Energy Estimates and Existence of Non-trivial Solutions for Robin Problems Involving $p$-Laplacian Operator

Abstract

This paper studies the existence of non-trivial solutions and
energy estimates for a nonlinear elliptic problem driven by the
$p$-Laplacian under Robin boundary conditions, which model various
physical phenomena such as heat transfer and fluid flow with
boundary interactions. Using a recent local minimum theorem, we
establish existence results under suitable growth and
Ambrosetti-Rabinowitz (AR) conditions. We identify intervals of
the parameter $\lambda$ where solutions exist and extend the
result to all $\lambda > 0$ under $(p - 1)$-sublinear growth at
zero and infinity. An illustrative example is also provided.