We are concerned with a nonlinear elliptic equation, involving a Kirchhoff type nonlocal term and a potential $V(x)$, on $\mathbb{R}^3$. As is well known that, even in $H^1_r(\mathbb{R}^3)$, the nonlinear term is a pure power form of $|u|^{p-1}u$ and $V(x)\equiv 1$, it seems very difficult to apply the mountain-pass theorem to get a solution (i.e., mountain-pass solution) to this kind of equation for all $p\in(1,5)$, due to the difficulty of verifying the boundedness of the Palais-Smale sequence obtained by the mountain-pass theorem when $p\in(1,3)$. In this paper, we find a new strategy to overcome this difficulty, and then get a mountain-pass solution to the equation for all $p\in(1,5)$ and for both $V(x)$ being constant and nonconstant. Also, we find a possibly optimal condition on $V(x)$.
Lifu WENG
,
Xu ZHANG
,
Huansong ZHOU
. MOUNTAIN-PASS SOLUTION FOR A KIRCHHOFF TYPE ELLIPTIC EQUATION[J]. Acta mathematica scientia, Series B, 2025
, 45(2)
: 385
-400
.
DOI: 10.1007/s10473-025-0207-6
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