Resonant semilinear Robin problems with a general potential

Resonant semilinear problems with nonlinear term depending on the derivative ✩

We study the existence of solution for a boundary value problem at resonance where the nonlinearity depends only on the derivative. In a sense, we can say that the problem considered is strongly resonant. Our proofs make use of the Lyapunov–Schmidt reduction; in so doing, we are led with the asymptotic estimate of the corresponding bifurcation equation.  2004 Elsevier Inc. All rights reserved.

Subcritical Perturbations of Resonant Linear Problems with Sign-changing Potential

We establish existence and multiplicity theorems for a Dirichlet boundary-value problem at resonance. This problem is a nonlinear subcritical perturbation of a linear eigenvalue problem studied by Cuesta, and includes a sign-changing potential. We obtain solutions using the Mountain Pass lemma and the Saddle Point theorem. Our paper extends some recent results of Gonçalves, Miyagaki, and Ma.

Inverse positivity for general Robin problems on Lipschitz domains

It is proved that elliptic boundary value problems in divergence form can be written in many equivalent forms. This is used to prove regularity properties and maximum principles for problems with Robin boundary conditions with negative or indefinite boundary coefficient on Lipschitz domains by rewriting them as a problem with positive coefficient. It is also shown that such methods cannot be ap...

Some Blow-Up Problems For A Semilinear Parabolic Equation With A Potential

The blow-up rate estimate for the solution to a semilinear parabolic equation ut = ∆u+V (x)|u|p−1u in Ω×(0, T ) with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data u(x, 0) = Mφ(x) as M goes to infinity, which have been found in [5], are improved under some reason...

Semilinear Problems with Bounded Nonlinear Term