The Ground State Solutions for Kirchhoff-Schrödinger Type Equations with Singular Exponential Nonlinearities in ℝ ^N

Abstract

In this paper, the authors consider the following singular Kirchhoff-Schrödinger problem$M\left( {\int_{{\mathbb{R}^N}} {{{\left| {\nabla u} \right|}^N} + V\left( x \right){{\left| u \right|}^N}{\rm{d}}x} } \right)\left( { - {\Delta _N}u + V\left( x \right){{\left| u \right|}^{N - 2}}u} \right) = {{f\left( {x,u} \right)} \over {{{\left| x \right|}^\eta }}}\,\,\,\,\,{\rm{in}}\,\,{\mathbb{R}^N},\,\,\,\,\,\,\,\left( {{P_\eta }} \right)$

where 0 < η < N, M is a Kirchhoff-type function and V(x) is a continuous function with positive lower bound, f(x, t) has a critical exponential growth behavior at infinity. Combining variational techniques with some estimates, they get the existence of ground state solution for (P _η)- Moreover, they also get the same result without the A-R condition.

Keywords

Ground state solutions / Singular elliptic equations / Critical exponential growth / Kirchhoff-Schrödinger equations / Singular Trudinger-Moser inequality

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