Multiple positive solutions for semilinear Schrödinger equations with critical growth in ℝN

2015 ◽  
Vol 56 (4) ◽  
pp. 041503 ◽  
Author(s):  
Jun Wang ◽  
Dianchen Lu ◽  
Junxiang Xu ◽  
Fubao Zhang
Keyword(s):  
Positive Solutions ◽  
Critical Growth ◽  
Schrodinger Equations ◽  
Multiple Positive Solutions ◽  
Schrödinger Equations

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.

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