Gluing approximate solutions of minimum type on the Nehari manifold.
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.
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 -Laplacian impulsive problem. As applications, we get unbounded sequences of solutions and sequences of arbitrarily small positive solutions of the -Laplacian impulsive problem.
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.