Existence of strong solutions to semilinear elliptic systems of second order

This paper investigates the existence of strong solutions to the first boundary value problem for semilinear elliptic systems of second order with summable right-hand sides. The analysis is based on an interpolation method originally developed by Pohozhaev and Laptev for elliptic and parabolic equations, which allows one to derive refined a priori estimates in Sobolev spaces. We establish sufficient conditions guaranteeing the boundedness of solutions in W²ₚ(Ω), expressed in terms of summability indices and growth restrictions on the nonlinear terms. A particular emphasis is placed on identifying the sharpness of the growth rate conditions, and we provide explicit counterexamples showing the unimprovability of the obtained bounds. Furthermore, the solvability of corresponding Dirichlet problems is established using the Leray–Schauder continuation principle. The results are then extended to semilinear elliptic systems, where compact embeddings of anisotropic Sobolev spaces and multiplicative inequalities play a crucial role. This work contributes to the broader theory of nonlinear elliptic equations and systems by clarifying the exact balance between coercivity conditions, summability properties, and nonlinear growth, ensuring the existence of strong solutions under minimal assumptions.