Abstract We consider Maxwell’s equations for Kerr-type optical materials, which are magnetically inactive and have a nonlinear response to electric fields. This response consists of a linear plus a cubic term, which are both inhomogeneous with bounded coefficients. The cubic term is temporally retarded while the linear term has instantaneous and retarded contributions. For slab waveguides we show existence of breathers, which are time-periodic, real-valued solutions that are localized in the direction perpendicular to the waveguide, and moreover they are traveling along one direction of the waveguide. We find these breathers using a variational method which relies on the assumption that an effective operator related to the linear part of Maxwell’s equations has a spectral gap about 0. We also give examples of material coefficients, including nonperiodic materials, where such a spectral gap is present.

Abstract We study the uniqueness of solutions to the stationary Schrödinger equation with potential on infinite graphs, within suitable weighted $$\ell ^p$$ ℓ p spaces. The potential is allowed to vanish at infinity at a controlled rate. Our results extend those in [20] by considering a broader class of potentials, by removing the assumption that the potential is bounded away from zero.

Abstract We study the following class of quasilinear degenerate elliptic equations with critical nonlinearity $$\begin{aligned} {\left\{ \begin{array}{ll}-\Delta _{\gamma ,p} u= \lambda |u|^{q-2}u+\left| u\right| ^{p_{\gamma }^{*}-2}u & \text { in } \Omega \subset \mathbb {R}^N,\\ u=0 & \text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$ - Δ γ , p u = λ | u | q - 2 u + u p γ ∗ - 2 u in Ω ⊂ R N , u = 0 on ∂ Ω , where $$\Delta _{\gamma , p}u:=\sum _{i=1}^N X_i(|\nabla _\gamma u|^{p-2}X_i u)$$ Δ γ , p u : = ∑ i = 1 N X i ( | ∇ γ u | p - 2 X i u ) is the Grushin p -Laplace operator, $$z:=(x, y) \in \mathbb {R}^N$$ z : = ( x , y ) ∈ R N , $$N=m+n,$$ N = m + n , $$m,n \ge 1$$ m , n ≥ 1 , where $$\nabla _\gamma =(X_1, \ldots , X_N)$$ ∇ γ = ( X 1 , … , X N ) is the Grushin gradient, defined as the system of vector fields $$X_i=\frac{\partial }{\partial x_i}, i=1, \ldots , m$$ X i = ∂ ∂ x i , i = 1 , … , m , $$X_{m+j}=|x|^\gamma \frac{\partial }{\partial y_j}, j=1, \ldots , n$$ X m + j = | x | γ ∂ ∂ y j , j = 1 , … , n , where $$\gamma >0$$ γ > 0 . Here, $$\Omega \subset \mathbb {R}^{N}$$ Ω ⊂ R N is a smooth bounded domain such that $$\Omega \cap \{x=0\}\ne \emptyset $$ Ω ∩ { x = 0 } ≠ ∅ , $$\lambda >0$$ λ > 0 , $$q \in [p,p_\gamma ^*)$$ q ∈ [ p , p γ ∗ ) , where $$p_{\gamma }^{*}=\frac{pN_\gamma }{N_\gamma -p}$$ p γ ∗ = p N γ N γ - p and $$N_\gamma =m+(1+\gamma )n$$ N γ = m + ( 1 + γ ) n denotes the homogeneous dimension attached to the Grushin gradient. The results extend to the p -case the Brezis-Nirenberg type results in Alves-Gandal-Loiudice-Tyagi [J. Geom. Anal. 2024, 34(2),52]. The main crucial step is to preliminarily establish the existence of the extremals for the involved Sobolev-type inequality $$ \int _{\mathbb {R}^N} |\nabla _{\gamma } u|^p dz \ge S_{\gamma ,p} \left( \int _{\mathbb {R}^N} |u|^{p_\gamma ^*} dz \right) ^{p/p_\gamma ^*} $$ ∫ R N | ∇ γ u | p d z ≥ S γ , p ∫ R N | u | p γ ∗ d z p / p γ ∗ and their qualitative behavior as positive entire solutions to the limit problem $$\begin{aligned} -\Delta _{\gamma ,p} u= u^{p_{\gamma }^{*}-1}\quad \hbox {on}\, \mathbb {R}^N, \end{aligned}$$ - Δ γ , p u = u p γ ∗ - 1 on R N , whose study has independent interest.