Existence of Solutions to Quasilinear Schrödinger Equations Involving Critical Sobolev Exponent

Wang, Youjun; Li, Zhouxin

doi:10.11650/tjm/8150

Cited by 14 publications

(2 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Obviously, I k has a good definition in D, and we can obtain the following lemma that are similar to [23].…”

Section: Preliminary Workmentioning

confidence: 88%

Nontrivial Solutions for a Class of Quasilinear Schrödinger Systems

Zhang,

Zhang

2024

Axioms

View full text Add to dashboard Cite

In this thesis, we research quasilinear Schrödinger system as follows in which 3<N∈R, 2<p<N, 2<q<N, V1(x),V2(x) are continuous functions, k,ι are parameters with k,ι>0, and nonlinear terms f,h∈C(RN×R2,R). We find a nontrivial solution (u,v) for all ι>ι1(k) by means of the mountain-pass theorem and change of variable theorem. Our main novelty of the thesis is that we extend Δ to Δp and Δq to find the existence of a nontrivial solution.

show abstract

“…Obviously, I k has a good definition in D, and we can obtain the following lemma that are similar to [23].…”

Section: Preliminary Workmentioning

confidence: 88%

Nontrivial Solutions for a Class of Quasilinear Schrödinger Systems

Zhang,

Zhang

2024

Axioms

View full text Add to dashboard Cite

show abstract

“…When κ = 0, the problem ( 1) is a class of semilinear Schrödinger equations which have been extensively studied by using variational methods in the past thirty years, see [4,14,18,26] and references therein. When κ > 0 is small enough, there are also many papers studied existence results of nontrivial solutions for (1) via variational methods, see [1,3,8,20] for λ = 1, µ = 0 with K(x) = 1 and [19] with K(x) = 0, [22] and [23] for K(x) = 1 and µ = 1 with small λ > 0 and λ = 1, respectively. Moreover, [25] obtained least energy nodal solution for (1) when κ > 0 is small enough, λ = 1, K(x) = 1 and µ > 0 is small.…”

Section: Introduction Consider the Following Quasilinear Schrödinger Equationsmentioning

confidence: 99%

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

Huang

2022

DCDS-B

View full text Add to dashboard Cite

In this paper, we study the existence of positive solutions for the following quasilinear Schrödinger equations<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*} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ \kappa>0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \lambda>0, \mu>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>0 $\end{document}</tex-math></inline-formula>. For the critical case we treat that <inline-formula><tex-math id="M8">\begin{document}$ \kappa>0 $\end{document}</tex-math></inline-formula> is small.

show abstract

Positive Solutions for a Class of Quasilinear Schrödinger Equations with Two Parameters

Chen

Huang

et al. 2019

Bull. Malays. Math. Sci. Soc.

View full text Add to dashboard Cite

Existence of Solutions to Quasilinear Schrödinger Equations Involving Critical Sobolev Exponent

Cited by 14 publications

References 25 publications

Nontrivial Solutions for a Class of Quasilinear Schrödinger Systems

Nontrivial Solutions for a Class of Quasilinear Schrödinger Systems

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

Positive Solutions for a Class of Quasilinear Schrödinger Equations with Two Parameters

Contact Info

Product

Resources

About