Existence results for a perturbed Dirichlet problem without sign condition in Orlicz spaces

H.  Moussa; M.  Rhoudaf; H.  Sabiki

doi:10.37863/umzh.v72i4.373

H. Moussa Univ. Sultan Moulay Slimane, Morocco https://orcid.org/0000-0002-4830-2900
M. Rhoudaf Univ. Ibn Tofail, Morocco
H. Sabiki Univ. Moulay Ismayil, Morocco https://orcid.org/0000-0001-7668-390X

DOI: https://doi.org/10.37863/umzh.v72i4.373

Анотація

УДК 517.5

Розглядається задача існування для нелінійних еліптичних рівнянь у формі
$$
Au + g(x, u,\nabla u) = f,
$$
де $-{\rm div}\Big(a(x,u,\nabla u)\Big)$ — оператор Лере–Ліонса з підмножини $W^{1}_{0}L_M(\Omega)$ у її дуальну множину. Умови зростання та коерцитивності в монотонному векторному полі $a$ визначаються $N$-функцією $M,$ яка не повинна задовольняти $\Delta_2$-умови.
Тому ми використовуємо простори Орліча–Соболєва, які не обов'язково є рефлексивними, і припускаємо, що нелінійність $g(x,u,\nabla u)$ є функцією Каратеодорі, що задовольняє лише умову зростання без умови знака. Права частина $f$ належить $W^{-1}E_{\overline{M}}(\Omega).$

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