DCDS-S
Noncoercive elliptic equations with subcritical growth
Vicenţiu D. Rădulescu
We study a class of nonlinear elliptic equations with subcritical growth and Dirichlet boundary condition. Our purpose in the present paper is threefold: (i) to establish the effect of a small perturbation in a nonlinear coercive problem; (ii) to study a Dirichlet elliptic problem with lack of coercivity; and (iii) to consider the case of a monotone nonlinear term with subcritical growth. This last feature enables us to use a dual variational method introduced by Clarke and Ekeland in the framework of Hamiltonian systems associated with a convex Hamiltonian and applied by Brezis to the qualitative analysis of large classes of nonlinear partial differential equations. Connections with the mountain pass theorem are also made in the present paper.
keywords: Mountain pass geometry. Nonlinear elliptic equation Dirichlet boundary condition Dual variational method
DCDS
Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities
Nikolaos S. Papageorgiou Vicenţiu D. Rădulescu
In this paper we deal with Robin and Neumann parametric elliptic equations driven by a nonhomogeneous differential operator and with a reaction that exhibits competing nonlinearities (concave-convex nonlinearities). For the Robin problem and without employing the Ambrosetti-Rabinowitz condition, we prove a bifurcation theorem for the positive solutions for small values of the parameter $\lambda>0$. For the Neumann problem with a different geometry and using the Ambrosetti-Rabinowitz condition we prove bifurcation for large values of $\lambda>0$.
keywords: nonhomogeneous differential operator bifurcation phenomena positive solutions Competing nonlinearities Robin and Neumann problems.
CPAA
A tribute to Professor Philippe G. Ciarlet on his 70th birthday
Hervé Le Dret Vicenţiu D. Rădulescu Roderick S. C. Wong
Professor Philippe G. Ciarlet was born on October 14, 1938, in Paris. He obtained his undergraduate degree in 1961 from the celebrated École Polytechnique in Paris, followed by graduate studies (1962-1964) at the École Nationale des Ponts et Chaussées in Paris. Professor Ciarlet received in 1966 his Ph.D. at the Case Institute of Technology, Cleveland, U.S.A., under the guidance of Professor Richard S. Varga. The title of his Ph.D. thesis is Variational Methods for Non-Linear Boundary-Value Problems. He continued with a Doctorat d'État Fonctions de Green Discrètes et Principe du Maximum Discret) at the University of Paris in 1971 and his advisor was Professor Jacques-Louis Lions.

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DCDS
Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential
Nikolaos S. Papageorgiou Vicenţiu D. Rădulescu Dušan D. Repovš

We study perturbations of the eigenvalue problem for the negative Laplacian plus an indefinite and unbounded potential and Robin boundary condition. First we consider the case of a sublinear perturbation and then of a superlinear perturbation. For the first case we show that for $ λ<\widehat{λ}_{1}$ ($ \widehat{λ}_{1}$ being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For $ λ≥q\widehat{λ}_{1}$ there are no positive solutions. In the superlinear case, for $ λ<\widehat{λ}_{1}$ we have at least two positive solutions and for $ λ≥q\widehat{λ}_{1}$ there are no positive solutions. For both cases we establish the existence of a minimal positive solution $ \bar{u}_{λ}$ and we investigate the properties of the map $ λ\mapsto\bar{u}_{λ}$.

keywords: Indefinite and unbounded potential Robin eigenvalue problem sublinear perturbation superlinear perturbation maximum principle positive solution minimal positive solution

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