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We consider parametric equations driven by the sum of a ▫$p$▫-Laplacian and a Laplace operator (the so-called ▫$(p, 2)$▫-equations). We study the existence and multiplicity of solutions when the parameter ▫$lambda > 0$▫ is near the principal eigenvalue ▫$hat{lambda}_1(p) > 0$▫ of ▫$(-Delta_p,W^{1-p}_0(Omega))$▫. We prove multiplicity results with precise sign information when the near resonance occurs from above and from below of ▫$hat{lambda}_1(p) > 0$▫.

We prove a uniqueness result for Nevanlinna functions. and this result is then used to give an elementary proof of the uniqueness in the inverse scattering problem for the equation $ u" + \frac{k^2}{c^2}u=0 $ on $\mathbb R$. Here $c$ is a real positive measurable function that is bounded from below by a positive constant, and is close to $1$ at $\pm \infty$., 22 pages