Pedro J. Martínez-Aparicio

Quasilinear equations with natural growth

We study the existence of positive solution w ∈ H1 0 (Ω) of the quasilinear equation −∆w + g(w)|∇w|2 = a(x), x ∈ Ω, where Ω is a bounded domain in RN , 0 ≤ a ∈ L∞(Ω) and g is a nonnegative continuous function on (0,+∞) which may have a singularity at zero.

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.

Advances in the Study of Singular Semilinear Elliptic Problems

In this paper we deal with some results concerning semilinear elliptic singular problems with Dirichlet boundary conditions. The problem becomes singular where the solution u vanishes. The model of this kind of problems is

A Singular Semilinear Elliptic Equation with a Variable Exponent

\displaystyle\begin{array}{rcl} \left \{\begin{array}{@{}l@{\quad }l@{}} u \geq 0 \quad &\mbox{ in }\varOmega, \\ -div\,A(x)Du = F(x,u)\quad &\mbox{ in}\;\varOmega, \\ u = 0 \quad &\mbox{ on}\;\partial \varOmega,\\ \quad \end{array} \right.& & {}\\ \end{array}

The sublinear problem for the 1 -homogeneous p -Laplacian

where Ω is a bounded open set of \(\mathbb{R}^{N}\), N ≥ 1, A is a coercive matrix with coefficients in \(L^{\infty }(\varOmega )\) and \(F: (x,s) \in \varOmega \times [0,+\infty [\rightarrow F(x,s) \in [0,+\infty ]\) is a Caratheodory function which is singular at s = 0. Our aim is to study the meaning of the assumptions made on the singular function F(x, s) in the papers [Giachetti et al., J. Math. Pures Appl. (2016, in press); Giachetti et al., Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0 (Preprint, 2016); Giachetti et al., Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u = 0 in a domain with many small holes (Preprint, 2016)], to extend some uniqueness results of the solution given in the same papers, and to prove the \(L^{\infty }\)-regularity of the solutions under some regularity assumption on the data.

Existence and nonexistence of solutions for singular quadratic quasilinear equations

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.

Singular quasilinear equations with quadratic growth in the gradient without sign condition

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 .