Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition

Abstract

In this paper, we study coupled elliptic systems with gradient dependent right-hand sides and nonlinear boundary conditions, where the left-hand sides are driven by so-called double phase operators. By applying a surjectivity result for pseudomonotone operators along with an equivalent norm in the function space, we prove that the system has at least one nontrivial solution under very general assumptions on the data. Under slightly stronger conditions, we are also able to show that this solution is unique. As a result of independent interest, we further prove the boundedness of solutions to such elliptic systems by employing Moser’s iteration scheme.

Keywords

• boundedness results
• convection term
• coupled elliptic systems
• equivalent norm
• gradient dependence
• nonlinear boundary condition
• pseudomonotone operators
• robin and steklov eigenvalues
• 35j20
• 35j25
• 35j47
• 35j57