Multiple solutions for elliptic systems with nonlinearities of arbitrary growth

Abstract

We prove the existence of infinitely many solutions for symmetric elliptic systems with nonlinearities of\ud arbitrary growth. Moreover, if the symmetry of the problem is broken by a small enough perturbation term,\ud we find at least three solutions. The proofs utilise a variational setting given by de Figueiredo and Ruf in\ud order to prove an existence’s result and the “algebraic” approach based on the Pohozaev’s fibering method

Infinitely many solutions for symmetric and non-symmetric elliptic systems

“…The following theorems improve the results stated in [14] and [24] concerning the study of the system below the critical hyperbola. Let us consider now the case of the breaking of symmetry h = 0; clearly, the non-linear term f (u) + h(x) does not verify ( f 2 ), then even the de Figueiredo and Ruf existence's result do not apply.…”

“…If f has a polinomial growth, they extend this existence result to the case N N−2 p 2 and N 4. In [24] the existence of infinitely many solutions has been proved if f is odd. Moreover, a multiplicity result has been stated if h is a non-trivial small enough term and also f is exactly a pure power term, i.e.…”

Infinitely many solutions of a Sturm-Liouville system with impulses

“…and functions G k by G k (u, v) = k(u p v q ) 1 2 log(log(uv)) sin 2 (log(log(log(uv))) u, v > 0, + 4 (u p v q ) 1 2 log(uv) , 0, u = 0 or v = 0, (21) and functions G k by G k (u, v) = k(u p v q ) …”

On a new class of elliptic systems with nonlinearities of arbitrary growth

“…Later on, Salvatore [10] guaranteed via the Pohozaev's fibering method the existence of a whole sequence of solutions to (S) in a similar context as [4] assuming in addition that the nonlinear term f is odd. Note that in the latter two papers (i.e., [4] and [10]) no further growth restriction is required on the nonlinear term f other than the Ambrosetti-Rabinowitz condition. This latter fact is not surprising taking into account that (1) is actually equivalent to…”

“…The aim of the present paper is to complete the works [3,4] and [10] by guaranteeing the existence of infinitely many pairs of distinct solutions to the system (S) when (1) holds and the nonlinear term f has an oscillatory behavior. Moreover, the nonlinear term f may enjoy an arbitrary growth at infinity (resp., at zero) whenever it oscillates near the origin (resp., at infinity) in a suitable way.…”