Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation

Shang, Xudong; Zhang, Jihui

doi:10.3934/cpaa.2018107

Cited by 2 publications

(2 citation statements)

References 15 publications

(29 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In recent years, many results regarding concentration phenomena for Equation () and its generalizations, under the assumption that

\inf_{ℝ^{N}} V > 0

, have arisen; see, for instance, previous studies 6–18 . In particular, in, Long et al and Shang and Zhang, 15,16,18 multipeak solutions were studied by overlapping single peaks that are sufficiently far away from one another so that one peak has no effect on the other peaks in the areas where decay occurs, avoiding interactions between peaks.…”

Section: Introductionmentioning

confidence: 99%

Existence and nonexistence of solutions for the fractional nonlinear Schrödinger equation

Alarcón

Ritorto

Silva

2021

Math Methods in App Sciences

View full text Add to dashboard Cite

We consider the equation ε2s(−Δ)su+V(x)u−f(u)=0inℝN, where s ∈ (0, 1), p∈()1,N+2sN−2s, N > 2s, ffalse(ufalse)=false|ufalse|p−1u, V∈L∞false(ℝNfalse) is such that infℝNV>0 and ε > 0 is small. We study the existence and nonexistence of solutions concentrating at a local minimum point of V as ε → 0 without using any symmetry assumption on V. First, we prove that certain type of positive solutions exhibiting peaks does not exist. Then, we study the existence of sign‐changing solutions under a suitable configuration of positive and negative peaks. To guarantee the existence, we cannot neglect the interaction between peaks. In particular, by using a minimization argument, we found solutions exhibiting peaks at the vertices of a 2ℓ‐regular polygon, such that two adjacent peaks have alternate sign.

show abstract

“…In recent years, many results regarding concentration phenomena for Equation () and its generalizations, under the assumption that

\inf_{ℝ^{N}} V > 0

Section: Introductionmentioning

confidence: 99%

Existence and nonexistence of solutions for the fractional nonlinear Schrödinger equation

Alarcón

Ritorto

Silva

2021

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…Notice that h u L 2 (R N ) = u L 2 (R N ) for all h ∈ R. Now we recall the notion of barycentre of a function u ∈ H s (R N ) \ {0} which has been introduced in [3,9,31]. Setting…”

mentioning

confidence: 99%

Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential

Peng¹,

Xia²

2021

CPAA

View full text Add to dashboard Cite

We are concerned with the following nonlinear fractional Schrödinger equation:<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\begin{equation} (-\Delta)^s u+V(x)u+\omega u = |u|^{p-2}u\quad {\rm{in}}\,\,{\mathbb{R}}^N,\;\;\;\;\;\;({\textbf{P}})\end{equation}$ \end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ s\in(0,1) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ p\in\left(2+4s/N,2^*_s\right) $\end{document}</tex-math></inline-formula>, that is, the mass supercritical and Sobolev subcritical. Under certain assumptions on the potential <inline-formula><tex-math id="M3">\begin{document}$ V:{\mathbb{R}}^N\rightarrow {\mathbb{R}} $\end{document}</tex-math></inline-formula>, positive and vanishing at infinity including potentials with singularities (which is important for physical reasons), we prove that there exists at least one <inline-formula><tex-math id="M4">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-normalized solution <inline-formula><tex-math id="M5">\begin{document}$ (u,\omega)\in H^s({\mathbb{R}}^N)\times{\mathbb{R}}^+ $\end{document}</tex-math></inline-formula> of equation (P). In order to overcome the lack of compactness, the proof is based on a new min-max argument and splitting lemma for nonlocal version.

show abstract

Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation

Cited by 2 publications

References 15 publications

Existence and nonexistence of solutions for the fractional nonlinear Schrödinger equation

Existence and nonexistence of solutions for the fractional nonlinear Schrödinger equation

Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential

Contact Info

Product

Resources

About