### Fractional Kirchhoff-Schrödinger equation with critical exponential growth in $\mathbb{R}^{N}$

Yanjun Liu, Lifeng Yin

DOI: http://dx.doi.org/10.12775/TMNA.2020.030

#### Abstract

In this paper, we consider the following fractional Kirchhoff-Schrödinger equation: \begin{equation*} \begin{cases} \displaystyle \Big(a+b\|u\|_{E}^{p(\theta-1)}\Big) \big[(-\triangle)_{p}^{s}u+V(x)|u|^{p-2}u\big]=f(x, u), & x \in \mathbb{R}^{N},\\ \noalign{\vskip5pt} \displaystyle \|u\|_{E}^p:=\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy +\int_{\mathbb{R}^{N}}V(x)|u|^{p}dx, \end{cases} \end{equation*} where $a> 0$, $b \geq 0$, $\theta \geq1$, dimension $N=sp$ with $s \in (0, 1)$ and $p\geq 1$. $V$ is a positive potential and $f$ is a critical nonlinearity with exponential growth. We derive a positive ground state solution by using minimax techniques combined with the fractional Truding-Moser inequality. Moreover, in the particular case of $a=1$ and $b=0$, we also obtain the existence of the ground state solution to the fractional Schrödinger equation.

#### Keywords

Fractional Kirchhoff-Schrödinger; critical exponential growth; fractional Trudinger-Moser inequality; the ground state solution

#### Full Text:

PREVIEW FULL TEXT

#### References

D. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. 128 (1988), 385–398.

D. Applebaum, Lévy processes from probability to finance quantum groups, Not. Amer. Math. Soc. 51 (2004), 1336–1347.

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.

L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symp. 7 (2012), 37–52.

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245–1260.

D.M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in R2 , Comm. Partial Differential Equations 17 (1992), 407–435.

X. Chang and Z.Q. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity 26 (2013), 479–494.

M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys. 53 (2012), 043507.

S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche (Catania) 68 (2013), 201–216.

D.G. de Figueiredo, O.H. Miyagaki and B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equtions 3 (1995), 139–153.

M. de Souza and Y.L. Araújo, On nonlinear perturbations of a periodic fractional Schrödinger equation with exponential critical growth, Math. Nachr. 289 (2016), 610–625.

M. de Souza and Y. L. Araújo, Semilinear elliptic equations for the fractional Laplacian involving critical exponential growth, Math. Methods Appl. Sci. 40 (2017), no. 5, 1757–1772.

J.M. do Ó, N -Laplacian equations in RN with critical growth, Abstr. Appl. Anal. 2 (1997), 301–315.

J.M. do Ó, O.H. Miyagaki and M. Squassina, Nonautonomous fractional problems with exponential growth, Nonlinear Differential Equations Appl. 22 (2015), 1395–1410.

J.M. do Ó, O.H. Miyagaki and M. Squassina, Ground states of nonlocal scalar field equations with Trudinger–Moser critical nonlinearity, Topol. Methods in Nonlinear Anal. 48 (2016), no. 2, 477–492.

P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), 1237–1262.

B. Feng, Ground states for the fractional Schrödinger equation, Electron. J. Differential Equations 127 (2013), 1–11.

J. Giacomoni, P. Mishra and K. Sreenadh, Fractional elliptic equations with critical exponential nonlinearity, Adv. Nonlinear Anal. 5 (2016), 57–74.

A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var. 9 (2016), 101–125.

A. Iannizzotto and M. Squassina, 1/2-Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl. 414 (2014), 372–385.

S. Iula, A note on the Moser–Trudinger inequality in Sobolev–Slobodeckij spaces in dimension one, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 (2017), no. 4, 871–884.

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000), 298–305.

Q. Li and Z. Yang, Multiple solutions for N -Kirchhoff type problems with critical exponential growth in RN , Nonlinear Anal. 117 (2015), 159–168.

J.L. Lions, On some questions in boundary value problems of mathematical physics, Contemporary Developments in Continuum Mechanics and Partial Differential Equations (P.G.M. De La, L.A.J. Medeiros, eds.), vol. 30, North-Holland Mathematics Studies, 1978, pp. 284–346.

P.L. Lions, The concentration-compactness principle in the calculus of variations, Part I, Rev. Mat. Iberoamericana 1 (1985), 145–201.

L. Martinazzi, Fractional Adams–Moser–Trudinger type inequalities, Nonlinear Anal. 127 (2015), 263–278.

S. Mosconi and M. Squassina, Recent progresses in the theory of nonlinear nonlocal problems, Bruno Pini Mathematical Analysis Seminar 2016, Bruno Pini Math. Anal. Semin. 2016, University Bologna, Bologna, 2016, pp. 147–164.

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970), 1077–1092.

E. Parini and B. Ruf, On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces, J. Anal. Math. 138 (2019), 281–300.

K. Perera and M. Squassina, Bifurcation results for problems with fractional Trudinger–Moser nonlinearity, Discrete Contin. Dyn. Syst. Ser. S 11 (2018), no. 3, 561–576.

P. Pucci, M. Xiang and B. Zhang, Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in RN , Calc. Var. 54 (2015), 2785–2806.

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in RN , J. Math. Phys. 54 (2013), 031501.

S. Secchi, On fractional Schrödinger equations in RN without Ambrosetti–Rabinowitz condition, Topol. Methods Nonlinear Anal. 47 (2016), no. 1, 19–41.

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2006), 67–112.

N.S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–484.

M. Xiang, V. Rădulescu, B. Zhang, Fractional Kirchhoff problems with critical Trudinger–Moser nonlinearity, Calc. Var. 57 (2019), 1–27.

J. Xiao and Z. Zhai, Fractional Sobolev, Moser–Trudinger, Morrey–Sobolev inequalities under Lorentz norms, J. Math. Sci. 166 (2010), 357–376.

M. Xiang, B. Zhang and X. Zhang, A nonhomogeneous fractional p-Kirchhoff type problem involving critical exponent in RN , Adv. Nonlinear Stud. 17 (2017), 611–640.

C. Zhang, Trudinger–Moser inequalities in fractional Sobolev–Slobodeckij spaces and multiplicity of weak solutions to the fractional-Laplacian equation, Adv. Nonlinear Stud. 19 (2019), 197–217.

### Refbacks

• There are currently no refbacks.