On principal eigenvalues of \(p\)-Laplacian-like operators
The authors consider both homogeneous and the nonhomogeneous boundary-value problems of the form \[ [r^{N-1}\phi(u')]'+\lambda r^{N-1}\psi(u)=0,\quad r\in(0,R),\quad u'(0)=0,\quad u(R)=0 \] and \[ [r^{N-1}\phi(u')]'+\lambda r^{N-1}\psi(u)=r^{N-1}h(r),\quad r\in(0,R),\quad u'(0)=0,\quad u(R)=0, \] where \(N\geq 1\) (not necessarily an integer), \(\phi\) is an increasing homeomorphism of \(R\) and \(\psi\) is nondecreasing with \(\phi(0)=\psi(0)=0\) while \(h\in L^\infty(0,R)\) is a given function, and discuss the existence of the values of \(\lambda\) for which these problems have solutions. Main results concerning the homogeneous problem claim that there exists a smallest \(\lambda_0>0\) such that for \(\lambda<\lambda_0\) this problem has no nontrivial solutions. As to the nonhomogeneous problem, they show that there exists a \(\lambda_0>0\) such that for every \(\lambda<\lambda_0\) the problem has a solution. The proofs are based on fixed point and continuation techniques.
homogeneous and nonhomogeneous boundary-value problems
scientific article; zbMATH DE number 936697