On a nonhomogeneous Kirchhoff type elliptic system with the singular Trudinger-Moser growth

The aim of this paper is to study the multiplicity of solutions for the following Kirchhoff type elliptic systems    −m (∑k j=1 ‖uj‖ ) ∆ui = fi(x,u1,...,uk) |x|β + εhi(x), in Ω, i = 1, . . . , k, u1 = u2 = · · · = uk = 0, on ∂Ω, where Ω is a bounded domain in R2 containing the origin with smooth boundary, β ∈ [0, 2), m is a Kirchhoff type function, ‖uj‖ = ∫ Ω |∇uj |2dx, fi behaves like eβs 2 when |s| → ∞ for some β > 0, and there is C1 function F : Ω × Rk → R such that ( ∂F ∂u1 , . . . , ∂F ∂uk ) = (f1, . . . , fk), hi ∈ (( H1 0 (Ω) )∗ , ‖ · ‖∗ ) . We establish sufficient conditions for the multiplicity of solutions of the above system by using variational methods with a suitable singular Trudinger-Moser inequality when ε > 0 is small.

[1]  F. Albuquerque,et al.  An Elliptic Equation Involving Exponential Critical Growth in ℝ2 , 2015 .

[2]  Giovany M. Figueiredo,et al.  Ground State Solution for a Kirchhoff Problem with Exponential Critical Growth , 2013, 1305.2571.

[3]  Adimurthi,et al.  Phenomena of critical exponent in ℝ2 , 1991, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[4]  Adimurthi,et al.  An Interpolation of Hardy Inequality and Trudinger–Moser Inequality in ℝN and Its Applications , 2009 .

[5]  M. Souza.  On a singular class of elliptic systems involving critical growth in R2 , 2011 .

[6]  S. Superiore,et al.  Multiplicity results for semilinear elliptic equations in a bounded domain of R2 involving critical exponents , 2019 .

[7]  Paulo Rabelo.  Elliptic systems involving critical growth in dimension two , 2009 .

[8]  Adimurthi,et al.  A singular Moser-Trudinger embedding and its applications , 2007 .

[9]  Elliott H. Lieb,et al.  A Relation Between Pointwise Convergence of Functions and Convergence of Functionals , 1983 .

[10]  Xiaoming He,et al.  Multiplicity of solutions for a class of Kirchhoff type problems , 2010 .

[11]  S. Superiore.  Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-laplacian , 2007 .

[12]  U. Severo,et al.  A nonhomogeneous elliptic problem involving critical growth in dimension two , 2008 .

[13]  Xiaoming He,et al.  Infinitely many positive solutions for Kirchhoff-type problems , 2009 .

[14]  B. Ruf,et al.  On an inequality by N. Trudinger and J. Moser and related elliptic equations , 2002 .

[15]  D. G. Figueiredo,et al.  Semilinear Elliptic Systems With Exponential Nonlinearities in Two Dimensions , 2006 .

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

[17]  Ó. JoãoMarcosdo,et al.  On a quasilinear nonhomogeneous elliptic equation with critical growth in R N , 2009 .

[18]  Wenjing Chen,et al.  On a nonhomogeneous Kirchhoff-type elliptic problem with critical exponential in dimension two , 2020, Applicable Analysis.

[19]  To Fu Ma,et al.  Positive solutions for a quasilinear elliptic equation of Kirchhoff type , 2005 .

[20]  Y. Yang.  Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space , 2011, 1106.4622.

[21]  Bitao Cheng,et al.  New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems , 2012 .

[22]  Bernhard Ruf,et al.  Elliptic Equations in R2 with Nonlinearities in the Critical Growth Range , 1995 .

[23]  Wenjing Chen.  Existence of solutions for fractional p-Kirchhoff type equations with a generalized Choquard nonlinearities , 2018, 1808.07996.

[24]  Neil S. Trudinger,et al.  On Imbeddings into Orlicz Spaces and Some Applications , 1967 .

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

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