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

Abstract

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.

Existence of least energy nodal solution for Kirchhoff–Schrödinger–Poisson system with potential vanishing

“…Inspired by the works mentioned, especially by [10,13,15,35,45], in this paper, we find the nodal solutions to system (1.1) under some weaker assumptions on f . As in [1], we say that (V , K) ∈ K if continuous functions V , K : R 3 → R satisfy the following conditions:…”

Existence of positive solutions for a class of quasilinear Schrödinger equation with critical exponent

“…Moreover, Aires and Souto [23] considered the nonlinearity 𝜌 satisfied subcritical growth, V can vanish at infinity, and proved the existence of nontrivial solutions by a penalization technique. By considering a minimizing problem restricted on a Nehari manifold, Yang et al [24] studied the existence of least action nodal solution via deformation flow argument and L ∞ -estimates for the following equation…”

“…Moreover, Aires and Souto [23] considered the nonlinearity $$$ \rho $$$ satisfied subcritical growth, $$$ V $$$ can vanish at infinity, and proved the existence of nontrivial solutions by a penalization technique. By considering a minimizing problem restricted on a Nehari manifold, Yang et al [24] studied the existence of least action nodal solution via deformation flow argument and $$$ {L}^{\infty } $$$‐estimates for the following equation $$$$ -\Delta u+V(x)u+\frac{\kappa }{2}\Delta \left({u}^2\right)u=g(u)+\lambda {\left|u\right|}^{p-2}, $$$$ where $$$ V(x) $$$ is radial symmetry and the nonlinear term $$$ g $$$ satisfies condition: $$$$ u\mapsto \frac{g(u)}{u^p},\kern0.30em u>0\kern0.50em \mathrm{is}\ \mathrm{non}\hbox{-} \mathrm{decreasing}\ \mathrm{for}\ \mathrm{some}\kern0.30em p>1. $$$$ Chen et al [25] used the method developed by Jeanjean [26] to prove the existence of positive solutions with superlinear condition.…”

Quasilinear Schrödinger equations with unbounded or decaying potentials in dimension 2

“…On the other hand, when 𝜅 > 0, the critical power is 𝑝 = 2 * , see [28], where the authors use Pohozaev identity to justify this fact. In the last years, some progress has been made in the case 𝜅 > 0, see [1,3,20,28,36]. Naturally, the above discussion was extended to the case 𝑁 = 2 and 2 * = ∞.…”