Vicenţiu D. Rădulescu

A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids

Abstract. We study the boundary value problem −div(a(x,∇u)) = λ(u − u) in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in R and div(a(x,∇u)) is a p(x)-Laplace type operator, with 1 < β < γ < infx∈Ω p(x). We prove that if λ is large enough then there exist at least two nonnegative weak solutions. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with adequate variational methods and a variant of Mountain Pass Lemma. 2000 Mathematics Subject Classification: 35D05, 35J60, 35J70, 58E05, 68T40, 76A02.

On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent

We consider the nonlinear eigenvalue problem -div (|∇ u | p(x) - 2 ∇ u ) = λ( u ( q(x) - 2 u in Ω, u = 0 on ∂Ω, where Ω is a bounded open set in R N with smooth boundary and p, q are continuous functions on Ω such that 1 0 sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekelands variational principle.

Eigenvalue problems for anisotropic discrete boundary value problems

In this paper, we prove the existence of a continuous spectrum for a family of discrete boundary value problems. The main existence results are obtained by using critical point theory. The equations studied in the paper represent a discrete variant of some recent anisotropic variable exponent problems, which deserve as models in different fields of mathematical physics.

Sublinear singular elliptic problems with two parameters

Abstract We establish several existence and nonexistence results for the boundary value problem −Δu+K(x)g(u)=λf(x,u)+μh(x) in Ω , u=0 on ∂Ω , where Ω is a smooth bounded domain in R N , λ and μ are positive parameters, h is a positive function, while f has a sublinear growth. The main feature of this paper is that the nonlinearity g is assumed to be unbounded around the origin. Our analysis shows the importance of the role played by the decay rate of g combined with the signs of the extremal values of the potential K(x) on Ω . The proofs are based on various techniques related to the maximum principle for elliptic equations.

EXISTENCE AND UNIQUENESS OF BLOW-UP SOLUTIONS FOR A CLASS OF LOGISTIC EQUATIONS

Let f be a non-negative C1-function on [0, ∞) such that f(u)/u is increasing and , where . Assume Ω ⊂ RN is a smooth bounded domain, a is a real parameter and b ≥ 0 is a continuous function on , b≢0. We consider the problem Δu+au =b(x)f(u) in Ω and we prove a necessary and sufficient condition for the existence of positive solutions that blow-up at the boundary. We also deduce several existence and uniqueness results for a related problem, subject to homogeneous Dirichlet, Neumann or Robin boundary condition.

Multi-parameter bifurcation and asymptotics for the singular Lane-Emden-Fowler equation with a convection term

to the second variable and g is unbounded around the origin. The asymptotic behaviour of the solution around the bifurcation point is also established, provided g(u) behaves like u −α around the origin, for some 0 <α< 1. Our approach relies on finding explicit sub- and supersolutions combined with various techniques related to the maximum principle for elliptic equations. The analysis we develop in this paper shows the key role played by the convection term |∇u|p.

Boundary blow-up in nonlinear elliptic equations of Bieberbach–Rademacher type

The research of the first author was carried out at Victoria University (Melbourne) with the support of the Australian Government through DETYA. The second author has been supported by Grant 2-CEX06-11-18/2006.

Entire solutions blowing up at infinity for semilinear elliptic systems

Abstract We consider the system Δu=p(x)g(v), Δv=q(x)f(u) in R N , where f,g are positive and non-decreasing functions on (0,∞) satisfying the Keller–Osserman condition and we establish the existence of positive solutions that blow-up at infinity.

On a class of sublinear singular elliptic problems with convection term

Abstract We establish several results related to existence, nonexistence or bifurcation of positive solutions for the boundary value problem − Δ u + K ( x ) g ( u ) + | ∇ u | a = λ f ( x , u ) in Ω, u = 0 on ∂Ω, where Ω ⊂ R N ( N ⩾ 2 ) is a smooth bounded domain, 0 a ⩽ 2 , λ is a positive parameter, and f is smooth and has a sublinear growth. The main feature of this paper consists in the presence of the singular nonlinearity g combined with the convection term | ∇ u | a . Our approach takes into account both the sign of the potential K and the decay rate around the origin of the singular nonlinearity g. The proofs are based on various techniques related to the maximum principle for elliptic equations.

Singular Phenomena in Nonlinear Elliptic Problems: From Blow-Up Boundary Solutions to Equations with Singular Nonlinearities

Abstract In this survey we report on some recent results related to various singular phenomena arising in the study of some classes of nonlinear elliptic equations. We establish qualitative results on the existence, nonexistence or the uniqueness of solutions and we focus on the following types of problems: (i) blow-up boundary solutions of logistic equations; (ii) Lane—Emden—Fowler equations with singular nonlinearities and subquadratic convection term. We study the combined effects of various terms involved in these problems: sublinear or superlinear nonlinearities, singular nonlinear terms, convection nonlinearities, as well as sign-changing potentials. We also take into account bifurcation nonlinear problems and we establish the precise rate decay of the solution in some concrete situations. Our approach combines standard techniques based on the maximum principle with nonstandard arguments, such as the Karamata regular variation theory.