Nonexistence of Positive Solutions for Quasilinear Equations with Decaying Potentials

Ohsang Kwon

doi:10.3390/math8030425

Abstract

In this paper, we consider a quasilinear Schrödinger equation, which arises from the study of the superfluid film equation in plasma physics. Our main goal is to find the growth condition for nonlinear term and decaying condition for the potential, which guarantee the nonexistence of positive solutions.

Highlights

In this paper, we consider the following quasilinear Schrödinger equation: (iεφt + ε2 ∆φ − W ( x )φ + ε2 φl 0 (|φ|2 )∆l (|φ|2 ) + ρ(|φ|2 )φ = 0 in (0, ∞) × R N, φ(0, x ) = a0 ( x ) in R N, N ≥ 1, (1)where φ : R × R N → C is a complex valued function, W : R N → R is a given potential, l, ρ : [0, ∞) → R are given functions, and i is the imaginary unit.Depending on the structure of the quasilinear term l (|φ|2 ) in (1), the above equation describes various physical phenomena
We refer to References [7,8] for the study of plasma physics and fluid mechanics, Reference [9] for Heisenberg ferromagnetic and magnon, and Reference [10] for the condensed matter theory
We are mainly interested in the type l (s) = s, which arises from the superfluid film equation in plasma physics and self-trapped electrons in quadratic or hexagonal lattices

Summary

Introduction

We consider the following quasilinear Schrödinger equation: ( Our main goal is to study the nonexistence of positive solution for the following equation with the potential V ( x ) decaying at infinity: ( Bae and Byeon in Reference [25] found almost optimal threshold on the decaying condition of V ( x ) at infinity between existence and nonexistence of positive solutions of (4).

Results

Conclusion