Some remarks on critical point theory for nondifferentiable functionals
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In this paper we study the existence of critical points of nondifferentiable functionals J of the kind \(J(v) =\int_\Omega A(x,v) |\nabla v|^2 - F(x,v)\) with A (x,z) a Carathéodory function bounded between positive constant and with bounded derivative respect to the variable z, and F (x,z) is the primitive of a (Carathéodory) nonlinearity f (x,z) satisfying suitable hipotheses. Since J is just differentible along bounded directions, a suitable compactness condition is introduced. Its connection with coercivity is discussed. In addition, the case of concave-convex nonlinearities f (x,z), unbounded coefficients A (x,z) and related problems are also studied.
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