Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities

Abstract

Abstract We are concerned with the following Kirchhoff-type equation with exponential critical nonlinearities − a + b ∫ R 2 ∣ ∇ u ∣ 2 d x Δ u + ( h ( x ) + μ V ( x ) ) u = K ( x ) f ( u ) in R 2 , -\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{2}}| \nabla u{| }^{2}{\rm{d}}x\right)\Delta u+\left(h\left(x)+\mu V\left(x))u=K\left(x)f\left(u)\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{2}, where a , b , μ > 0 a,b,\mu \gt 0 , the potential V V has a bounded set of zero points and decays at infinity as ∣ x ∣ − γ | x{| }^{-\gamma } with γ ∈ ( 0 , 2 ) \gamma \in \left(0,2) , the weight K K has finite singular points and may have exponential growth at infinity. By using the truncation technique and working in some weighted Sobolev space, we obtain the existence of a mountain pass solution for μ > 0 \mu \gt 0 large and the concentration behavior of solutions as μ → + ∞ \mu \to +\infty .