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Sublinear eigenvalue problems on compact Riemannian manifolds with applications in Emden-Fowler equations

Alexandru Kristály;  Vicenţiu Rădulescu — 2009

Studia Mathematica

Let (M,g) be a compact Riemannian manifold without boundary, with dim M ≥ 3, and f: ℝ → ℝ a continuous function which is sublinear at infinity. By various variational approaches, existence of multiple solutions of the eigenvalue problem - Δ g ω + α ( σ ) ω = K ̃ ( λ , σ ) f ( ω ) , σ ∈ M, ω ∈ H₁²(M), is established for certain eigenvalues λ > 0, depending on further properties of f and on explicit forms of the function K̃. Here, Δ g stands for the Laplace-Beltrami operator on (M,g), and α, K̃ are smooth positive functions. These multiplicity...

Explosive solutions of semilinear elliptic systems with gradient term.

Marius Ghergu;  Vicentiu Radulescu — 2003

RACSAM

Estudiamos la existencia de soluciones del sistema elíptico no lineal Δu + |∇u| = p(|x|)f(v), Δv + |∇v| = q(|x|)g(u) en Ω que explotan en el borde. Aquí Ω es un dominio acotado de R o el espacio total. Las nolinealidades f y g son funciones continuas positivas mientras que los potenciales p y q son funciones continuas que satisfacen apropiadas condiciones de crecimiento en el infinito. Demostramos que las soluciones explosivas en el borde dejan de existir si f y g son sublineales. Esto se tiene...

Variational analysis for a nonlinear elliptic problem on the Sierpiński gasket

Gabriele Bonanno;  Giovanni Molica Bisci;  Vicenţiu Rădulescu — 2012

ESAIM: Control, Optimisation and Calculus of Variations

Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinear term, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem on the Sierpiński gasket is proved. Our approach is based on variational methods and on some analytic and geometrical properties of the Sierpiński fractal. The abstract results are illustrated by explicit examples.

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