Volume 4, Issue 4

Some existence theorems are obtained for infinitely many periodic solutions of ordinary p -Laplacian systems by minimax methods in critical point theory.

In this article, we consider a vibrating nonlinear Timoshenko system with thermoelasticity with second sound. We discuss the well-posedness and the regularity of solutions using the semi-group theory. Moreover, we establish an explicit and general decay result for a wide class of relaxation functions, which depend on a stability number μ.

In the present paper, by using variational principle, we obtain the existence and multiplicity of solutions of a nonlocal problem involving p ( x )-Laplacian. The problem is settled in the variable exponent Sobolev space W 0 1, p ( x ) (Ω), and the main tools are the Mountain-Pass theorem and Fountain theorem.

We consider the reaction-diffusion problem -Δ g u = f ( u ) in ℬ R with zero Dirichlet boundary condition, posed in a geodesic ball ℬ R with radius R of a Riemannian model ( M , g ). This class of Riemannian manifolds includes the classical space forms, i.e., the Euclidean, elliptic, and hyperbolic spaces. For the class of semistable solutions we prove radial symmetry and monotonicity. Furthermore, we establish L ∞ , L p , and W 1, p estimates which are optimal and do not depend on the nonlinearity f . As an application, under standard assumptions on the nonlinearity λ f ( u ), we prove that the corresponding extremal solution u * is bounded whenever n ≤ 9. To establish the optimality of our regularity results we find the extremal solution for some exponential and power nonlinearities using an improved weighted Hardy inequality.

A boundary control problem for the pure Cahn–Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first-order necessary conditions for optimality are proved.