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A result on the bifurcation from the principal eigenvalue of the \(Ap\)-Laplacian. (English) Zbl 0933.35151

Summary: We study the following bifurcation problem in any bounded domain \(\Omega\) in \(\mathbb{R}^N\): \[ \begin{aligned} A_pu:= & -\sum^N_{i,j=1} {\partial\over \partial x_i}\left [\left(\sum^N_{m,k=1} a_{mk}(x){\partial u\over\partial x_m}{\partial u\over \partial x_k}\right)^{p-2\over 2}a_{ij} (x){\partial u\over\partial x_j} \right]=\\ & \lambda g(x)|u|^{p-2} u+f(x,u,\lambda), \quad u\in W_0^{1,p} (\Omega). \end{aligned} \] We prove that the principal eigenvalue \(\lambda_1\) of the eigenvalue problem \[ A_pu=\lambda g(x)|u|^{p-2}u, \quad u\in W_0^{1,p} (\Omega), \] is a bifurcation point of the problem mentioned above.

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B32 Bifurcations in context of PDEs
35J70 Degenerate elliptic equations
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