scholarly journals Positive solutions for a class of generalized quasilinear Schrödinger equation involving concave and convex nonlinearities in Orilicz space

2021 ◽  
pp. 1-26
Author(s):  
Yan Meng ◽  
Xianjiu Huang ◽  
Jianhua Chen
Keyword(s):  
Schrödinger Equation ◽  
Positive Solutions ◽  
Schrodinger Equation ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Quasilinear Schrödinger Equation ◽  
Existence Of Positive Solutions ◽  
Change Of Variable ◽  
Concave And Convex Nonlinearities

In this paper, we study the following generalized quasilinear Schrödinger equation − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = λ f ( x , u ) + h ( x , u ) , x ∈ R N , where λ > 0 , N ≥ 3 , g ∈ C 1 ( R , R + ) . By using a change of variable, we obtain the existence of positive solutions for this problem with concave and convex nonlinearities via the Mountain Pass Theorem. Our results generalize some existing results.

scholarly journals Multiplicity of Quasilinear Schrödinger Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Xiaorong Luo ◽  
Anmin Mao ◽  
Xiangxiang Wang
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Negative Energy ◽  
Mountain Pass ◽  
Type Solution ◽  
Fountain Theorem ◽  
Quasilinear Schrödinger Equation ◽  
New Techniques ◽  
Dual Fountain Theorem ◽  
Concave And Convex Nonlinearities

In this paper, we study the quasilinear Schrödinger equation involving concave and convex nonlinearities. When the pair of parameters belongs to a certain subset of ℝ2, we establish the existence of a nontrivial mountain pass-type solution and infinitely many negative energy solutions by using some new techniques and dual fountain theorem. Recent results from the literature are improved and extended.

Positive solutions for a quasilinear Schrödinger equation involving Hardy potential and critical exponent

2014 ◽  
Vol 16 (06) ◽  
pp. 1450034 ◽  
Author(s):  
Xiaoyu Zeng ◽  
Yimin Zhang ◽  
Huan-Song Zhou
Keyword(s):  
Schrödinger Equation ◽  
Critical Exponent ◽  
Positive Solutions ◽  
Schrodinger Equation ◽  
Quasilinear Equation ◽  
Hardy Potential ◽  
Quasilinear Schrödinger Equation ◽  
Semilinear Equation ◽  
Existence Of Positive Solutions ◽  
Nehari Method

We are concerned with positive solutions of a quasilinear Schrödinger equation with Hardy potential and critical exponent. Different from the semilinear equation, the Hardy term in our equation is not only singular, but also nonlinear. It seems unlikely to get solutions for our equation in H1(ℝN) ∩ L∞(ℝN) by using Nehari method as Liu–Liu–Wang [Ground states for quasilinear Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations 46 (2013) 641–669]. In this paper, by transforming the quasilinear equation to a semilinear equation, we established the existence of positive solutions for the quasilinear Schrödinger equation in H1(ℝN) under suitable conditions.

scholarly journals Nonexistence of Positive Solutions for Quasilinear Equations with Decaying Potentials

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 425
Author(s):  
Ohsang Kwon
Keyword(s):  
Schrödinger Equation ◽  
Plasma Physics ◽  
Growth Condition ◽  
Positive Solutions ◽  
Schrodinger Equation ◽  
Nonlinear Term ◽  
Quasilinear Equations ◽  
Quasilinear Schrödinger Equation ◽  
Superfluid Film

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.

Bifurcation and standing wave solutions for a quasilinear Schrödinger equation

2018 ◽  
Vol 149 (04) ◽  
pp. 939-968
Author(s):  
Guowei Dai
Keyword(s):  
Schrödinger Equation ◽  
Positive Solution ◽  
Standing Wave ◽  
Positive Solutions ◽  
Schrodinger Equation ◽  
Quasilinear Schrödinger Equation ◽  
Topological Methods ◽  
Wave Solutions ◽  
Multiplicity Of Positive Solutions ◽  
Image Position

AbstractWe use bifurcation and topological methods to investigate the existence/nonexistence and the multiplicity of positive solutions of the following quasilinear Schrödinger equation$$\left\{ {\matrix{ {-\Delta u-\kappa \Delta \left( {u^2} \right)u = \beta u-\lambda \Phi \left( {u^2} \right)u{\mkern 1mu} {\mkern 1mu} } \hfill & {{\rm in}\;\Omega ,} \hfill \cr {u = 0} \hfill & {{\rm on}\;\partial \Omega } \hfill \cr } } \right.$$involving sublinear/linear/superlinear nonlinearities at zero or infinity with/without signum condition. In particular, we study the changes in the structure of positive solution withκas the varying parameter.

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