We are concerned with the followingmodified nonlinear Schrödinger system: −Δu+u−(1/2)uΔ(u2) = (2α/(α+β))|u||V|u, x ∈ Ω, −ΔV+V−(1/2)VΔ(V2) = (2β/(α+β))|u||V|V, x ∈ Ω, u = 0, V = 0, x ∈ ∂Ω, whereα > 2, β > 2, α+β < 2⋅2, 2∗ = 2N/(N−2) is the critical Sobolev exponent, andΩ ⊂ RN (N ≥ 3) is a bounded smooth domain. By using the perturbationmethod, we establish the existence of both positive and nega...

We consider the problem vt = ∆v + |v|p−1v in Ω× (0, T ), v = 0 on ∂Ω× (0, T ), v > 0 in Ω× (0, T ). In a domain Ω ⊂ Rd, d ≥ 7 enjoying special symmetries, we find the first example of a solution with type II blow-up for a power p less than the JosephLundgren exponent pJL(d) = { ∞, if 3 ≤ d ≤ 10, 1 + 4 d−4−2 √ d−1 , if d ≥ 11. No type II radial blow-up is present for p < pJL(d). We take p = d+1 ...

We study the Choquard equation with a local perturbation −Δu=λu+(Iα∗|u|p)|u|p−2u+μ|u|q−2u,x∈RN having prescribed mass ∫RN|u|2dx=a2. For L2-critical or L2-supercritical μ|u|q−2u, we prove nonexistence, existence and symmetry of normalized ground states, by using mountain pass lemma, Pohožaev constraint method, Schwartz symmetrization rearrangements some theories polarizations. In particular, our...

In this paper, we are concerned with the semilinear Schrödinger equation (1.1) −∆Au− V (x)u = |u| ∗−2u , x ∈ R , where −∆A = (−i∇+A)2, u : R → C, N ≥ 3, 2∗ = 2N N−2 denotes the critical Sobolev exponent, A = (A1, A2, ..., AN ) : R N → R is the vector (or magnetic) potential, the coefficient V is the scalar (or electric) potential and may be signchanging. The nonlinear Schrödinger equation arise...

We study the existence of ground state standing waves, prescribed mass, for nonlinear Schrödinger equation with mixed power nonlinearitiesi?tv+?v+?v|v|q?2+v|v|2??2=0,(t,x)?R×RN, where N?3, v:R×RN?C, ?>0, 2<q<2+4/N and 2?=2N/(N?2) is critical Sobolev exponent. show that all states correspond to local minima associated Energy functional. Next, despite fact nonlinearity critical, we set orbitally ...

In this paper we consider the following problem: where Q c Rn is a bounded domain and We prove the existence of a nontrivial solution of (1) for any ~, &#x3E; 0, RESUME. Soient Q un sous-ensemble ouvert borne de Rn et À un nombre positif, le but de cette note c’est de montrer que le probleme suivant : admet, au moins, une solution non triviale, si r~ &#x3E; 4. Work supported by G. N. A. F. A. o...

and Applied Analysis 3 The following Hardy-Sobolev inequality is due to Caffarelli et al. 12 , which is called Caffarelli-Kohn-Nirenberg inequality. There exist constants S1, S2 > 0 such that (∫ RN |x|−bp |u|pdx )p/p∗ ≤ S1 ∫ RN |x|−ap|∇u|pdx, ∀u ∈ C∞ 0 ( R N ) , 1.8 ∫ RN |x|− a 1 |u|dx ≤ S2 ∫ RN |x|−ap|∇u|pdx, ∀u ∈ C∞ 0 ( R N ) , 1.9 where p∗ Np/ N − pd is called the Sobolev critical exponent. ...