Abstract This paper studies the existence of solutions for Robin problems involving p ( x )-Laplacian-like operators which arise from capillarity phenomena. When we only consider the convective term, due to the lack of a variational structure, the well-known variational methods are not applicable. Using Galerkin method together with Brouwer’s fixed point theorem, we obtain the existence of finite-dimensional approximate solution and generalized solution. On the other hand, utilizing local linking theorem without Ambrosetti-Rabinowitz ((A-R) for short) condition, we obtain the existence of a nontrivial solution under some conditions. The main difficulties and innovations of the present article are that we consider the convective term, the weaker assumptions on the nonlinear term, and p ( x )-Laplacian-like operators with Robin boundary condition.
Weichun Bu (Wed,) studied this question.