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

Abstract

We are concerned with the following Kirchhoff-type equation with exponential critical nonlinearities −a+b∫R2∣∇u∣2dxΔu+(h(x)+μV(x))u=K(x)f(u)inR2,-\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>0a,b,\mu \gt 0, the potential VV 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 KK 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 .

Keywords

• kirchhoff-type equation
• steep potential well
• vanishing potential
• trudinger-moser inequality
• variational method
• 35a15
• 35j60