scholarly journals Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Guofa Li ◽  
Yisheng Huang
Keyword(s):  
Positive Solutions ◽  
Critical Case ◽  
Existence Of Solutions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Subcritical Case ◽  
Existence Of Positive Solutions

<p style='text-indent:20px;'>In this paper, we study the existence of positive solutions for the following quasilinear Schrödinger equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} -\triangle u+V(x)u+\frac{\kappa}{2}[\triangle|u|^{2}]u = \lambda K(x)h(u)+\mu|u|^{2^*-2}u, \quad x\in\mathbb{R}^{N}, \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \kappa&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \lambda&gt;0, \mu&gt;0, h\in C(\mathbb{R}, \mathbb{R}) $\end{document}</tex-math></inline-formula> is superlinear at infinity, the potentials <inline-formula><tex-math id="M3">\begin{document}$ V(x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ K(x) $\end{document}</tex-math></inline-formula> are vanishing at infinity. In order to discuss the existence of solutions we apply minimax techniques together with careful <inline-formula><tex-math id="M5">\begin{document}$ L^{\infty} $\end{document}</tex-math></inline-formula>-estimates. For the subcritical case (<inline-formula><tex-math id="M6">\begin{document}$ \mu = 0 $\end{document}</tex-math></inline-formula>) we can deal with large <inline-formula><tex-math id="M7">\begin{document}$ \kappa&gt;0 $\end{document}</tex-math></inline-formula>. For the critical case we treat that <inline-formula><tex-math id="M8">\begin{document}$ \kappa&gt;0 $\end{document}</tex-math></inline-formula> is small.</p>

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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