Least action nodal solutions for a quasilinear defocusing Schrödinger equation with supercritical nonlinearity

2019 ◽  
Vol 21 (05) ◽  
pp. 1850026 ◽  
Author(s):  
Minbo Yang ◽  
Carlos Alberto Santos ◽  
Jiazheng Zhou
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nehari Manifold ◽  
Nodal Solution ◽  
Nodal Solutions ◽  
Subcritical Growth ◽  
Least Action ◽  
Continuous Potential ◽  
Supercritical Nonlinearity

In this paper, we consider the existence of least action nodal solutions for the quasilinear defocusing Schrödinger equation in [Formula: see text]: [Formula: see text] where [Formula: see text] is a positive continuous potential, [Formula: see text] is of subcritical growth, [Formula: see text] and [Formula: see text] are two non-negative parameters. By considering a minimizing problem restricted on a partial Nehari manifold, we prove the existence of least action nodal solution via deformation flow arguments and [Formula: see text]-estimates.

Least Energy Nodal Solutions for a Defocusing Schrödinger Equation with Supercritical Exponent

2018 ◽  
Vol 62 (1) ◽  
pp. 1-23 ◽  
Author(s):  
Minbo Yang ◽  
Carlos Alberto Santos ◽  
Jiazheng Zhou
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Positive Function ◽  
Nehari Manifold ◽  
Nodal Solution ◽  
Nodal Solutions ◽  
Quasilinear Schrödinger Equation ◽  
Subcritical Growth ◽  
Image Position ◽  
Least Energy

AbstractIn this paper we consider the existence of least energy nodal solution for the defocusing quasilinear Schrödinger equation $$-\Delta u - u \Delta u^2 + V(x)u = a(x)[g(u) + \lambda \vert u \vert ^{p-2}u] \hbox{in} {\open R}^N,$$ where λ≥0 is a real parameter, V(x) is a non-vanishing function, a(x) can be a vanishing positive function at infinity, the nonlinearity g(u) is of subcritical growth, the exponent p≥22*, and N≥3. The proof is based on a dual argument on Nehari manifold by employing a deformation argument and an $L</italic>^{\infty}({\open R}^{N})$-estimative.

scholarly journals Concentration phenomena for fractional magnetic NLS equations

2021 ◽  
pp. 1-39
Author(s):  
Vincenzo Ambrosio
Keyword(s):  
Schrödinger Equation ◽  
Electric Potential ◽  
Schrodinger Equation ◽  
Nehari Manifold ◽  
Local Condition ◽  
Subcritical Growth ◽  
Concentration Phenomena ◽  
Penalization Technique ◽  
Complex Valued ◽  
Kato’S Inequality

We study the multiplicity and concentration of complex-valued solutions for a fractional magnetic Schrödinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik–Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems.

scholarly journals On Jensen-type Inequalities for Nonsmooth Radial Scattering Solutions of a Loglog Energy-Supercritical Schrödinger Equation

2018 ◽  
Vol 2020 (8) ◽  
pp. 2501-2541
Author(s):  
Tristan Roy
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
A Priori ◽  
Time Interval ◽  
Type Estimate ◽  
A Priori Bound ◽  
Long Time ◽  
Supercritical Nonlinearity ◽  
Gamma S ◽  
Radial Scattering

Abstract We prove scattering of solutions of the loglog energy-supercritical Schrödinger equation $i \partial _{t} u + \triangle u = |u|^{\frac{4}{n-2}} u g(|u|)$ with $g(|u|) := \log ^{\gamma } {( \log{(10+|u|^{2})} )}$, $0 &lt; \gamma &lt; \gamma _{n}$, n ∈ {3, 4, 5}, and with radial data $u(0) := u_{0} \in \tilde{H}^{k}:= \dot{H}^{k} (\mathbb{R}^{n})\,\cap\,\dot{H}^{1} (\mathbb{R}^{n})$, where $\frac{n}{2} \geq k&gt; 1 \left(\text{resp.}\,\frac{4}{3}&gt; k &gt; 1\right)$ if n ∈ {3, 4} (resp. n = 5). The proof uses concentration techniques (see e.g., [ 2, 12]) to prove a long-time Strichartz-type estimate on an arbitrarily long time interval J depending on an a priori bound of some norms of the solution, combined with an induction on time of the Strichartz estimates in order to bound these norms a posteriori (see e.g., [ 8, 10]). We also revisit the scattering theory of solutions with radial data in $\tilde{H}^{k}$, $k&gt; \frac{n}{2}$, and n ∈ {3, 4}; more precisely, we prove scattering for a larger range of $\gamma$ s than in [ 10]. In order to control the barely supercritical nonlinearity for nonsmooth solutions, that is, solutions with data in $\tilde{H}^{k}$, $k \leq \frac{n}{2}$, we prove some Jensen-type inequalities.

Ground state solution for the Schrödinger equation with Hardy potential and critical Sobolev exponent

2021 ◽  
pp. 1-19
Author(s):  
Jing Zhang ◽  
Lin Li
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Critical Sobolev Exponent ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
State Solution ◽  
Hardy Potential ◽  
Asymptotically Periodic ◽  
Sobolev Exponent

In this paper, we consider the following Schrödinger equation (0.1) − Δ u − μ u | x | 2 + V ( x ) u = K ( x ) | u | 2 ∗ − 2 u + f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where N ⩾ 4, 0 ⩽ μ < μ ‾, μ ‾ = ( N − 2 ) 2 4 , V is periodic in x, K and f are asymptotically periodic in x, we take advantage of the generalized Nehari manifold approach developed by Szulkin and Weth to look for the ground state solution of (0.1).

Positive and nodal solutions for an elliptic equation with critical growth

2016 ◽  
Vol 18 (02) ◽  
pp. 1550021 ◽  
Author(s):  
Marcelo F. Furtado ◽  
Bruno N. Souza
Keyword(s):  
Elliptic Equation ◽  
Positive Solution ◽  
Critical Growth ◽  
Smooth Domain ◽  
Nodal Solution ◽  
Nodal Solutions ◽  
Bounded Smooth Domain ◽  
Continuous Potential ◽  
Bounded Smooth

We consider the problem [Formula: see text] where [Formula: see text] is a bounded smooth domain, [Formula: see text], [Formula: see text], [Formula: see text]. Under some suitable conditions on the continuous potential [Formula: see text] and on the parameter [Formula: see text], we obtain one nodal solution for [Formula: see text] and one positive solution for [Formula: see text].

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