Liouville Results for <Emphasis Type="Italic">m</Emphasis>-Laplace Equations in Half-Space and Strips with Mixed Boundary Value Conditions and Finite Morse Index

We consider sign-changing solutions of the equation \(-\Delta _m u= |u|^{p-1}u\) where \(m\ge 2\) and \(p>1\) in half-space and strips with nonlinear mixed boundary value conditions. We prove Liouville type theorems for stable solutions or for solutions which are stable outside a compact set. The main methods used are the integral estimates, the Pohozaev-type identity and the monotonicity formula.     

A priori estimates for superlinear and subcritical elliptic equations: the Neumann boundary condition case

We consider here solutions of a nonlinear Neumann elliptic equation Δu +?f (x, u) =?0 in Ω, ?u/?ν =?0 on ?Ω, where Ω is a bounded open smooth domain in ${\mathbb{R}^N, N\geq2}$ and f satisfies super-linear and subcritical growth conditions. We prove that L ^∞?bounds on solutions are equivalent to bounds on their Morse indices.     

On the Palais–Smale condition

We present a new sufficient assumption weaker than the classical Ambrosetti–Rabinowitz condition which guarantees the boundedness of (PS) sequences. Moreover, we relax the standard subcritical polynomial growth condition ensuring the compactness of a bounded (PS) sequences. We also revise the Costa–Magalhaes condition [8] to obtain Cerami condition. As a consequence, some existence results derived by minimax methods were proved. Finally, we establish the existence of positive solution under the subcritical polynomial growth condition, while the strong superlinear condition is only required along an unbounded sequence. In other words, a certain degraded oscillation is allowed.     

Criteria determination to choose piezoelectric materials for BAW resonator applications via colored picosecond acoustics

A great number of piezoelectric materials which could be used in the fabrication of BAW resonators were investigated via colored picosecond acoustics technique in order to study the required parameters for designing and fabricating improved devices. These parameters concerns acoustic longitudinal velocity, v_L, elastic stiffness constant, C^D₃₃, intrinsic mechanical loss, tanδ, and electromechanical coupling coefficient, k²_t. We first quantify the effect of the ratio between the wavelength pulse of a femtosecond laser and the period of Brillouin oscillations, λ_p/T. It is found that C^D₃₃ depends linearly on λ_p/T. Then, we deduced novel relations for stiffness constant and mechanical coupling coefficient. Moreover, the determination of different parameters (density, refractive index and v_L) of piezoelectric film is achieved, with good agreement with literature. The optimized conditions for the parameter choice of BAW resonators are found to be: 300?GPa?<?C^D₃₃?<?500?GPa and 1.7?GPa?<?e²₃₃/ε^s₃₃?<?28.5?GPa with λ_p/T?>?37 10³?m/s where e₃₃ and ε^s₃₃ are the piezoelectric constant and the materials permittivity in the direction 3, respectively.     

Existence of radial solutions with prescribed number of zeros for elliptic equations and their Morse index

In this paper we consider radially symmetric solutions of the nonlinear Dirichlet problem Δu+f(|x|,u)=0 in Ω, where Ω is a ball in RN, N?3 and f satisfies some appropriate assumptions. We prove existence of radially symmetric solutions with k prescribed number of zeros. Moreover, when f(|x|,u)=K(|x|)|u|^p−1u, using the uniqueness result due to Tanaka (2008) [21], we verify that these solutions are non-degenerate and we prove that their radial Morse index is exactly k.