José Carmona

Elliptic Obstacle Problems With Natural Growth on the Gradient and Singular Nonlinear Term

Abstract Given a bounded, open set Ω in ℝN (N ≥ 3), ψ∈ W1,p(Ω) (p > N) such that υ̸+ ∈ H01(Ω) ∩ L∞(Ω) and a suitable strictly positive (see (1.4)) function a ∈ Lq(Ω) with q > N/2, we prove the existence of positive solution w ∈ H01(Ω) of some variational inequality with a singular nonlinearity whose typical model is where the set of test functions K1 consists of all functions υ ∈ H01(Ω) ∩ L∞(Ω) such that υ(x) ≥ ψ(x) a.e. x ∈ Ω and supp (υ - ψ+) ⊂⊂ Ω. Bigger classes of test functions are also studied. We also recover the case in which the variational inequality reduces to an equation.

Bifurcation for Quasilinear Elliptic Singular BVP

For a continuous function g ≥ 0 on (0, + ∞) (which may be singular at zero), we confront a quasilinear elliptic differential operator with natural growth in ∇u, − Δu + g(u)|∇u|2, with a power type nonlinearity, λu p + f 0(x). The range of values of the parameter λ for which the associated homogeneous Dirichlet boundary value problem admits positive solutions depends on the behavior of g and on the exponent p. Using bifurcations techniques we deduce sufficient conditions for the boundedness or unboundedness of the cited range.

Quasilinear elliptic problems interacting with its asymptotic spectrum

Under suitable assumptions on the coefficients of the matrix A(x, u) and on the nonlinear term f(x, u), we study the quasilinear problem in bounded domains Ω ⊂ RN -div(A(x, u)∇u) = f(x, u), x ∈ Ω, u = 0, x∈∂Ω. We extend the semilinear results of Landesman-Lazer (J. Math. Mech. 19 (1970) 609) and of Ambrosetti-Prodi (in: A Primer on Nonlinear Analysis, Cambridge University Press, Cambridge, 1993) for resonant problems. The existence of positive solution is also considered extending to the quasilinear case the classical result by Ambrosetti-Rabinowitz (J. Funct. Anal. 14 (1973) 349). In this case, the result is obtained as a corollary of the previous multiplicity result in the Ambrosetti-Prodi framework.

Mii School: New 3D Technologies Applied in Education to Detect Drug Abuses and Bullying in Adolescents

Mii School is a 3D school simulator developed with Blender and used by psychology researchers for the detection of drugs abuses, bullying and mental disorders in adolescents. The school simulator created is an interactive video game where the players, in this case the students, have to choose, along 17 scenes simulated, the options that better define their personalities. In this paper we present a technical characteristics description and the first results obtained in a real school.

Comparison principle for elliptic equations in divergence with singular lower order terms having natural growth

In this paper, we are concerned with the zero Dirichlet boundary value problem associated to the quasilinear elliptic equation −div(a(u)M(x)∇u) + H(x,u,∇u) = f(x),x ∈ Ω, where Ω is an open and bounded set in ℝN (N ≥ 3), a is a continuously differentiable real function in (0, +∞), M(x) is an elliptic, bounded and symmetric matrix, H(x,⋅,ξ) is non-negative and may be singular at zero and f ∈ L1(Ω). We give sufficient conditions on H, M and a in order to have a comparison principle and, as a consequence, uniqueness of positive solutions being continuous up to the boundary.

A Singular Semilinear Elliptic Equation with a Variable Exponent

Abstract In this paper we consider singular semilinear elliptic equations with a variable exponent whose model problem is - Δ ⁢ u = f ⁢ ( x ) u γ ⁢ ( x )   in ⁢ Ω , u = 0   on ⁢ ∂ ⁡ Ω .

Regularity and Morse Index of the Solutions to Critical Quasilinear Elliptic Systems

-\Delta u=\frac{f(x)}{u^{\gamma(x)}}\quad\text{in }\Omega,\qquad u=0\quad\text% {on }\partial\Omega.

Existence and nonexistence of solutions for singular quadratic quasilinear equations

Here Ω is an open bounded set of ℝ N

Bifurcation for some quasilinear operators

{\mathbb{R}^{N}}

MII-School: A 3d videogame for the early detection of abuse of substances, bullying, and mental disorders in adolescents

, γ ⁢ ( x )