Alimohammady, MohsenRezvani, AsiehTunc, Cemil2025-05-102025-05-1020231225-17632234-302410.4134/CKMS.c2203082-s2.0-85176330287https://doi.org/10.4134/CKMS.c220308https://hdl.handle.net/20.500.14720/967Using variational methods, Krasnoselskii's genus theory and symmetric mountain pass theorem, we introduce the existence and multiplicity of solutions of a parameteric local equation. At first, we consider the following equation( -div[a(x, | backward difference u|) backward difference u] = mu(b(x)|u|s(x)-2 - |u|r(x)-2)u in S2, u = 0 on partial differential S2, where S2 subset of RN is a bounded domain, mu is a positive real parameter, p, r and s are continuous real functions on S2 over bar and a(x, xi) is of type |xi|p(x)-2. Next, we study boundedness and simplicity of eigenfunction for the case a(x, | backward difference u|) backward difference u = g(x)| backward difference u|p(x)-2 backward difference u, where g is an element of L infinity(S2) and g(x) >= 0 and the case a(x, | backward difference u|) backward difference u = (1 + backward difference u|2) p(x)-2 2 backward difference u such that p(x) equivalent to p.eninfo:eu-repo/semantics/closedAccessP(X)-LaplacianModular FunctionGenus TheoryArticle384N/AQ410451061WOS:001127420800012