Splitting the Fučík spectrum and the number of solutions to a quasilinear ODE (Q2914233)

The authors study the two-point boundary value problem NEWLINE\[NEWLINE(\Phi(u'))'+f(u)=0,\quad u(0)=u(L)=0,NEWLINE\]NEWLINE where \(\Phi\) is an increasing homeomorphism of \({\mathbb R}\), \(\Phi(0)=0\) and \(f\) is continuous in \({\mathbb R}\), \(f(0)=0\), \(xf(x)>0\) \(\forall x\neq0\) and \(\int_0^{\pm\infty}f=\infty\).NEWLINENEWLINEBecause the equation is autonomous, there is a certain conservation of energy that allows a reduction to first order. Assuming additional conditions on \(\Phi\) that ensure a convenient behaviour of the time map, the procedure goes as follows. A positive pseudo-Fučík spectrum (PPFS) is defined, to account for solutions that start at \(t=0\) with \(u(0)=0\) and positive slope, consisting in countably many curves in the first quadrant of the \((\mu,\nu)\)-plane, related to problem (1) when \(f\) is replaced by \(\mu\Phi(u^+)-\nu\Phi(u^-)\). The complement of these curves is the union of a sequence of regions \(Z^+_i\), \(i\in{\mathbb N}\). More precisely, there are two kinds of such sets of curves and regions so that we refer to \(Z^+_i(\infty)\) and \(Z^+_i(0)\) according to whether one is interested in ``big'' or ``small'' solutions, respectively.NEWLINENEWLINEA typical result of the paper is then the following one.NEWLINENEWLINESuppose that \(\frac{f(s)}{\Phi(s)}\) has limits \(a^+\), \(a^-\), \(A^+\), \(A^-\) as \(s\) tends to \(0^+\), \(0^-\), \(+\infty\) or \(-\infty\), respectively. Then, if \((a^+,a^-)\in Z^+_k(0)\) and \((A^+,A^-)\in Z^+_l(0)\), the problem \((1)\) has at least \(|k-l|\) solutions with positive slope at \(0\).NEWLINENEWLINEA similar construction and result is given for solutions with negative slope at \(0\).NEWLINENEWLINEThe article contains also examples, and useful properties of time-maps.