Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension

We study the existence of ground states to a nonlinear fractional Kirchhoff equation with an external potential V. Under suitable assumptions on V, using the monotonicity trick and the profile decomposition, we prove the existence of ground states. In particular, the nonlinearity does not satisfy the Ambrosetti–Rabinowitz type condition or monotonicity assumptions.

[1]  Luis Silvestre,et al.  Holder estimates for solutions of integra-differential equations like the fractional laplace , 2006 .

[2]  W. Zou,et al.  A BERESTYCKI-LIONS THEOREM REVISITED , 2012 .

[3]  Rupert L. Frank,et al.  Uniqueness and Nondegeneracy of Ground States for (−Δ)^sQ+Q−Q^(α+1)=0 in R , 2015 .

[4]  Pierre-Louis Lions,et al.  Nonlinear scalar field equations, I existence of a ground state , 1983 .

[5]  Giuseppina Autuori,et al.  Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity , 2014, 1410.6762.

[6]  N. Nyamoradi.  Existence of three solutions for Kirchhoff nonlocal operators of elliptic type , 2013 .

[7]  Enrico Valdinoci,et al.  A critical Kirchhoff type problem involving a nonlocal operator , 2013 .

[8]  L. Caffarelli,et al.  An Extension Problem Related to the Fractional Laplacian , 2006, math/0608640.

[9]  Enrico Valdinoci,et al.  Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian , 2012, 1202.0576.

[10]  Shangjiang Guo,et al.  Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent , 2014, Zeitschrift für angewandte Mathematik und Physik.

[11]  Mingqi Xiang,et al.  Infinitely many solutions for a fractional Kirchhoff type problem via Fountain Theorem , 2015 .

[12]  Xiaojun Chang,et al.  Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity , 2013 .

[13]  Enrico Valdinoci,et al.  The Brezis-Nirenberg result for the fractional Laplacian , 2014 .

[14]  Xiaoming He,et al.  Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3 , 2012 .

[15]  Louis Jeanjean,et al.  On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝN , 1999, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[16]  Antonio Azzollini,et al.  The elliptic Kirchhoff equation in $\mathbb R^N$ perturbed by a local nonlinearity , 2012, Differential and Integral Equations.

[17]  Kanishka Perera,et al.  Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow , 2006 .

[18]  Gongbao Li,et al.  Existence of positive solutions for nonlinear Kirchhoff type problems in R3 with critical Sobolev exponent , 2014 .

[19]  W. Rother,et al.  Nonlinear scalar field equations , 1992, Differential and Integral Equations.

[20]  S. Secchi.  Ground state solutions for nonlinear fractional Schrödinger equations in RN , 2012, 1208.2545.

[21]  Patrizia Pucci,et al.  Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setl , 2015, Calculus of Variations and Partial Differential Equations.

[22]  A. Quaas,et al.  Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian , 2012, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[23]  Jean Van Schaftingen.  Symmetrization and minimax principles , 2005 .

[24]  Shangjiang Guo,et al.  Existence of positive ground state solutions for Kirchhoff type problems , 2015 .

[25]  Rupert L. Frank,et al.  Uniqueness of non-linear ground states for fractional Laplacians in $${\mathbb{R}}$$R , 2013 .

[26]  Jaime E. Muñoz Rivera,et al.  Positive solutions for a nonlinear nonlocal elliptic transmission problem , 2003, Appl. Math. Lett..

[27]  Jacques-Louis Lions,et al.  On Some Questions in Boundary Value Problems of Mathematical Physics , 1978 .

[28]  Hongyu Ye,et al.  Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3 , 2014 .

[29]  Rupert L. Frank,et al.  Uniqueness of non-linear ground states for fractional Laplacians in $${\mathbb{R}}$$R , 2010, 1009.4042.

[30]  Michel Chipot,et al.  Some remarks on non local elliptic and parabolic problems , 1997 .

[31]  Kanishka Perera,et al.  Nontrivial solutions of Kirchhoff-type problems via the Yang index , 2006 .

[32]  K. Teng.  Existence of ground state solutions for the nonlinear fractional Schrödinger–Poisson system with critical Sobolev exponent , 2016 .

[33]  Patrizia Pucci,et al.  Critical stationary Kirchhoff equations in RN involving nonlocal operators , 2016 .

[34]  Jian Zhang,et al.  The critical case for a Berestycki-Lions theorem , 2014 .

[35]  V. Ambrosio,et al.  A multiplicity result for a fractional Kirchhoff equation in ℝN with a general nonlinearity , 2016, Communications in Contemporary Mathematics.

[36]  P. Pucci,et al.  Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations , 2016 .

[37]  Haroldo V. Ribeiro,et al.  Fractional Schrödinger equation with noninteger dimensions , 2012, Appl. Math. Comput..

[38]  Haim Brezis,et al.  Positive Solutions of Nonlinear Elliptic Equations Involving Critical Sobolev Exponents Haim Brezis , 2022 .

[39]  Xian Wu,et al.  Existence and multiplicity of positive solutions for fractional Schrödinger equations with critical growth , 2017 .

[40]  M. Squassina.  On the Struwe—Jeanjean—Toland monotonicity trick , 2010, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[41]  P. Pucci,et al.  p-fractional Kirchhoff equations involving critical nonlinearities , 2017 .

[42]  Xian Wu,et al.  Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in RN☆ , 2011 .

[43]  P. Pucci,et al.  Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations , 2016 .