David Arcoya

Critical points for multiple integrals of the calculus of variations

AbstractIn this paper we deal with the existence of critical points of functional defined on the Sobolev space W01,p(Ω), p>1, by

BIFURCATION THEORY AND RELATED PROBLEMS: ANTI-MAXIMUM PRINCIPLE AND RESONANCE*

J(u) = \int\limits_\Omega {\vartheta (x,u,Du)dx,} {\text{ }}

Quasilinear equations with natural growth

where Ω is a bounded, open subset of ℝN. Even for very simple examples in ℝN the differentiability of J(u) can fail. To overcome this difficulty we prove a suitable version of the Ambrosetti-Rabinowitz Mountain Pass Theorem applicable to functionals which are not differentiable in all directions. Existence and multiplicity of nonnegative critical points are also studied through the use of this theorem.

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

In this work we study the existence of a solution for the problem − Δ p u = f(u) + tΦ(x) + h(x), with homogeneous Dirichlet boundary conditions. Here the nonlinear term f(u) is a so-called jumping nonlinearity. In the proofs we use topological arguments and the sub-supersolutions method, together with comparison principles for the p-Laplacian.

Existence of critical points for some noncoercive functionals

For a smooth bounded domain Ω ⊂ IR N , we consider the b.v.p. where m ∈ L r (Ω) for some r ∈ (max {1, N/2}, + ∞], with m + ≢ 0 and g is a Carathéodory function. We deduce sufficient and sharp conditions to have subcritical (“to the left”) or supercritical (“to the right”) bifurcations (either from zero or from infinity) at an eigenvalue λ k (m) of the associated linear weighted eigenvalue problem. Furthermore, as a consequence, we also point out the bifurcation nature of some classical results like the (local) Antimaximum Principle of Clement and Peletier and the Landesman-Lazer theorem for resonant problems. In addition, we see that the bifurcation viewpoint allows to obtain also local maximum principle and more general results for some classes of strongly resonant problems. In addition, we extend the above technique to handle quasilinear b.v.p. *Supported by Acción Integrada Spain-Italy HI1997-0049, by D.G.E.S. Ministerio de Educación y Ciencia (Spain) PB98-1283 and by E.E.C. contract n. ERBCHRXCT940494. A preliminary communication of some of the results in this paper was presented at Nichtlineare Eigenwertaufgaben, held in Oberwolfach, Germany, 15–21 December 1996.

Bifurcation for Quasilinear Elliptic Singular BVP

Notation.- Preliminaries.- Some Fixed Point Theorems.- Local and Global Inversion Theorems.- Leray-Schauder Topological Degree.- An Outline of Critical Points.- Bifurcation Theory.- Elliptic Problems and Functional Analysis.- Problems with A Priori Bounds.- Asymptotically Linear Problems.- Asymmetric Nonlinearities.- Superlinear Problems.- Quasilinear Problems.- Stationary States of Evolution Equations.- Appendix A Sobolev Spaces.- Exercises.- Index.- Bibliography.

Asymmetric modes on symmetric nonlinear optical waveguides

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.

Quasilinear elliptic problems interacting with its asymptotic spectrum

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.