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Information × Registration Number 0226U002330, (0125U001647) , R & D reports Title Development of modern methods for studying the behavior of solutions to elliptic equations with a complex mathematical structure popup.stage_title Розробка сучасних методів дослідження поведінки розв’язків еліптичних рівнянь складної математичної структури Head Savchenko Mariia O., Кандидат фізико-математичних наук Registration Date 23-02-2026 Organization Institute of Applied Mathematics and Mechanics National Academy of Science of Ukraine popup.description1 Development of new theoretical approaches and the adaptation of modern methods to study the qualitative properties of solutions to elliptic equations with nonstandard growth conditions. popup.description2 The object of the study is non-uniformly elliptic equations with non-standard growth conditions and non-logarithmic continuity assumptions on the coefficients. The aim of the work is to develop new theoretical approaches and to adapt modern analytical methods for investigating qualitative properties of solutions to elliptic equations with non-standard growth. The research methods include adapted positivity expansion techniques, sharp pointwise estimates of nonlinear potential type, De Giorgi’s method, and energy estimate techniques. The results are of a fundamental nature and consist in adapting modern nonlinear analysis methods – specifically, the positivity expansion method, De Giorgi’s technique, and a modified approach to sharp pointwise estimates based on nonlinear potentials – to the study of elliptic equations with non-uniform ellipticity and spatially dependent nonlinearities. Key local properties of non-negative weak solutions have been established. In particular, weak and classical Harnack inequalities have been proved, together with a local clustering lemma. The problem of removable isolated singularities for elliptic equations with spatially dependent nonlinearities has been investigated. The nonlinearity is described via a variable exponent p(x) or a coefficient a(x), assumed to be bounded and to satisfy the log-Hölder continuity condition. Sufficient conditions for removability have been obtained; in the case of a constant exponent p(x)=p, these conditions agree with classical criteria known in the literature. Product Description popup.authors popup.nrat_date 2026-02-23 Close