An eigenvalue problem for generalized Laplacian in Orlicz—Sobolev spaces

Source abstract

Let m : [ 0, ∞) → [ 0, ∞) be an increasing continuous function with m ( t ) = 0 if and only if t = 0, m ( t ) → ∞ as t → ∞ and Ω C ℝ N a bounded domain. In this note we show that for every r > 0 there exists a function u r solving the minimization problem where Moreover, the function u r is a weak solution to the corresponding Euler–Lagrange equation for some λ > 0. We emphasize that no Δ 2 -condition is needed for M or M ; so the associated functionals are not continuously differentiable, in general.