Bifurcation and multiplicity of positive solutions for nonhomogeneous fractional Schrödinger equations with critical growth

2020 ◽  
Vol 63 (8) ◽  
pp. 1571-1612
Author(s):  
Xiaoming He ◽  
Wenming Zou
Keyword(s):  
Positive Solutions ◽  
Critical Growth ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Fractional Schrödinger Equations ◽  
Multiplicity Of Positive Solutions

scholarly journals A Unified Approach to Singularly Perturbed Quasilinear Schrödinger Equations

2020 ◽  
Vol 88 (2) ◽  
pp. 507-534
Author(s):  
Daniele Cassani ◽  
Youjun Wang ◽  
Jianjun Zhang
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Real Function ◽  
Critical Growth ◽  
External Potential ◽  
Singularly Perturbed ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Unified Approach

AbstractIn this paper we present a unified approach to investigate existence and concentration of positive solutions for the following class of quasilinear Schrödinger equations, $$-\varepsilon^2\Delta u+V(x)u\mp\varepsilon^{2+\gamma}u\Delta u^2=h(u),\ \ x\in \mathbb{R}^N, $$ - ε 2 Δ u + V ( x ) u ∓ ε 2 + γ u Δ u 2 = h ( u ) , x ∈ R N , where $$N\geqslant3, \varepsilon > 0, V(x)$$ N ⩾ 3 , ε > 0 , V ( x ) is a positive external potential,h is a real function with subcritical or critical growth. The problem is quite sensitive to the sign changing of the quasilinear term as well as to the presence of the parameter $$\gamma>0$$ γ > 0 . Nevertheless, by means of perturbation type techniques, we establish the existence of a positive solution $$u_{\varepsilon,\gamma}$$ u ε , γ concentrating, as $$\varepsilon\rightarrow 0$$ ε → 0 , around minima points of the potential.

A class of asymptotically periodic fractional Schrödinger equations with critical growth

2018 ◽  
Vol 20 (03) ◽  
pp. 1750011
Author(s):  
Manassés de Souza ◽  
Yane Lísley Araújo
Keyword(s):  
Mountain Pass Theorem ◽  
Critical Sobolev Exponent ◽  
Critical Growth ◽  
Mountain Pass ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Positive Potential ◽  
Fractional Schrödinger Equations ◽  
Asymptotically Periodic ◽  
Sobolev Exponent

In this paper, we study a class of fractional Schrödinger equations in [Formula: see text] of the form [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] is the critical Sobolev exponent, [Formula: see text] is a positive potential bounded away from zero, and the nonlinearity [Formula: see text] behaves like [Formula: see text] at infinity for some [Formula: see text], and does not satisfy the usual Ambrosetti–Rabinowitz condition. We also assume that the potential [Formula: see text] and the nonlinearity [Formula: see text] are asymptotically periodic at infinity. We prove the existence of at least one solution [Formula: see text] by combining a version of the mountain-pass theorem and a result due to Lions for critical growth.

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