Groundstates and radial solutions to nonlinear Schr"odinger-Poisson-Slater equations at the critical frequency (original) (raw)

Non-radial normalized solutions for a nonlinear Schrodinger equation

Electronic Journal of Differential Equations

This article concerns the existence of multiple non-radial positive solutions of the L2-constrained problem \displaylines{-\Delta{u}-Q(\varepsilon x)|u|^{p-2}u=\lambda{u},\quad \text{in }\mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=1,}$$ where $Q(x)$ is a radially symmetric function, ε>0 is a small parameter, $N\geq 2$, and $p \in (2, 2+4/N)$ is assumed to be mass sub-critical. We are interested in the symmetry breaking of the normalized solutions and we prove the existence of multiple non-radial positive solutions as local minimizers of the energy functional.

A note on quasilinear Schrödinger equations with singular or vanishing radial potentials

Differential and Integral Equations

In this note we complete the study of [3], where we got existence results for the quasilinear elliptic equation −∆w + V (|x|) w − w ∆w 2 = K(|x|)g(w) in R N , with singular or vanishing continuous radial potentials V (r), K(r). In [3] we assumed, for technical reasons, that K(r) was vanishing as r → 0, while in the present paper we remove this obstruction. To face the problem we apply a suitable change of variables w = f (u) and we find existence of non negative solutions by the application of variational methods. Our solutions satisfy a weak formulations of the above equation, but they are in fact classical solutions in R N \ {0}. The nonlinearity g has a double-power behavior, whose standard example is g(t) = min{t q 1 −1 , t q 2 −1 } (t > 0), recovering the usual case of a singlepower behavior when q1 = q2.

Finite time blow-up of non-radial solutions for some inhomogeneous Schr\"{o}dinger equations

arXiv (Cornell University), 2023

This work studies the inhomogeneous Schrödinger equation i∂ t u − K s,λ u + F (x, u) = 0, u(t, x) : R × R N → C. Here, s ∈ {1, 2}, N > 2s and λ > − (N −2) 2 4. The linear Schrödinger operator reads K s,λ := (−∆) s + (2 − s) λ |x| 2 and the focusing source term is local or non-local F (x, u) ∈ {|x| −2τ |u| 2(q−1) u, |x| −τ |u| p−2 (J α * | • | −τ |u| p)u}. The Riesz potential is J α (x) = C N,α |x| −(N −α) , for certain 0 < α < N. The singular decaying term |x| −2τ , for some τ > 0 gives a inhomogeneous non-linearity. One considers the inter-critical regime, namely 1 + 2(1−τ) N < q < 1 + 2(1−τ) N −2s and 1 + 2−2τ +α N < p < 1 + 2−2τ +α N −2s. The purpose is to prove the finite time blow-up of solutions with datum in the energy space, non necessarily radial or with finite variance. The assumption on the data is expressed in terms of non-conserved quantities. This is weaker than the ground state threshold standard condition. The blow-up under the ground threshold or with negative energy are consequences. The proof is based on Morawetz estimates and a non-global ordinary differential inequality. This work extends the recent paper by R. Bai and B. Li (Blow-up for the inhomogeneous nonlinear Schrödinger equation, Nonlinear Anal. (2023)). The extensions is in many directions. First, one considers the non-local source term. Second, one expresses the assumptions on term of non-conserved quantities, which gives some blow-up criteria. Third, one treats the INLS with inverse square potential. Fourth, one investigates the bi-harmonic case. The authors try to consider all these cases in a unified approach. This makes more clear some shared proprieties of the different variants of Schrödinger equations. Furthermore, this work gives a natural complement of the paper by T. Saanouni (Energy scattering for radial focusing inhomogeneous bi-harmonic Schrödinger equations, Calc. Var., 2021, 60-113), where the author deals with the scattering of the bi-harmonic Schrödinger equation in the inter-critical focusing regime under the ground state threshold. The result of this note don't extend to the limiting case τ = 0, which is still an open problem. This gives an essential difference between the NLS and the INLS.

Existence of weak solutions to some stationary Schr "odinger equations with singular nonlinearity

2013

We prove some existence (and sometimes also uniqueness) of weak solutions to some stationary equations associated to the complex Schr\"{o}dinger operator under the presence of a singular nonlinear term. Among other new facts, with respect some previous results in the literature for such type of nonlinear potential terms, we include the case in which the spatial domain is possibly unbounded (something which is connected with some previous localization results by the authors), the presence of possible non-local terms at the equation, the case of boundary conditions different to the Dirichlet ones and, finally, the proof of the existence of solutions when the right-hand side term of the equation is beyond the usual L2L^2L2-space.

Global existence and scattering for the inhomogeneous nonlinear Schr\"odinger equation

arXiv (Cornell University), 2021

In this paper we consider the inhomogeneous nonlinear Schrödinger equation i∂tu + ∆u = K(x)|u| α u, u(0) = u0 ∈ H s (R N), s = 0, 1, N ≥ 1, |K(x)| + |x| s |∇ s K(x)| |x| −b , 0 < b < min(2, N − 2s), 0 < α < (4 − 2b)/(N − 2s). We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted L 2-space for a new range α0(b) < α < (4 − 2b)/N. The value α0(b) is the positive root of N α 2 + (N − 2 + 2b)α − 4 + 2b = 0, which extends the Strauss exponent known for b = 0. Our results improve the known ones for K(x) = µ|x| −b , µ ∈ C and apply for more general potentials. In particular, we show the impact of the behavior of the potential at the origin and infinity on the allowed range of α. Some decay estimates are also established for the defocusing case. To prove the scattering results, we give a new criterion taking into account the potential K. Here u = u(t, x) ∈ C, t ∈ R, x ∈ R N , N ≥ 1, s = 0 or s = 1 and α > 0. The potential K is a complex valued function satisfying some hypothesis. In particular, K(x) = µ|x| −b and K(x) = µ(1 + |x| 2) − b 2 , b > 0, µ ∈ C, will be considered. Equation (1.1) with a constant function K, corresponds to the standard nonlinear Schrödinger equation. The case where K is non constant and bounded is considered in [33, 35]. The unbounded potential case is also treated in [11, 12, 41], where K(x) = |x| b. Here we consider (1.1) with potential having a decay like |x| −b at infinity and may be singular at the origin. This kind

Groundstates and radial solutions to nonlinear Schr"odinger-Poisson-Slater equations at the critical frequency (original) (raw)

Related papers