Volume 14, Issue 4

G. Stampacchia introduced the geometric boundary value problem for ODEs in his doctoral thesis and published four papers related to it. Here we point out that the proof of his last theorem on the subject is incorrect and we provide a substitute for it as well as a generalizations of some of his earlier results.

We consider the semilinear elliptic problem −Δu = f (x, u), posed in a smooth bounded domain Ω of ℝ N with Dirichiel data u|∂Ω = 0, where f : Ω × [0, α f ) → ℝ + (0 < α f ≤ +∞) is a function of appropriate regularity which blows up at α f . We give pointwise lower bounds for the supersolutions under some appropriate conditions on f , and apply them to eigenvalue problem −Δu = λ f (x, u), by giving upper and lower bounds for the extremal parameter λ∗ and the extremal solution u∗. To demonstrate the sharpness of our results, we consider the eigenvalue problem −Δu = λ f (u p ) (p ≥ 1) with Dirichlet boundary condition, and show that for every increasing, convex and superlinear C 2 function f: ℝ + →ℝ + with , where ψΩ is the maximum of the torsion function of Ω. Also, we consider the eigenvalue problem −Δu = λρ(x) f (u), where f is either a regular singularity such as f (u) = e u , or a singular one such as and give explicit estimates on λ∗ and u∗, that improve and extend several results in the literature, by Payne[17], Sperb [21], Brezis-Vasquez [3], Guo-Pan-Ward [11], Ghoussoub-Guo [10], Cowan-Ghoussoub [6], and others.

In this paper, which is a mixture of original and expository ideas, the author shares some sketches of his research ideas and projects on certain important areas in Nonlinear Analysis. In addition to stating some open conjectures and suggestions for directions, the paper also contains an erratum to a previously published article.

Given a C 2 nonnegative potential F with a discrete set of zeros we consider the existence of heteroclinics connecting two arbitrary zeros of F for the Swift-Hohenberg equation, i.e. uʺʺ− βuʺ + Fʹ (u) = 0, for β ≤ 0. The emphasis is on the case where F has more than two zeros. We prove that when Fʹ has a sign near the zeros of F, |β| is sufficiently small in a variational sense, and then given any two different zeros z – and z + , there exists a solution to the Swift-Hohenberg equation with boundary values . We also consider heteroclinics constrained in a strip of the form {l < u(x) < r}, and consider some local conditions on F which produce heteroclinics regardless of the behavior of F outside [l, r].

We consider a nonlinear Neumann problem driven by a nonhomogeneous differential operator (special case is the p-Laplacian) with a (p − 1)-superlinear Carathéodory reaction term, which need not satisfy the usual in such cases Ambrosetti-Rabinowitz condition. Using variational methods based on the critical point theory coupled with suitable truncation techniques, we show that the problem has at least five nontrivial smooth solutions.

In this paper, existence and multiplicity results for a class of second-order difference equations are established. In particular, the existence of at least one positive solution without requiring any asymptotic condition at infinity on the nonlinear term is presented and the existence of two positive solutions under a superlinear growth at infinity of the nonlinear term is pointed out. The approach is based on variational methods and, in particular, on a local minimum theorem and its variants. It is worth noticing that, in this paper, some classical results of variational methods are opportunely rewritten by exploiting fully the finite dimensional framework in order to obtain novel results for discrete problems.

In this paper we are concerned with the quasilinear Schrödinger equation −Δu + V(x)u − Δ(u 2 )u = q(x)|u| p−2 u, x ∈ ℝ N , where N ≥ 3, 4 < p < 4N/(N − 2), and V(x) and q(x) go to some positive limits V ∞ and q ∞ respectively as |x| → ∞. Under a weak one-sided asymptotic estimate for V(x) and q(x) at infinity, we obtain the existence of a ground state solution.

In this paper, we study a nonlinear Schrödinger equation with magnetic fields involving subcritical growth. Applying the finite reduction method, we prove that the equation has multi-peak solutions under some suitable conditions which are given in section 1.

The asymptotic behavior of the sequence {u n } of positive first eigenfunctions for a class of inhomogeneous eigenvalue problems is studied in the setting of Orlicz-Sobolev spaces. After possibly extracting a subsequence, we prove that u n → u ∞ uniformly in Ω as n→∞, where u ∞ is a nontrivial viscosity solution of a nonlinear PDE involving the ∞-Laplacian.

We introduce and study a family of integral operators in the Kantorovich sense acting on functions defined on locally compact topological groups. We obtain convergence results for the above operators with respect to the pointwise and uniform convergence and in the setting of Orlicz spaces with respect to the modular convergence. Moreover, we show how our theory applies to several classes of integral and discrete operators, as sampling, convolution and Mellin type operators in the Kantorovich sense, thus obtaining a simultaneous approach for discrete and integral operators. Further, we obtain general convergence results in particular cases of Orlicz spaces, as L p −spaces, interpolation spaces and exponential spaces. Finally we construct some concrete examples of our operators and we show some graphical representations.

In this paper, we investigate the following quasilinear elliptic system (P) with explosive boundary conditions: Δ p u = f 1 (x, u, v) in Ω; u| ∂Ω = +∞, u >0 in Ω, Δ q v = f 2 (x, u, v) in Ω; v| ∂Ω = +∞, v >0 in Ω, where Ω is a smooth bounded domain of ℝ N , N ≥ 2, 1 < p, q < +∞ and f 1 , f 2 are two Carathéodory functions in Ω × (ℝ + * × ∗ ℝ + *). Under rather general conditions on f 1 and f 2 and assuming the existence of a sub and supersolutions pair, we prove the existence of a large solution to (P) by a fixed point approach. Then, we apply this result considering particular systems arising in Biology.

We consider a jumping problem for singular semilinear elliptic equations. Existence of two distinct solutions is proved via a truncation argument and exploiting minimax methods to prove the existence of solutions to the truncated problem. In the final step, to pass to the limit, a refined analysis about the behavior of the solutions is required, because of the singular nature of the problem.

Since the work of Hörmander on linear pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, harmonic analysis, theory of several complex variables and other branches of modern analysis (e.g., they are used to construct parametrices and establish the regularity of solutions to PDEs such as the ∂̅ problem, etc.). The work of Coifman and Meyer on multi-linear Fourier multipliers and pseudo-differential operators has stimulated further such applications. In [2], the authors developed a fairly satisfactory theory of symbolic calculus for multi-linear pseudo-differential operators. Motivated by this work [2] and L p estimates of [34, 35] on multi-parameter and multi-linear Fourier multipliers and of [12] on multi-parameter and multi-linear pseudo-differential operators, we study and carry out the theory of symbolic calculus for multi-parameter and multi-linear pseudo-differential operators. Our results include the symbol estimates of the adjoints, asymptotic behavior, kernel estimates and boundedness properties and extend those in [2] to the multi-parameter and multi-linear setting. The estimates of the distributional kernel of associated multi-parameter and multilinear pseudo-differential operators can be found useful in establishing the boundedness of such multi-parameter and multi-linear pseudo-differential operators.