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Variational Methods Examine SolutionMultiplicity in Local Equations, Indicating New Findings in Genus Theory.
Using 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 (Formula presented) where Ω (Formula presented) (Formula presented) is a bounded domain, µ, is a positive real parameter, p, r and s are continuous real functions on (Formula presented) and a(x,(Formula presented)) is of type |(Formula presented)|p(x)—2. Next, we study boundedness and simplicity of eigenfunction for the case a(x, |(Formula presented)u|)(Formula presented)u = g(x)|(Formula presented)u|p(x)-2(Formula presented)u, where g (Formula presented) L∞(Q) and g(x) > 0 and the case a(x, |(Formula presented)u|)(Formula presented)u = (1 + (Formula presented)u|2)(Formula presented) (Formula presented)u such that p(x) = p.
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Tunç et al. (2023) studied this question.
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