Concentration phenomena in nonlinear eigenvalue problems with variable exponents and sign-changing potential
This paper deals with nonlinear eigenvalue problems of the type \(Au=\mu Bu\), generalizing problems with the usual \(p\)-Laplacian for \(p> 1\) constant. Here \(A\) (resp. \(B\)) is a differential operator( resp. a zero order term ) which is a sum of two \(p\)-Laplacians with different \(p\)'s ( resp. two zero order terms with different powers). Now \(p(x)> 0\) is a non-constant function. These problems have been studied in the last years in the frame of variable exponent Lebesgue spaces. Moreover, the operator A may include a zero order term where an indefinite changing sign weight \(V(x)\) arises. Under suitable assumptions it is proved that for given \(V\), an eigenvalue \(E(V)\) exists such that any \(\mu> E(V)\) is an eigenvalue and any \(\mu< E(V)\) is not an eigenvalue. It is also proved that \(E(V)\) can be minimized on closed, convex, bounded subsets in a suitable space. Since the problem has not a variational character, most of the usual arguments should be modified or replaced by different ones.
scientific article; zbMATH DE number 5864342
scientific article