In this paper we study the semiclassical limit for the pseudo-relativistic Hartree equation √ −ε2∆ +m2u+ V u = (Iα ∗ |u|) |u|p−2u, in R , where m > 0, 2 ≤ p < 2N N−1 , V : R N → R is an external scalar potential, Iα(x) = cN,α |x|N−α is a convolution kernel, cN,α is a positive constant and (N − 1)p−N < α < N . For N = 3, α = p = 2, our equation becomes the pseudo-relativistic Hartree equation wi...

In this note we prove the nonexistence of stable solutions to the p-Laplace equation −∆pu = eu on the entire Euclidean space RN , where p > 2 and N < p(p+3) p−1 .

By using a nonassociative algebra argument, we prove that u ≡ 0 is the only cubic homogeneous polynomial solution to the p-Laplace equation div|Du|p−2Du(x) = 0 in Rn for any n ≥ 2 and p 6∈ {1, 2}.

We prove a well-posedness result for two pseudo-parabolic problems, which can be seen as models the same electrical conduction phenomenon in heterogeneous media, neglecting magnetic field. One of problems is concentration limit other one, when thickness dielectric inclusions goes to zero. The concentrated problem involves transmission condition through interfaces, mediated by suitable Laplace-B...