A result on the bifurcation from the principal eigenvalue of the $A_p$-Laplacian

Abstract

We study the following bifurcation problem in any bounded domain $\Omega$ in ${ℝ}^N$: $$\{\begin{array}{l}{A}_{p}u:=-\sum _{i,j=1}^N\frac{\partial }{\partial x_i}\left[{\left(\sum _{m,k=1}^N{a}_{mk}\left(x\right)\frac{\partial u}{\partial x_m}\frac{\partial u}{\partial x_k}\right)}^{\frac{p-2}{2}}{a}_{ij}\left(x\right)\frac{\partial u}{\partial x_j}\right]=\\ \lambda g\left(x\right){\left|u\right|}^{p-2}u+f\left(x,u,\lambda \right),\\ \\ u\in W_0^{1,p}\left(\Omega \right).\end{array}$$. We prove that the principal eigenvalue ${\lambda }_1$ of the eigenvalue problem $$\{\begin{array}{l}{A}_{p}u=\lambda g\left(x\right){\left|u\right|}^{p-2}u,\\ u\in W_0^{1,p}\left(\Omega \right),\end{array}$$ is a bifurcation point of the problem mentioned above.