Sturm-Liouville type problems for the \(p\)-Laplacian under asymptotic non-resonance conditions
An analogue of Sturm's oscillation theorem is proved for the eigenfunctions of the boundary problem which in the regular case looks as follows \[ L_p^{\alpha}u+(q(r)+\lambda s(r))u^{(p-1)}=0, \quad x\in [a_1,a_2], \quad 0\leq a_1<a_2<\infty, \] \[ \gamma_{1j}u^{(p-1)}(a_j)+\gamma_{2j}\left(r^{\alpha}u^{\prime (p-1)}\right) (a_j)=0, \quad j=1,2, \tag{1} \] with \(L_p^{\alpha}u=r^{-\alpha}\left(r^{\alpha}u^{\prime (p-1)}\right)'\) , \(\gamma_{ij}\in\mathbb{R}\) (\(i=1,2; j=1,2\)), \(\gamma_{1j}^2+\gamma_{2j}^2>0\) and \(u^{(q)}=|u|^{q-1}u\) is the odd power function. Also the two-parameter problem is considered generated by the following equation \[ L_p^{\alpha}u+q(r)u^{(p-1)}+s(r)\left(\mu(u^{+})^{p-1}-\nu(u^{-})^{p-1} \right)=0 \tag{2} \] with boundary conditions (1). Here, \(u^{+}(x)=\max(u(x),0)\) and \(u=u^{+}-u^{-}\). The pair \((\mu,\nu)\) is called a Fučik-eigenvalue and \(u_{\mu,\nu} \not\equiv 0\) a Fučik-eigenfunction if \(\{\mu,\nu,u_{\mu,\nu}\}\) solves (2), (1). Using the Prüfer-transformation the authors obtained a complete global description of the Fučik-spectrum for (2), (1). This enables them to prove the existence of a solution to the corresponding nonlinear problem.
Sturm-Liouville type boundary conditions
scientific article; zbMATH DE number 1339358