Nontrivial solutions for an asymptotically linear Dirichlet problem.
The existence of a nontrivial critical point is proved for a functional containing an area-type term. Techniques of nonsmooth critical point theory are applied.
In this article, we study the existence of nontrivial weak solutions for the following boundary value problem: where is a bounded domain with smooth boundary in , for some , is a subelliptic linear operator of the type where satisfies certain homogeneity conditions and degenerates at the coordinate hyperplanes and the nonlinearity is of subcritical growth and does not satisfy the Ambrosetti-Rabinowitz (AR) condition.
It is well-known that the “standard” oblique derivative problem, in , on ( is the unit inner normal) has a unique solution even when the boundary condition is not assumed to hold on the entire boundary. When the boundary condition is modified to satisfy an obliqueness condition, the behavior at a single boundary point can change the uniqueness result. We give two simple examples to demonstrate what can happen.
Models introduced by R. F. Streater describe the evolution of the density and temperature of a cloud of self-gravitating particles. We study nonuniqueness of steady states in annular domains in , d ≥ 2.
The purpose of this paper is to derive norm inequalities for potentials of the formTf(x) = ∫(Rn) f(y)K(x,y)dy, x ∈ Rn,when K is a Kernel which satisfies estimates like those that hold for the Green function associated with the degenerate elliptic equations studied in [3] and [4].