Existence and concentration of solutions to Kirchhoff-type equations in \(\mathbb{R}^2\) with steep potential well vanishing at infinity and exponential critical nonlinearities. (English) Zbl 1519.35163
Summary: We are concerned with the following Kirchhoff-type equation with exponential critical nonlinearities
\[
-\left(a+b \int\limits_{\mathbb{R}^2} | \nabla u|^2 \mathrm{d} x\right)\Delta u+(h(x)+\mu V(x))u=K(x)f(u) \text{ in } \mathbb{R}^2,
\]
where \(a,b,\mu > 0\), the potential \(V\) has a bounded set of zero points and decays at infinity as \(| x |^{-\gamma}\) with \(\gamma \in (0,2)\), the weight \(K\) 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 \(\mu > 0\) large and the concentration behavior of solutions as \(\mu \to +\infty\).
MSC:
| 35J62 | Quasilinear elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A15 | Variational methods applied to PDEs |
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