Ground states of nonlocal scalar field equations with Trudinger-Moser critical nonlinearity

We investigate the existence of ground state solutions for a class of nonlinear scalar field equations defined on whole real line, involving a fractional Laplacian and nonlinearities with Trudinger-Moser critical growth. We handle the lack of compactness of the associated energy functional due to the unboundedness of the domain and the presence of a limiting case embedding.Comment: 13 page

Asymptotics of ground states for fractional H\'enon systems

We investigate the asymptotic behavior of positive ground states for H\'enon type systems involving a fractional Laplacian on a bounded domain, when the powers of the nonlinearity approach the Sobolev critical exponent.Comment: 18 page

Existence results for quasilinear elliptic exterior problems involving convection term and nonlinear Robin boundary conditions

In this paper, the authors establish the existence of solutions for a class of elliptic exterior problems involving convection terms and nonlinear Robin boundary conditions. The proof of the result is made by combining Galerkin method with a priori estimates for this kind of problem

Soliton solutions for quasilinear Schrödinger equations with critical growth

In this paper we establish the existence of standing wave solutions for quasilinear Schrödinger equations involving critical growth. By using a change of variables, the quasilinear equations are reduced to semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution v. In the proof that v is nontrivial, the main tool is the concentration–compactness principle due to P.L. Lions together with some classical arguments used by H. Brezis and L. Nirenberg (1983) in [9]

Soliton solutions for quasilinear Schrödinger equations: the critical exponential case

Quasilinear elliptic equations in R2 of second order with critical exponential growth are considered. By using a change of variable, the quasilinear equations are reduced to semilinear equations, whose respective associated functionals are well defined in H 1 (R2) and satisfy the geometric hypotheses of the mountain pass theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution v. In the proof that v is nontrivial, the main tool is the concentration–compactness principle [P.L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Part I and II, Ann. Inst. H. Poincar ́ Anal. Non. Lineaire 1 (1984) 109–145, 223–283] combined with test functions connected with optimal Trudinger–Moser inequality