We deal with the boundary value problem $$\begin{array}{ccccccc}\hfill -\Delta u\left(x\right)& ={\lambda }_{1}u\left(x\right)+g\left(\nabla u\left(x\right)\right)+h\left(x\right),\hfill & & x\in \Omega u\left(x\right)\hfill & \hfill =0,& & \hfill x\in \partial \Omega \end{array}$$ where $\Omega \subset {ℝ}^N$ is an smooth bounded domain, ${\lambda }_1$ is the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions on $\Omega$, $h\in L^{max\{2,N/2\}}\left(\Omega \right)$ and $g:{ℝ}^N⟶ℝ$ is bounded and continuous. Bifurcation theory is used as the right framework to show the existence of solution provided that $g$ satisfies certain conditions on the origin and at infinity.

The aim of this paper is to study the existence of variational solutions to a nonhomogeneous elliptic equation involving the $N$-Laplacian $$-{\Delta }_{N}u\equiv -{div\left(\right|\nabla u|}^{N-2}\nabla u)=e(x,u)+h\left(x\right)\text{in}\Omega$$ where $u\in W_0^{1,N}\left({ℝ}^N\right)$, $\Omega$ is a bounded smooth domain in ${ℝ}^N$, $N\ge 2$, $e(x,u)$ is a critical nonlinearity in the sense of the Trudinger-Moser inequality and $h\left(x\right)\in {\left(W_0^{1,N}\right)}^{*}$ is a small perturbation.

We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrödinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of $h$, and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we...