Normalized ground state for the Sobolev critical Schrödinger equation involving Hardy term with combined nonlinearities
Abstract: In this paper, we study the existence and properties of normalized solutions for the following Sobolev critical Schrödinger equation involving Hardy term:where 2 * is the Sobolev critical exponent. For a 𝐿 2 -subcritical, 𝐿 2 -critical, or 𝐿 2 -supercritical perturbation 𝜈|𝑢| 𝑝−2 𝑢, we prove several existence results of normalized ground state when 𝜈 ≥ 0 and non-existence results when 𝜈 ≤ 0. Furthermore, we also consider the asymptotic behavior of the normalized solutions 𝑢 as 𝜇 → 0 or 𝜈 → 0.
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Cited by 3 publications
References 31 publications
Normalized Solutions to at Least Mass Critical Problems: Singular Polyharmonic Equations and Related Curl–Curl Problems
We are interested in the existence of normalized solutions to the problem $$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^m u+\frac{\mu }{|y|^{2m}}u + \lambda u = g(u), \quad x = (y,z) \in \mathbb {R}^K \times \mathbb {R}^{N-K}, \\ \int _{\mathbb {R}^N} |u|^2 \, dx = \rho > 0, \end{array}\right. } \end{aligned}$$ ( - Δ ) m u + μ | y | 2 m u + λ u = g ( u ) , x = ( y , z ) ∈ R K × R N - K , ∫ R N | u | 2 d x = ρ > 0 , in the so-called at least mass critical regime. We utilize recently introduced variational techniques involving the minimization on the $$L^2$$ L 2 -ball. Moreover, we find also a solution to the related curl–curl problem $$\begin{aligned} {\left\{ \begin{array}{ll} \nabla \times \nabla \times \textbf{U}+\lambda \textbf{U}=f(\textbf{U}), \quad x \in \mathbb {R}^N,\\ \int _{\mathbb {R}^N}|\textbf{U}|^2\,dx=\rho ,\\ \end{array}\right. } \end{aligned}$$ ∇ × ∇ × U + λ U = f ( U ) , x ∈ R N , ∫ R N | U | 2 d x = ρ , which arises from the system of Maxwell equations and is of great importance in nonlinear optics.
Normalized Solutions to at Least Mass Critical Problems: Singular Polyharmonic Equations and Related Curl–Curl Problems
We are interested in the existence of normalized solutions to the problem $$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^m u+\frac{\mu }{|y|^{2m}}u + \lambda u = g(u), \quad x = (y,z) \in \mathbb {R}^K \times \mathbb {R}^{N-K}, \\ \int _{\mathbb {R}^N} |u|^2 \, dx = \rho > 0, \end{array}\right. } \end{aligned}$$ ( - Δ ) m u + μ | y | 2 m u + λ u = g ( u ) , x = ( y , z ) ∈ R K × R N - K , ∫ R N | u | 2 d x = ρ > 0 , in the so-called at least mass critical regime. We utilize recently introduced variational techniques involving the minimization on the $$L^2$$ L 2 -ball. Moreover, we find also a solution to the related curl–curl problem $$\begin{aligned} {\left\{ \begin{array}{ll} \nabla \times \nabla \times \textbf{U}+\lambda \textbf{U}=f(\textbf{U}), \quad x \in \mathbb {R}^N,\\ \int _{\mathbb {R}^N}|\textbf{U}|^2\,dx=\rho ,\\ \end{array}\right. } \end{aligned}$$ ∇ × ∇ × U + λ U = f ( U ) , x ∈ R N , ∫ R N | U | 2 d x = ρ , which arises from the system of Maxwell equations and is of great importance in nonlinear optics.
Normalized ground state solutions for critical growth Schrödinger equations with Hardy potential
In this article, we study the following Schrödinger equation \begin{align*} \begin{cases} -\Delta u -\frac{\mu}{|x|^2} u+\lambda u =f(u), &\text{in}~ \mathbb{R}^N\backslash\{0\},\\ \int_{\mathbb{R}^{N}}|u|^{2}\mathrm{d} x=a, & u\in H^1(\mathbb{R}^{N}), \end{cases} \end{align*} where $N\geq 3$ , a > 0, and $\mu \lt \frac{(N-2)^2}{4}$ . Here $\frac{1}{|x|^2} $ represents the Hardy potential (or ‘inverse-square potential’), λ is a Lagrange multiplier, and the nonlinearity function f satisfies the general Sobolev critical growth condition. Our main goal is to demonstrate the existence of normalized ground state solutions for this equation when $0 \lt \mu \lt \frac{(N-2)^2}{4}$ . We also analyse the behaviour of solutions as $\mu\to0^+$ and derive the existence of normalized ground state solutions for the limiting case where µ = 0. Finally, we investigate the existence of normalized solutions when µ < 0 and analyse the asymptotic behaviour of solutions as $\mu\to 0^-$ .
Global existence, blow-up and mass concentration for the inhomogeneous nonlinear Schrödinger equation with inverse-square potential
<abstract><p>In the current paper, the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation including inverse-square potential is considered. First, some criteria of global existence and finite-time blow-up in the mass-critical and mass-supercritical settings with $ 0 < c\leq c^{*} $ are obtained. Then, by utilizing the potential well method and the sharp Sobolev constant, the sharp condition of blow-up is derived in the energy-critical case with $ 0 < c < \frac{N^{2}+4N}{(N+2)^{2}}c^{*} $. Finally, we establish the mass concentration property of explosive solutions, as well as the dynamic behaviors of the minimal-mass blow-up solutions in the $ L^{2} $-critical setting for $ 0 < c < c^{*} $, by means of the variational characterization of the ground-state solution to the elliptic equation, scaling techniques and a suitable refined compactness lemma. Our results generalize and supplement the ones of some previous works.</p></abstract>
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