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Hausdorff measures and the Morse-Sard theorem.

Carlos Gustavo T. de A. Moreira (2001)

Publicacions Matemàtiques

Let F : U ⊂ Rn → Rm be a differentiable function and p < m an integer. If k ≥ 1 is an integer, α ∈ [0, 1] and F ∈ Ck+(α), if we set Cp(F) = {x ∈ U | rank(Df(x)) ≤ p} then the Hausdorff measure of dimension (p + (n-p)/(k+α)) of F(Cp(F)) is zero.

Infinitely many solutions of a second-order p -Laplacian problem with impulsive condition

Libo Wang, Weigao Ge, Minghe Pei (2010)

Applications of Mathematics

Using the critical point theory and the method of lower and upper solutions, we present a new approach to obtain the existence of solutions to a p -Laplacian impulsive problem. As applications, we get unbounded sequences of solutions and sequences of arbitrarily small positive solutions of the p -Laplacian impulsive problem.

Infinitely many weak solutions for a non-homogeneous Neumann problem in Orlicz--Sobolev spaces

Ghasem A. Afrouzi, Shaeid Shokooh, Nguyen T. Chung (2019)

Commentationes Mathematicae Universitatis Carolinae

Under a suitable oscillatory behavior either at infinity or at zero of the nonlinear term, the existence of infinitely many weak solutions for a non-homogeneous Neumann problem, in an appropriate Orlicz--Sobolev setting, is proved. The technical approach is based on variational methods.

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