We consider the existence of a non-trivial weak solution for the equation
$$ \left\{ \begin{array}{@{}ll}
-\Delta_p u= f(x,u) & \text{in}\enskip\ \Omega\,, \\
u=0 & \text{on}\enskip\ \partial\Omega\,,
\end{array}\right. $$
where $f$ satisfies $f(x,u)=a u_+^{p-1} -bu_-^{p-1} + o(|u|^{p-1})$ ($p>1$) at 0 or $\infty$.
By using Morse theory and calculating the critical groups, we show the existence of a non-trivial weak solution to the equation under mild auxiliary conditions.
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