Bifurcation diagrams of a \(p\)-Laplacian Dirichlet problem with Allee effect and an application to a diffusive logistic equation with predation
The authors study bifurcation diagrams of positive solutions for the \(p\)-Laplacian Dirichlet problem \[ (\varphi_p(u'(x)))'+\lambda f(u)=0,\quad u(-1)=u(1)=0, \] where \(p>1\), \(\varphi_p(y) = |y|^{p-2} y,\) \(\lambda> 0\) is a bifurcation parameter, \(f(y)=|y|^{p-1}g(y)\), and \(g\) is of Allee effect type. Assuming a suitable condition on \(g\), they prove that, on the (\(\lambda\), \(\|u\|_\infty\))-plane, the bifurcation diagram consists of exactly one continuous curve with exactly one turning point where the curve turns to the right. Hence, the problem has at most two positive solutions for each \(\lambda>0\). More precisely, they prove the exact multiplicity of positive solutions. Also, they give an application to a \(p\)-Laplacian diffusive logistic equation with predation of Holling type II functional response.
scientific article; zbMATH DE number 5835299
scientific article