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 $-{\mathrm{d}iv\left(\right(\left|\nabla u\right|}^{p_1\left(x\right)-2}+{\left|\nabla u\right|}^{p_2\left(x\right)-2}\left)\nabla u\right)=f(x,u)$ in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega$ is a smooth bounded domain in ${ℝ}^N$. Our attention is focused on two cases when $f(x,u)=\pm (-\lambda |u{|}^{m\left(x\right)-2}u+\left|u{|}^{q\left(x\right)-2}u\right)$, where $m\left(x\right)=max\{p_1\left(x\right),p_2\left(x\right)\}$ for any $x\in \overline{\Omega }$ or $m\left(x\right)<q\left(x\right)<\frac{N·m\left(x\right)}{(N-m(x\left)\right)}$ for any $x\in \overline{\Omega }$. In the former case we show the existence of infinitely many weak solutions for any $\lambda >0$. In the latter we prove that if $\lambda$ 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 ${\mathbb{Z}}_2$-symmetric version for even functionals...

Let $\Omega \subset {ℝ}^n$ be a bounded starshaped domain and consider the $(p,q)$-Laplacian problem $$\begin{array}{c}\hfill -{\Delta }_{p}u-{\Delta }_{q}u=\lambda \left(𝐱\right){\left|u\right|}^{p^{☆}-2}u+\mu {\left|u\right|}^{r-2}u\end{array}$$ where $\mu$ is a positive parameter, $1<q\le p<n$, $r\ge p^{☆}$ and $p^{☆}:=\frac{np}{n-p}$ is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the $(p,q)$-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.