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Solving Strongly Sublinear p(x)-Laplacian Dirichlet Problems: Existence via Fixed-Point Theorems
Resolución de problemas de Dirichlet para p(x)-Laplacianos fuertemente sublineales: existencia utilizando teoremas de punto fijo
DOI:
https://doi.org/10.15446/recolma.v59n2.125950Palabras clave:
Strongly nonlinear elliptic problem, p(x)-Laplacian, generalized Lebesgue-Sobolev spaces, fixed point (en)Problemas elípticos fuertemente no lineales, p(x)-Laplaciano, espacios generalizados de Lebesgue-Sobolev, teoremas de punto fijo (es)
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We study the existence of weak solutions to the nonlinear elliptic problem −Δp(x) u = λ|u|s(x)−2 u + f(x, u,∇u) in a bounded domain Ω ⊂ RN with smooth boundary ∂Ω, under homogeneous Dirichlet conditions. The equation features a variable exponent p(x) in the p(x)-Laplacian operator, an eigenvalue term λ|u|s(x)−2 u, and a Carathéodory perturbation f with sublinear growth, depending on both u and ∇u. Using a topological approach based on fixed-point theorems, we establish the existence of weak solutions to the problem.
Estudiamos la existencia de soluciones débiles para el problema elíptico no lineal −Δp(x) u = λ|u|s(x)−2 u + f(x, u,∇u) en un dominio acotado Ω ⊂ RN con frontera suave ∂Ω, bajo condiciones de Dirichlet homogéneas. La
ecuación involucra una variable exponencial p(x) en el operador Laplaciano, un término λ|u|s(x)−2 u, y una perturbación de Carath´eodory f con crecimiento sublineal, que depende tanto de u como de ∇u. Usando un método topológico de puntos fijos, probamos la existencia de soluciones débiles al problema.
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