Existence and multiplicity of weak solutions for a nonlocal Robin system with variable exponents

This paper studies a nonlocal system of elliptic equations involving variable-exponent p j ( x ) -Laplacian operators subject to Robin boundary conditions. Each of the n equations contains a nonlocal source term of the form λ f j ( x , u j ) ⎛ ∫ D F j ( x , u j ) d x ⎞ h j , where D ⊂ R N is a bounded smooth domain, λ > 0 is a parameter, and each p j is a continuous variable exponent satisfying 1 < p j − ≤ p j ( x ) ≤ p j + < N . Working in the framework of variable-exponent Sobolev spaces, we first establish the existence of at least one nontrivial weak solution. We then apply a three-critical-points theorem of Bonanno–Marano type to prove that the energy functional has at least three critical points, hence that the system has at least two nontrivial weak solutions, for a suitable range of the parameter λ . A key step is the verification of the Palais–Smale condition, obtained under a structural growth condition linking the exponents β j , the nonlocal exponents h j and the lower bounds p j − ; this condition guarantees subcritical growth and compactness of the associated energy functional. An illustrative example set in the unit ball of R 3 confirms the applicability of the abstract results.