<p>We prove the existence of weak solutions for a nonlinear double phase Schrödinger–Kirchhoff problem with Robin boundary conditions. The leading part contains a nonlocal Kirchhoff coefficient depending on the double phase energy. The lower-order nonlinearity is treated as a compact perturbation, using the compact Sobolev embedding below the critical exponent. The proof is based on a precise decomposition of the operator, compact Sobolev embeddings, and the Browder–Minty theorem for pseudomonotone operators. A uniqueness criterion is also given under a separate smallness condition.</p>

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A PSEUDOMONOTONE APPROACH TO A DOUBLE PHASE SCHRÖDINGER–KIRCHHOFF PROBLEM WITH ROBIN BOUNDARY CONDITIONS

  • El-mahdi Nafia,
  • Zahra Mansouri,
  • M’hamed El Omari

摘要

We prove the existence of weak solutions for a nonlinear double phase Schrödinger–Kirchhoff problem with Robin boundary conditions. The leading part contains a nonlocal Kirchhoff coefficient depending on the double phase energy. The lower-order nonlinearity is treated as a compact perturbation, using the compact Sobolev embedding below the critical exponent. The proof is based on a precise decomposition of the operator, compact Sobolev embeddings, and the Browder–Minty theorem for pseudomonotone operators. A uniqueness criterion is also given under a separate smallness condition.