On a fractional magnetic Schrödinger equation in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" id="d1e19" altimg="si1.gif"><mml:mi mathvariant="double-struck">R</mml:mi></mml:math>with exponential critical growth

Ambrosio, Vincenzo

doi:10.1016/j.na.2019.01.016

Cited by 27 publications

(12 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Nonlinear problems with exponential growth have been considered recently by Alves and de Freitas [1], Alves and Santos [2], Ambrosio [3], Figueiredo and Severo [12], Li, Santos and Yang [23], Medeiros, Severo and Silva [24], etc.…”

Section: Introductionmentioning

confidence: 99%

Positive solutions of the prescribed mean curvature equation with exponential critical growth

Figueiredo

Rădulescu

2021

Annali di Matematica

View full text Add to dashboard Cite

In this paper, we are concerned with the problem $$\begin{aligned} -\text{ div } \left( \displaystyle \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) = f(u) \ \text{ in } \ \Omega , \ \ u=0 \ \text{ on } \ \ \partial \Omega , \end{aligned}$$ - div ∇ u 1 + | ∇ u | 2 = f ( u ) in Ω , u = 0 on ∂ Ω , where $$\Omega \subset {\mathbb {R}}^{2}$$ Ω ⊂ R 2 is a bounded smooth domain and $$f:{\mathbb {R}}\rightarrow {\mathbb {R}}$$ f : R → R is a superlinear continuous function with critical exponential growth. We first make a truncation on the prescribed mean curvature operator and obtain an auxiliary problem. Next, we show the existence of positive solutions of this auxiliary problem by using the Nehari manifold method. Finally, we conclude that the solution of the auxiliary problem is a solution of the original problem by using the Moser iteration method and Stampacchia’s estimates.

show abstract

Section: Introductionmentioning

confidence: 99%

Positive solutions of the prescribed mean curvature equation with exponential critical growth

Figueiredo

Rădulescu

2021

Annali di Matematica

View full text Add to dashboard Cite

show abstract

“…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 [4], the authors obtain the existence and the multiplicity of solutions for a nonlinear fractional magnetic Schrödinger equation by using variational methods and Ljusternick-Schnirelmann theory. In [5], the authors obtain the existence and the multiplicity of solutions for a nonlinear fractional magnetic Schrödinger equation with exponential critical growth. And in [6], the author obtains the existence of nontrivial solutions for a class of fractional magnetic Schrödinger equations via penalization techniques.…”

Section: Introductionmentioning

confidence: 99%

Infinitely many high energy solutions for fractional Schrödinger equations with magnetic field

Yang

Zuo

2019

Bound Value Probl

View full text Add to dashboard Cite

In this paper we investigate the existence of infinitely many solutions for nonlocal Schrödinger equation involving a magnetic potential continuous function, and (-) sA is the fractional magnetic operator. Under suitable assumptions for the potential function V and nonlinearity f , we obtain the existence of infinitely many nontrivial high energy solutions by using the variant fountain theorem. MSC: 35R11; 35A15; 35J60; 47G20

show abstract

On a fractional magnetic Schrödinger equation inRwith exponential critical growth

Cited by 27 publications

References 46 publications

Positive solutions of the prescribed mean curvature equation with exponential critical growth

Positive solutions of the prescribed mean curvature equation with exponential critical growth

Existence and nonexistence of solutions for the fractional nonlinear Schrödinger equation

Infinitely many high energy solutions for fractional Schrödinger equations with magnetic field

Contact Info

Product

Resources

About