Bifurcation theory and related problems: anti-maximum principle and resonance. (English) Zbl 1086.35010

Arcoya, David; Gámez, José L.

Commun. Partial Differ. Equations 26, No. 9-10, 1879-1911 (2001).

Summary: For a smooth bounded domain \(\Omega\subset\mathbb R^N\), we consider the boundary value problem \[ \begin{gathered} -\Delta u=\lambda m(x)u+g(\lambda,x,u)\quad\text{if }x\in\Omega,\\ u(x)=0\quad\text{if }x\in\partial\Omega,\end{gathered} \] where \(m\in L^r(\Omega)\) for some \(r\in(\max\{1,N/2\},+\infty]\), with \(m^+\not\equiv 0\), and \(g\) is a Carathéodory function. We deduce sharp sufficient conditions to have subcritical (‘to the left’) or supercritical (‘to the right’) bifurcations (either from zero or from infinity) at an eigenvalue \(\lambda_k(m)\) of the associated linear weighted eigenvalue problem. Furthermore, as a consequence, we also point out the bifurcation nature of some classical results such as the (local) antimaximum principle of Clement and Peletier and the Landesman-Lazer theorem for resonant problems. In addition, we see that the bifurcation viewpoint also makes it possible to obtain a local maximum principle and more general results for some classes of strongly resonant problems. In addition, we extend the above technique to handle quasilinear boundary value problems.

MSC:

35B32	Bifurcations in context of PDEs
35J65	Nonlinear boundary value problems for linear elliptic equations
35J60	Nonlinear elliptic equations
35B50	Maximum principles in context of PDEs
35J25	Boundary value problems for second-order elliptic equations
47J15	Abstract bifurcation theory involving nonlinear operators
58E07	Variational problems in abstract bifurcation theory in infinite-dimensional spaces