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We consider a Kirchhoff problem of Brezis-Nirenberg type in a smooth bounded domain of $\mathbb{R}^4$ with Dirichlet boundary conditions. Our approach, novel in this framework and based upon approximation arguments, allows us to cope with the interaction between the higher order Kirchhoff term and the critical nonlinearity, typical of the dimension four. We derive several existence results of positive solutions, complementing and improving earlier results in the literature. In particular, we provide explicit bounds of the parameters $b$ and $\lambda$ coupled, respectively, with the higher order Kirchhoff term and the subcritical nonlinearity, for which the existence of solutions occurs.

Wireless SAW reflective delay line (R-DL) technology is very powerful to carry out remote measurements of various parameters under harsh environments, while enabling the identification of a given sensor among several of them. However, R-DL technology is currently limited to 350$^\circ$C for long-term applications, likely because of aluminium electrodes oxidation and/or congruent lithium niobate segregation process. In this study, an innovative alloy, namely NiAl, is investigated as an alternative to Al to make R-DLs able to withstand high temperatures up to 500$^\circ$C on the long term. Indeed, NiAl gathers, in the bulk state, all the necessary properties (fairly low electrical resistivity and density, high melting temperature and resistance to oxidation). The study also examines the extent of the congruent LiNbO$_3$ segregation process, to determine its impact on the NiAl/LiNbO$_3$ R-DLs performances. The obtained results are very promising. NiAl electrodes self-passivate during the first 50h of annealing at 500$^\circ$C: 20 nm-thick Al$_2$O$_3$ layers form at the surface and in between the electrodes and the substrate, protecting the remaining NiAl layer from further oxidation. Besides, the segregation process occurs mainly in the same time. It is located in the first 150-200 nm of the substrate. Both phenomena have no significant impact on the performance of NiAl/LiNbO$_3$ R-DLs working at 433 MHz. Continuous in-situ electrical monitoring of such devices shows a standard deviation of the operating frequency of only 1.04 ppm during an annealing process of 250h at 500$^\circ$C. Moreover, the time-resolved S11 response of the device at the end of this treatment is not degraded at all. Thus, 433 MHz NiAl/LiNbO$_3$ R-DL sensors can operate with high fidelity for at least 10 days at 500$^\circ$C under air atmosphere, and there are strong signs that their lifetime is actually much longer.

In this paper, by variational and topological arguments based on linking and $\nabla$-theorems, we prove the existence of multiple solutions for the following nonlocal problem with mixed Dirichlet-Neumann boundary data, $$ \left\{ \begin{array}{lcl} (-\Delta)^su=\lambda u+f(x,u) & &\text{in } \Omega, \\[2pt] \mkern+39mu u=0& &\text{on } \Sigma_{\mathcal{D}}, \\[2pt] \mkern+26mu \displaystyle \frac{\partial u}{\partial \nu}=0& &\text{on } \Sigma_{\mathcal{N}}, \end{array} \right. $$ where $(-\Delta)^s$, $s\in (1/2,1)$, is the spectral fractional Laplacian operator, $\Omega\subset\mathbb{R}^N$, $N>2s$, is a smooth bounded domain, $\lambda>0$ is a real parameter, $\nu$ is the outward normal to $\partial\Omega$, $\Sigma_{\mathcal{D}}$, $\Sigma_{\mathcal{N}}$ are smooth $(N-1)$-dimensional submanifolds of $\partial\Omega$ such that $\Sigma_{\mathcal{D}}\cup\Sigma_{\mathcal{N}}=\partial\Omega$, $\Sigma_{\mathcal{D}}\cap\Sigma_{\mathcal{N}}=\emptyset$ and $\Sigma_{\mathcal{D}}\cap\overline{\Sigma}_{\mathcal{N}}=\Gamma$ is a smooth $(N-2)$-dimensional submanifold of $\partial\Omega$.

By using variational techniques we provide new existence results for Yamabe-type equations with subcritical perturbations set on a compact $d$-dimensional ($d\geq 3$) Riemannian manifold without boundary. As a direct consequence of our main theorems, we prove the existence of at least one solution to the following singular Yamabe-type problem $$ \left\lbrace \begin{array}{ll} -\Delta_g w + \alpha(\sigma)w = \mu K(\sigma) w^\frac{d+2}{d-2} +\lambda \left( w^{r-1} + f(w)\right), \quad \sigma\in\mathcal{M} &\\ &\\ w\in H^2_\alpha(\mathcal{M}), \quad w>0 \ \ \mbox{in} \ \ \mathcal{M} & \end{array} \right.$$ where, as usual, $\Delta_g$ denotes the Laplace-Beltrami operator on $(\mathcal{M},g)$, $\alpha, K:\mathcal{M}\to\mathbb{R}$ are positive (essentially) bounded functions, $r\in(0,1)$, and $f:[0,+\infty)\to[0,+\infty)$ is a subcritical continuous function. Restricting ourselves to the unit sphere ${\mathbb{S}}^d$ via the stereographic projection, we also solve some parametrized Emden-Fowler equations in the Euclidean case.

In this paper we examine the existence of multiple solutions of parametric fractional equations involving the square root of the Laplacian $A_{1/2}$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$ ($n\geq 2$) and with Dirichlet zero-boundary conditions, i.e. \begin{equation*} \left\{ \begin{array}{ll} A_{1/2}u=\lambda f(u) & \mbox{ in } \Omega\\ u=0 & \mbox{ on } \partial\Omega. \end{array}\right. \end{equation*} The existence of at least three $L^{\infty}$-bounded weak solutions is established for certain values of the parameter $\lambda$ requiring that the nonlinear term $f$ is continuous and with a suitable growth. Our approach is based on variational arguments and a variant of Caffarelli-Silvestre's extension method.

We are concerned with existence results for a critical problem of Brezis-Nirenberg Type involving an integro-differential operator. Our study includes the fractional Laplacian. Our approach still applies when adding small singular terms. It hinges on appropriate choices of parameters in the mountain-pass theorem

A solution of the old problem raised by Tolman, Ehrenfest, Podolsky and Wheeler, concerning the lack of attraction of two light pencils "moving parallel", is proposed, considering that the light can be source of nonlinear gravitational waves corresponding (in the would be quantum theory of gravity) to spin-1 massless particles., Comment: Style is changed in standard latex, abstract has been reduced and the order of sections has been changed