifold depends on the topological properties of the manifold. In particular we consider the Lusternik–Schnirelmann category and the Poincare polynomial of the manifold.

Let N ≥ 3 , 2 ∗ = 2 N / ( N − 2 ) . Our purpose in this paper is to consider the existence and multiplicity of solutions of the problem { − Δ u = | u | 2 ∗ − 2 u + f on R N u ∈ H 1 ( R N ) where f ∈ L 2 ∗ / ( 2 ∗ − 1 ) ( R N ) ∩ L ∞ ( R N ) , f ≥ 0 and f ≢ 0 .

We consider a semilinear problem of the type Lu = f ( b , u ) in L 2 ( R N ) , where f ( b , u )≃ bu as u →0 and f ( b , u )≃ b ∞ u as | u |→∞ ( b b ∞ ). We assume that the essential spectrum of L is bounded below by a number greater than b ∞ and that there exist a finite number of eigenvalues between b and b ∞ . We prove existence and multiplicity results for nontrivial solutions under suitable assumptions on g . We give an application to the Schrodinger equation.

The main purpose of this paper is to construct a family of positive solutions for both the slightly subcritical and slightly supercritical equations -Δu + V(x)u = n(n - 2)(u+)(n+2)/(n-2)±e in Rn, which blow-up and concentrate at a single point as e goes to 0, under certain conditions on the potential V.

Let ( M , g ) be a smooth, compact Riemannian manifold with smooth boundary, with n = dim M = 2 , 3 . We suppose the boundary ∂ M to be a smooth submanifold of M with dimension n − 1 . We consider a singularly perturbed nonlinear system, namely Klein–Gordon–Maxwell–Proca system, or Klein–Gordon–Maxwell system of Schroedinger–Maxwell system on M . We prove that the number of low energy solutions, when the perturbation parameter is small, depends on the topological properties of the boundary ∂ M , by means of the Lusternik–Schnirelmann category. Also, these solutions have a unique maximum point that lies on the boundary.

We describe the asymptotic profile of sign changing solutions with small energy to the singularly perturbed nonlinear elliptic equation $-\varepsilon ^{2}\Delta_{g}u+u=|u|^{p-2}u$ on an $n$-dimensional compact Riemannian manifold $M$ without boundary, $p\in(2,2^{\ast}),$ as $\varepsilon \rightarrow0.$

We study periodic flexings of a floating beam via critical point theory. The beam is described by the nonlinear equation with free-end boundary conditions described in the Introduction. We give a theorem of existence of two nontrivial solutions and a theorem of existence of four nontrivial solutions, depending on the position of a suitable parameter with respect to the eigenvalues of the linear part.

Given a smooth Riemannian manifold ( M , g ) we investigate the existence of positive solutions to the equation − e 2 Δ g u + u = u p − 1 on M which concentrate at some submanifold of M as e → 0 , for supercritical nonlinearities. We obtain a positive answer for some manifolds, which include warped products. Using one of the projections of the warped product or some harmonic morphism, we reduce this problem to a problem of the form − e 2 div h ( c ( x ) ∇ h u ) + a ( x ) u = b ( x ) u p − 1 , with the same exponent p, on a Riemannian manifold ( M , h ) of smaller dimension, so that p turns out to be subcritical for this last problem. Then, applying Lyapunov–Schmidt reduction, we establish existence of a solution to the last problem which concentrates at a point as e → 0 .

We show a result of genericity for non degenerate critical points of the Robin function with respect to deformations of the domain.

Given (M, g 0), we consider the problem with . Here ℬρ is a ball centred at 0 with radius ρ in the Banach space of all C k symmetric covariant 2-tensors on M. Using the Poincare polynomial of M, we give an estimate on the number of nonconstant solutions with low energy for (ϵ, h) belonging to an residual subset of , for , ρ small enough.

Given that $(M,g)$ is a smooth compact and symmetric Riemannian $n$ -manifold, $n\ge 2$ , we prove a multiplicity result for antisymmetric sign changing solutions of the problem $-{\epsilon }^{2}{\Delta }_{g}u+u={|u|}^{p-2}u$ in $M$ . Here $p>2$ if $n=2$ and $2

Given (M, g) a smooth compact Riemannian N-manifold, N ≥ 2, we show that positive solutions to the problem $${-\varepsilon^2\Delta_g u + u = u^{p-1}\,{\rm in}\,M,}$$ are generated by stable critical points of the scalar curvature of g, provided $${\varepsilon}$$ is small enough. Here p > 2 if N = 2 and $${2 < p < 2^{*} = {2N \over N-2}}$$ if N ≥ 3.

In this paper we investigate the dynamics of solitons occurring in the nonlinear Schroedinger equation when a parameter h->0. We prove that under suitable assumptions, the the soliton approximately follows the dynamics of a point particle, namely, the motion of its barycenter $q_h(t)$ satisfies the equation $ddot{q}_h(t)+\nabla V(q_h(t))=H_h(t)$ where $\sup_{t\in R} |H_h(t)| -> 0$ as h->0.

We study the existence of sign changing solutions to the slightly subcritical problem $$-\Delta u=|u|^{p-1-\epsilon} u\ {\rm in}\ \Omega,\quad u=0\ {\rm on}\ \partial \Omega,$$ where ω is a smooth bounded domain in ℝ N , N ≥ 3, p = (N + 2)/(N − 2) and ɛ > 0. We prove that, for ɛ small enough, there exist N pairs of solutions which change sign exactly once. Moreover, the nodal surface of these solutions intersects the boundary of ω, provided some suitable conditions are satisfied.

We study the existence of the ground state solution of the problem $$ \begin{cases} -\Delta u+V(x)u=f'(u) & x\in\mathbb R^N, \\ u(x)> 0, \end{cases} $$ under the assumption that $\lim_{x\to\infty}V(x)=0$.

We study the field equation $$ -\Delta u+V(x)u+\varepsilon^r(-\Delta_pu+W'(u))=\mu u $$ on $\mathbb R^n$, with $\varepsilon$ positive parameter. The function $W$ is singular in a point and so the configurations are characterized by a topological invariant: the topological charge. By a min-max method, for $\varepsilon$ sufficiently small, there exists a finite number of solutions $(\mu(\varepsilon),u(\varepsilon))$ of the eigenvalue problem for any given charge $q\in{\mathbb Z}\setminus\{0\}$.

Given a connected compact $C^\infty$ manifold M of dimension $n\ge2$ without boundary, a Riemannian metric $g$ on M and an eigenvalue $\lambda^*(M,g)$ of multiplicity $\nu\ge2$ of the Laplace-Beltrami operator $\Delta_g,$ we provide a sufficient condition such that the set of the deformations of the metric $g,$ which preserve the multiplicity of the eigenvalue, is locally a manifold of codimension $1/2 \nu(\nu+1)-1$ in the space of $C^k$ symmetric covariant 2-tensors on M. Furthermore we prove that such a condition is fulfilled when $n=2$ and $\nu=2.$

where Ω is a bounded smooth domain, u + =m ax(u, 0) and u − = − min(u, 0). The study of Σ turns out to be difficult except when Ω is an interval in R. Therefore it is interesting to have some information about the resonance set, as precise as possible. In [GK] the authors showed that if λk is a simple eigenvalue of − ∆t hen Σ∩]λk−1 ,λ k+1[ 2 coincides with two continuous curves through the point (λk ,λ k). In [DeFG] the authors characterized a curve γ through the point (λ2 ,λ 2 )w hich belongs to Σ such that Σ∩{(α, β) ∈ R 2 | λ1 λ 1} = ∅. Finally, in [MMP] and [M] the following result was shown: if k ≥ 2 is such that λk

We prove the existence of two nontrivial solutions for the fourth order problem and u = 0 on ∂ω when λ1≥c>λi+1 and either b λj(λj-c) and b is close to λj(λj-c) where j≥i+1. (Here (λi)i≥1 is the sequence of the eigenvalues of –Δin H1 0(ω)). Moreover if c>λ1, c is close to λ1, b>λj(λj-c) and b is close to λj(λj-c) where j≥2 we get three non trivial solutions