We study the following Neumann boundary problem associated with the stationary solutions of the Keller-Segel system, a basic model of chemotaxis phenomena: { -Δ _g u +β u =λ( Ve^u/∫ _Σ Ve^u dv_g-1) in Σ ∂ _ν _g u =0 on ∂Σ. , on a compact Riemann surface (Σ , g) of unit area, with interior Σ and smooth boundary ∂Σ . Here, Δ _g denotes the Laplace-Beltrami operator, dv_g is the area element of (Σ , g) , ν _g is the unit outward normal to ∂Σ , λ and β are non-negative parameters, and V is non-negative with finite zero set. For any m∈ℕ and k,l∈ℕ∪{ 0} with m=2k+l , we establish a sufficient condition on V for the existence of a sequence of blow-up solutions concentrating at k points in the interior and l points on the boundary as λ approaches the resonant values 4π m . Moreover, our analysis extends to the corresponding singular problem.

On a compact Riemann surface (Σ ,g) with smooth boundary ∂Σ , we consider the following mean field equation with homogeneous Neumann boundary conditions: { -Δ _g u = λ (Ve^u/∫ _Σ Ve^u dv_g - 1/|Σ |_g ) in Σ , ∂ _ν _g u = 0 on ∂Σ . . Given integers k,l≥ 0 , we find conditions on the potential functions V such that solutions exist for the parameter λ in a small left (or right) neighborhood of a critical value 4π (2k+l) , and these solutions blow up at k points in Σ:=Σ∖∂Σ and l points on ∂Σ .

On a compact Riemann surface $(\Sigma, g)$ with a smooth boundary $\partial \Sigma$, we consider the following mean field equations with Neumann boundary conditions: $$ -\Delta_g u = \lambda \left(\frac{Ve^u}{\int_{\Sigma} Ve^u \, dv_g} - \frac{1}{|\Sigma|_g}\right) \text{ in } \Sigma \text{ with } \partial_{\nu_g} u = 0 \text{ on } \partial \Sigma, $$ We find conditions on the potential function $V: \Sigma \to \mathbb{R}^+$ such that solutions exist for the parameter $\lambda$ when it is in a small right (or left) neighborhood of a critical value $4\pi(m+k)$ for $k \leq m \in \mathbb{N}_+$ and blow up as $\lambda$ approaches the critical parameter. The blow-up occurs exactly at $k$ points in the interior of $\Sigma$ and $(m-k)$ points on the boundary $\partial \Sigma$.

In this paper, we study the Morse property for functions related to limit functions of mean field equations on a smooth, compact surface Σ with boundary ∂Σ . Given a Riemannian metric g on Σ we consider functions of the form where σ _i 0 for i=1,… ,m , G^g is the Green function of the Laplace-Beltrami operator on (Σ ,g) with Neumann boundary conditions, R^g is the corresponding Robin function, and h ∈𝒞^2(Σ ^m,ℝ) is arbitrary. We prove that for any Riemannian metric g, there exists a metric g which is arbitrarily close to g and in the conformal class of g such that f_g is a Morse function. Furthermore we show that, if all σ _i>0 , then the set of Riemannian metrics for which f_g is a Morse function is open and dense in the set of all Riemannian metrics.

The paper addresses an open problem raised in Bartsch et al. (Commun Partial Differ Equ 46(9):1729–1756, 2021) on the existence of normalized solutions to Schrödinger equations with potentials and inhomogeneous nonlinearities. We consider the problem -Δ u+V(x)u+λ u = |u|^q-2u+β |u|^p-2u, ‖ u‖ ^2_2=∫ |u|^2dx = α both on ℝ^N as well as on domains rΩ where Ω⊂ℝ^N is a bounded smooth star-shaped domain and r>0 is large. The exponents satisfy 2

This work studies the partial blow-up phenomena for the SU(3) Toda system on compact Riemann surfaces with smooth boundary. We consider the following coupled Liouville system with Neumann boundary conditions: -Δ_g u_1 = 2ρ_1( V_1 e^u_1/∫_Σ V_1 e^u_1 dv_g - 1/|Σ|_g) - ρ_2( V_2 e^u_2/∫_Σ V_2 e^u_2 dv_g - 1/|Σ|_g) in Σ and -Δ_g u_2 = 2ρ_2( V_2 e^u_2/∫_Σ V_2 e^u_2 dv_g - 1/|Σ|_g) - ρ_1( V_1 e^u_1/∫_Σ V_1 e^u_1 dv_g - 1/|Σ|_g) in Σ with boundary conditions ∂_ν_g u_1 = ∂_ν_g u_2 = 0 on ∂Σ, where (Σ, g) is a compact Riemann surface with the interior Σ and smooth boundary ∂Σ, ρ_i is a non-negative parameter and V_i is a smooth positive function for i=1,2. We construct a family of blow-up solutions via the Lyapunov-Schmidt reduction and variational methods, wherein one component remains uniformly bounded from above, while the other exhibits partial blow-ups at a prescribed number of points, both in the interior and on the boundary. This construction is based on the existence of a non-degeneracy solution of a so-called shadow system. Moreover, we establish the existence of partial blow-up solutions in three cases: (i) for any ρ_2>0 sufficiently small; (ii) for generic V_1, V_2 and any ρ_2∈ (0,2π); (iii) for generic V_1, V_2, the Euler characteristic χ(Σ)<1 and any ρ_2∈ (2π,+∞)∖ 2πℕ_+.

We prove the existence of critical points of vortex type Hamiltonians \[ H(p_1,\ldots, p_N) = \sum_{{i,j=1}\atop{i\ne j}} ^N \Gamma_i\Gamma_jG(p_i,p_j)+\Psi(p_1,\dots,p_N) \] on a closed Riemannian surface $(\Sigma,g)$ which is not homeomorphic to the sphere or the projective plane. Here $G$ denotes the Green function of the Laplace-Beltrami operator in $\Sigma$, $\Psi\colon \Sigma^N\to\mathbb{R}$ may be any function of class ${\mathcal C}^1$, and $\Gamma_1,\dots,\Gamma_N\in\mathbb{R}\setminus\{0\}$ are the vorticities. The Kirchhoff-Routh Hamiltonian from fluid dynamics corresponds to $\Psi(p) = -\sum\limits_{i=1}^N \Gamma_i^2h(p_i,p_i)$ where $h\colon \Sigma\times\Sigma\to\mathbb{R}$ is the regular part of the Laplace-Beltrami operator. We obtain critical points $p=(p_1,\dots,p_N)$ for arbitrary $N$ and vorticities $(\Gamma_1,\dots,\Gamma_N)$ in $\mathbb{R}^N\setminus V$ where $V$ is an explicitly given algebraic variety of codimension 1.

Semiclassical approximations provide a connection between quantum mechanics and classical mechanics. That relationship is problematic when classical orbits are chaotic. Highly excited states of atoms have provided a laboratory for the study of this relationship. When classical trajectories are orderly, corresponding quantum spectra and wave-functions display the same type of order. When they are chaotic, the quantum spectrum is very complex, but short closed classical orbits are connected to the large-scale structure of the density of states and the absorption spectrum. This chapter is restricted to a discussion of Rydberg atoms in strong static fields. Related information on atoms in strong fields, on Rydberg atoms, and on the interaction of atoms with strong laser fields can be found elsewhere in the book.

The paper addresses an open problem raised in [Bartsch, Molle, Rizzi, Verzini: Normalized solutions of mass supercritical Schrödinger equations with potential, Comm. Part. Diff. Equ. 46 (2021), 1729-1756] on the existence of normalized solutions to Schrödinger equations with potentials and inhomogeneous nonlinearities. We consider the problem -Δ u+V(x)u+λ u = |u|^q-2u+β |u|^p-2u, u^2_2=∫|u|^2dx = α, both on ℝ^N as well as on domains rΩ where Ω⊂ℝ^N is an open bounded convex domain and r>0 is large. The exponents satisfy 2<p<2+4/N<q<2^*=2N/N-2, so that the nonlinearity is a combination of a mass subcritical and a mass supercritical term. Due to the presence of the potential a by now standard approach based on the Pohozaev identity cannot be used. We develop a robust method to study the existence of normalized solutions of nonlinear Schrödinger equations with potential and find conditions on V so that normalized solutions exist. Our results are new even in the case β=0.

Let $$(M_1,\textit{g}^{(1)})$$ , $$(M_2,\textit{g}^{(2)})$$ be closed Riemannian spin manifolds. We study the existence of solutions of the Spinorial Yamabe problem on the product $$M_1\times M_2$$ equipped with a family of metrics $$\varepsilon ^{-2}\textit{g}^{(1)}\oplus \textit{g}^{(2)}$$ , $$\varepsilon >0$$ . Via variational methods and blow-up techniques, we prove the existence of solutions which depend only on the factor $$M_1$$ , and which exhibit a spike layer as $$\varepsilon \rightarrow 0$$ . Moreover, we locate the asymptotic position of the peak points of the solutions in terms of the curvature tensor on $$(M_1,\textit{g}^{(1)})$$ .

The paper is concerned with the existence and asymptotic properties of normalized ground states of the following nonlinear Schrödinger system with critical exponent: { -Δ u+λ _1 u=|u|^2^*-2u+να |u|^α -2|v|^β u, in ℝ^N, -Δ v+λ _2 v=|v|^2^*-2v+νβ |u|^α |v|^β -2v, in ℝ^N, ∫ _ℝ^Nu^2=a^2, ∫ _ℝ^Nv^2=b^2, . where N=3,4 , α ,β >1 , 2<α +β <2^*=2N/N-2 . We prove that a normalized ground state does not exist for ν <0 . When ν >0 and α +β≤ 2+4/N , we show that the system has a normalized ground state solution for 0<ν <ν _0 , the constant ν _0 will be explicitly given. In the case α +β >2+4/N we prove the existence of a threshold ν _1≥ 0 such that a normalized ground state solution exists for ν >ν _1 , and does not exist for ν <ν _1 . We also give conditions for ν _1=0 . Finally we obtain the asymptotic behavior of the minimizers as ν→ 0^+ or ν→ +∞ .

We study semiclassical states of the nonlinear Dirac equation -i (h) over bar partial derivative(t)psi = ic (h) over bar Sigma(3)(k=1) alpha(k)partial derivative(k)psi - mc(2)beta psi - M(x)psi + f(vertical bar psi vertical bar)psi, t is an element of R, x is an element of R-3, where V is a bounded continuous potential function and the nonlinear term f (vertical bar psi vertical bar)psi is superlinear, possibly of critical growth. Our main result deals with standing wave solutions that concentrate near a critical point of the potential. Standard methods applicable to nonlinear Schrodinger equations, like Lyapunov-Schmidt reduction or penalization, do not work, not even for the homogeneous nonlinearity f(s) = s(p). We develop a variational method for the strongly indefinite functional associated to the problem.

This paper is concerned with the existence of normalized solutions of the nonlinear Schrodinger equation -Delta u + V(x)u + lambda u = vertical bar u vertical bar(p-2)u in R-N in the mass supercritical and Sobolev subcritical case 2 + 4/N < p < 2*. We prove the existence of a solution (u, lambda) is an element of H-1 (R-N) x R+ with prescribed L-2-norm parallel to u parallel to(2) = rho under various conditions on the potential V : R-N -> R, positive and vanishing at infinity, including potentials with singularities. The proof is based on a new min-max argument.

Let (M, g, sigma) be a compact Riemannian spin manifold of dimension m >= 2, let S(M) denote the spinor bundle on M, and let Dbe the Atiyah-Singer Dirac operator acting on spinors psi : M -> S(M). We study the existence of solutions of the nonlinear Dirac equation with critical exponent D psi = lambda psi + f(vertical bar psi vertical bar)psi + vertical bar psi vertical bar(2/m-1) psi (NLD) where lambda is an element of R and f(vertical bar psi vertical bar)psi is a subcritical nonlinearity in the sense that f(s) = o(s(2/m-1)) as s -> infinity. A model nonlinearity is f(s) = alpha s(p-2) with 2 < p < 2m/m-1, alpha is an element of R. In particular we study the nonlinear Dirac equation D psi = lambda psi + vertical bar psi vertical bar(2/m-1)psi, lambda is an element of R. (BND) This equation is a spinorial analogue of the Brezis-Nirenberg problem. As corollary of our main results we obtain the existence of least energy solutions (lambda, psi) of (BND) for every lambda > 0, even if lambda is an eigenvalue of D. For some classes of nonlinearities fwe also obtain solutions of (NLD) for every lambda is an element of R, except for non-positive eigenvalues. If m not equivalent to 3(mod 4) we obtain solutions of (BND) for every lambda is an element of R, except for a finite number of non-positive eigenvalues. In certain parameter ranges we obtain multiple solutions of (NLD) and (BND), some near the trivial branch, others away from it. The proofs of our results are based on variational methods using the strongly indefinite energy functional associated to (NLD). (C) 2021 Elsevier Inc. All rights reserved.

We prove the existence of a least energy solution to the problem $$\begin{aligned} -\Delta u-(I_{\alpha }*F(u))f(u)=\lambda u\ \text { in }\ {\mathbb {R}}^{N},\quad \int _{{\mathbb {R}}^N}u^2(x)dx = a^2, \end{aligned}$$ where $$N\ge 1$$ , $$\alpha \in (0,N)$$ , $$F(s):=\int _{0}^{s}f(t)dt$$ , and $$I_{\alpha }:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}$$ is the Riesz potential. If f is odd in u then we prove the existence of infinitely many normalized solutions.

The talk will be concerned with the existence of L 2 normalized solutions to nonlinear elliptic equations and systems.A model problem is the system of nonlinear Schrödinger equations(−∆u + λ1u = µ1u 3 + βuv2 in R 3−∆ v + λ2v = µ2v 3 + βu 2 v in R 3 with normalization constraints Z R3 u 2 = a 2 and Z R3 v 2 = b 2 .Whereas nonlinear elliptic equations and systems have been investigated intensively since the 1960 s, in comparison surprisingly little is known about solutions with prescribed L 2 norms.We discuss this problem and survey recent results.The talk is based on joint work with Louis Jeanjean, Yanyan Liu, Zhaoli Liu, Nicola Soave, Xuexiu Zhong, Wenming Zou.

In the present paper, we prove the existence of solutions $$(\lambda _1,\lambda _2,u,v)\in \mathbb {R}^2\times H^1(\mathbb {R}^3,\mathbb {R}^2)$$ to systems of coupled Schrödinger equations $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda _1u=\mu _1 u^3+\beta uv^2\quad &{}\hbox {in}\;\mathbb {R}^3\\ -\Delta v+\lambda _2v=\mu _2 v^3+\beta u^2v\quad &{}\hbox {in}\;\mathbb {R}^3\\ u,v>0&{}\hbox {in}\;\mathbb {R}^3 \end{array}\right. } \end{aligned}$$ satisfying the normalization constraint $$ \int _{\mathbb {R}^3}u^2=a^2\quad \hbox {and}\;\int _{\mathbb {R}^3}v^2=b^2, $$ which appear in binary mixtures of Bose–Einstein condensates or in nonlinear optics. The parameters $$\mu _1,\mu _2,\beta >0$$ are prescribed as are the masses $$a,b>0$$ . The system has been considered mostly in the case of fixed frequencies $$\lambda _1,\lambda _2$$ . When the masses are prescribed, the standard approach to this problem is variational with $$\lambda _1,\lambda _2$$ appearing as Lagrange multipliers. Here we present a new approach based on the fixed point index in cones, bifurcation theory, and the continuation method. We obtain the existence of normalized solutions for any given $$a,b>0$$ for $$\beta $$ in a large range. We also have a result about the nonexistence of positive solutions which shows that our existence theorem is almost optimal. Especially, if $$\mu _1=\mu _2$$ we prove that normalized solutions exist for all $$\beta >0$$ and all $$a,b>0$$ .

We study the possible blow-up behavior of solutions to the slightly subcritical elliptic problem with Hardy term $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u-\mu \frac{u}{|x|^2} = |u|^{2^{*}-2-\varepsilon }u &{} \text {in } \Omega , \\ u = 0&{} \text {on } \partial \Omega , \end{array}\right. } \end{aligned}$$in a bounded domain $$\Omega \subset {\mathbb {R}}^N (N\ge 7)$$ with $$0\in \Omega$$, as $$\mu ,\epsilon \rightarrow 0^+$$. In [6], we obtained the existence of nodal solutions that blow up positively at the origin and negatively at a different point as $$\mu =O(\epsilon ^\alpha )$$ with $$\alpha >\frac{N-4}{N-2}$$, $$\epsilon \rightarrow 0^+$$. Here we prove the existence of nodal bubble tower solutions, i.e. superpositions of bubbles of different signs, all blowing up at the origin but with different blow-up order, as $$\mu =O(\epsilon )$$, $$\epsilon \rightarrow 0^+$$.

We prove the existence of infinitely many solutions λ _1, λ _2 ∈ℝ , u,v ∈ H^1(ℝ^3) , for the nonlinear Schrödinger system {[ -Δ u - λ _1 u = μ u^3+ β u v^2 in ℝ^3; -Δ v- λ _2 v = μ v^3 +β u^2 v in ℝ^3; u,v>0 in ℝ^3; ∫ _ℝ^3 u^2 = a^2 and ∫ _ℝ^3 v^2 = a^2, ]. where a,μ >0 and β≤ -μ are prescribed. Our solutions satisfy u v so they do not come from a scalar equation. The proof is based on a new minimax argument, suited to deal with normalization conditions.