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We study the first eigenvalue of the $p$-Laplacian under the Dirichlet boundary condition. For a convex domain, we give an a priori estimate for the first eigenvalue in terms of the radius $d$ of the maximum ball contained in the domain. As a consequence, we prove that the first eigenvalue diverges to infinity as $p\to\infty$ if the domain is convex and $d\leq 1$. Moreover, we show that in the annulus domain $a < |x| < b$, the first eigenvalue diverges to infinity if $b-a\leq 2$ and converges to zero if $b-a>2$.

In this paper, we study one-dimensional $p$-Laplace equation, whose weight function has a critical power at the origin. We show the existence of infinitely many bifurcation branches emanating from the same single point and also study the existence and multiplicity of sign-changing solutions. Moreover, we investigate the global shape of bifurcation branches.