Via a sub-supersolution method and a perturbation argument, we study the Lane-Emden-Fowler equation −∆u = p(x)[g(u) + f(u) + |∇u| ] in RN (N ≥ 3), where 0 < q < 1, p is a positive weight such that R∞ 0 rφ(r)dr < ∞, where φ(r) = max|x|=r p(x), r ≥ 0. Under the hypotheses that both g and f are sublinear, which include no monotonicity on the functions g(u), f(u), g(u)/u and f(u)/u, we show the exi...

Bellemans and De Leener) have studied the ground state energy of an electron gas moving in a lattice of positive charges. This picture is ex~ pected to correspond roughly to actual metals or metallic solutions. The purpose of this paper is to extend their treatment to a study of spin paramagnetism. The magnetic susceptibility of the system is determined by the change in energy as the electron s...

The D dimensional quasi exact solutions for the singular even power anharmonic potential V (q) = aq2 + bq−4 + cq−6 are reported. We show that whilst Dong and Ma’s [5] quasi exact ground state solution (in D=2) is beyond doubt, their solution for the first excited state is exotic. Quasi exact solutions for the ground and first excited states are also given for the above potential confined to an ...

The D dimensional quasi exact solutions for the singular even power anharmonic potential V (q) = aq2 + bq−4 + cq−6 are reported. We show that whilst Dong and Ma’s [5] quasi exact ground state solution (in D=2) is beyond doubt, their solution for the first excited state is exotic. Quasi exact solutions for the ground and first excited states are also given for the above potential confined to an ...

We are concerned with singular elliptic equations of the form −∆u = p(x)(g(u) + f (u) + |∇u| a) in R N (N ≥ 3), where p is a positive weight and 0 < a < 1. Under the hypothesis that f is a nondecreasing function with sublinear growth and g is decreasing and unbounded around the origin, we establish the existence of a ground state solution vanishing at infinity. Our arguments rely essentially on...

We consider solutions u(t) to the 3d focusing NLS equation i∂tu+∆u+ |u|2u = 0 such that ‖xu(t)‖L2 = ∞ and u(t) is nonradial. Denoting by M [u] and E[u], the mass and energy, respectively, of a solution u, and byQ(x) the ground state solution to −Q+∆Q+ |Q|2Q = 0, we prove the following: if M [u]E[u] < M [Q]E[Q] and ‖u0‖L2‖∇u0‖L2 > ‖Q‖L2‖∇Q‖L2 , then either u(t) blows-up in finite positive time o...