Bifurcation for nonlinear eigenvalue $p$-Lapalcian problems involving $L^{q}$-norm

Abdelouahed El Khalil, My Driss Morchid Alaoui

Abstract


In this paper, we study a global bifurcation phenomenon associated with the non-linear problem
$$
-\Delta_{p}u= \lambda \|u\|_{q}^{p-q} |u|^{q-2}u + f(x, u, \lambda)\;\; \mbox{in}\;\; \Omega, \leqno({E_f})
$$
wehre, the unknown $u\in W_0^{1,p}(\Omega)$. Under some natural hypotheses on the nonlinear perturbation $f$, we prove that $(\lambda_{1}, 0 )$
is a global bifurcation point of the above problem, where $\lambda_{1}$ stands the first eigenvalue of $(E_{\{f\equiv 0\}})$.


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DOI: https://doi.org/10.52846/ami.v52i1.1977