Nonlinear oscillations of completely resonant wave equations
We present recent existence results of small amplitude periodic and quasi-periodic solutions of completely resonant nonlinear wave equations. Both infinite-dimensional bifurcation phenomena and small divisors difficulties occur. The proofs rely on bifurcation theory, Nash-Moser implicit function theorems, dynamical systems techniques and variational methods.
Asymptotically quadratic functions defined on Hilbert spaces are studied by using some results of the theory of Morse-Conley index. Applications are given to existence of nontrivial weak solutions for asymptotically linear elliptic partial and ordinary differential equations at resonances.
The existence of a nontrivial critical point is proved for a functional containing an area-type term. Techniques of nonsmooth critical point theory are applied.
Let X be a finite CW complex, and ρ: π 1(X) → GL(l, ℂ) a representation. Any cohomology class α ∈ H 1(X, ℂ) gives rise to a deformation γ t of ρ defined by γ t (g) = ρ(g) exp(t〈α, g〉). We show that the cohomology of X with local coefficients γ gen corresponding to the generic point of the curve γ is computable from a spectral sequence starting from H*(X, ρ). We compute the differentials of the spectral sequence in terms of the Massey products and show that the spectral sequence degenerates in case...
We study the boundary value problem in , on , where is a smooth bounded domain in . Our attention is focused on two cases when , where for any or for any . In the former case we show the existence of infinitely many weak solutions for any . In the latter we prove that if is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a -symmetric version for even functionals...
Let be a bounded starshaped domain and consider the -Laplacian problem where is a positive parameter, , and is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the -Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.
In this paper we mainly introduce a min-max procedure to prove the existence of positive solutions for certain semilinear elliptic equations in RN.