Existence of eigenvalues for reflected Tricomi operators and applications to multiplicity of solutions for sublinear and asymptotically linear nonlocal Tricomi problems

Abstract

The dual variational method is applied to establish results on the multiplicity of solutions for nonlocal semilinear Tricomi problems of the type introduced in [9]. For classes of odd sublinear nonlinearities, the existence of infinitely many solutions as preimages of critical points of the dual functional at minimax values is established with the aid of the Krasnoselski genus. For classes of asymptotically linear nonlinearities the existence of at least one nontrivial solution is captured as the preimage of a mountain pass or a linking critical point of the dual functional. In all the cases, information on the eigenvalues of the inverse of the linear operator is needed and demonstrated. An analogous problem involving a second order ordinary differential operator and with homogeneous Cauchy conditions on one endpoint of an interval is resolved in the same way and the role of the nonlocal effect is studied in terms of the question of the uniqueness of the trivial solution.