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We show that each point of the principal eigencurve of the nonlinear problem
$$-{{\Delta }_{p}}u-\text{ }\lambda m(x){{\left| u \right|}^{p-2}}u=\mu {{\left| u \right|}^{p-2}}u\,\,\text{in}\Omega ,$$
is stable (continuous) with respect to the exponent 
$p$ varying in 
$\left( 1,\infty \right)$; we also prove some convergence results of the principal eigenfunctions corresponding.
