scholarly journals Positive solutions of quasilinear Schrödinger equations with critical growth

2015 ◽  
Vol 45 ◽  
pp. 52-57 ◽  
Author(s):  
Ke Wu
Keyword(s):  
Positive Solutions ◽  
Critical Growth ◽  
Schrodinger Equations ◽  
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.

scholarly journals Infinitely many positive solutions of nonlinear Schrödinger equations

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
Riccardo Molle ◽  
Donato Passaseo
Keyword(s):  
Positive Solutions ◽  
Radial Symmetry ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

AbstractThe paper deals with the equation $$-\Delta u+a(x) u =|u|^{p-1}u $$ - Δ u + a ( x ) u = | u | p - 1 u , $$u \in H^1({\mathbb {R}}^N)$$ u ∈ H 1 ( R N ) , with $$N\ge 2$$ N ≥ 2 , $$p> 1,\ p< {N+2\over N-2}$$ p > 1 , p < N + 2 N - 2 if $$N\ge 3$$ N ≥ 3 , $$a\in L^{N/2}_{loc}({\mathbb {R}}^N)$$ a ∈ L loc N / 2 ( R N ) , $$\inf a> 0$$ inf a > 0 , $$\lim _{|x| \rightarrow \infty } a(x)= a_\infty $$ lim | x | → ∞ a ( x ) = a ∞ . Assuming that the potential a(x) satisfies $$\lim _{|x| \rightarrow \infty }[a(x)-a_\infty ] e^{\eta |x|}= \infty \ \ \forall \eta > 0$$ lim | x | → ∞ [ a ( x ) - a ∞ ] e η | x | = ∞ ∀ η > 0 , $$ \lim _{\rho \rightarrow \infty } \sup \left\{ a(\rho \theta _1) - a(\rho \theta _2) \ :\ \theta _1, \theta _2 \in {\mathbb {R}}^N,\ |\theta _1|= |\theta _2|=1 \right\} e^{\tilde{\eta }\rho } = 0 \quad \text{ for } \text{ some } \ \tilde{\eta }> 0$$ lim ρ → ∞ sup a ( ρ θ 1 ) - a ( ρ θ 2 ) : θ 1 , θ 2 ∈ R N , | θ 1 | = | θ 2 | = 1 e η ~ ρ = 0 for some η ~ > 0 and other technical conditions, but not requiring any symmetry, the existence of infinitely many positive multi-bump solutions is proved. This result considerably improves those of previous papers because we do not require that a(x) has radial symmetry, or that $$N=2$$ N = 2 , or that $$|a(x)-a_\infty |$$ | a ( x ) - a ∞ | is uniformly small in $${\mathbb {R}}^N$$ R N , etc. ....

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