Jean-Pierre Gossez

Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients

Variational boundary value problems for quasilinear elliptic systems in divergence form are studied in the case where the nonlinearities are nonpolynomial. Monotonicity methods are used to derive several existence theorems which generalize the basic results of Browder and Leray-Lions. Some features of the mappings of monotone type which arise here are that they act in nonreflexive Banach spaces, that they are unbounded and not everywhere defined, and that their inverse is also unbounded and not everywhere defined.

Local superlinearity and sublinearity for indefinite semilinear elliptic problems

In this paper the usual notions of superlinearity and sublinearity for semilinear problems like _Du ¼ f ðx; uÞ are given a local form and extended to indefinite nonlinearities. Here f ðx; sÞ is allowed to change sign or to vanish for s near zero as well as for s near infinity. Some of the well-known results of Ambrosetti–Bre′zis–Cerami are partially extended to this context.

Strict monotonicity of eigenvalues and unique continuation

In this paper, we show that the strict monotonicity of the eigenvalues of an uniformly elliptic operator of second order is equivalent to a unique continuation property.

Periodic solutions of a second order ordinary differential equation: A necessary and sufficient condition for nonresonance

The nonlinearity g in (1.1) is a continuous function from R to R and the forcing term h is taken in L”(O,27r). Nonresonance means that (1.1) admits at least one solution x for any given h. Integrating Eq. (1.1) over a period, one immediately sees that a necessary condition for nonresonance is that the function g be unbounded from above and from below on R. It will appear later (cf. (1.6)) that this unboundedness can be looked at as a condition relating the behaviour at infinity of the nonlinearity g with respect to the first eigenvalue 1, = 0 of the associated linear problem: -xf’= Ax in [0, 27r], x(0) = x(27r), x’(0) = x1(271). (1.2)

Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity

In this paper we study the existence, nonexistence and multiplicity of positive solutions for the family of problems

Nonlinear Perturbations of a Linear Elliptic Problem Near Its First Eigenvalue

-\Delta u = f_\lambda (x,u)

On the antimaximum principle for the p-Laplacian with indefinite weight

,

ASYMMETRIC ELLIPTIC PROBLEMS WITH INDEFINITE WEIGHTS

u \in H^1_0(\Omega)

On the range of a coercive maximal monotone operator in a nonreflexive Banach space

, where

A nodal domain property for the p-Laplacian

\Omega