Existence of solutions for semilinear elliptic problems without (PS) condition

Abstract

Abstract. We establish an existence result for semilinear elliptic problems with the associated functional not satisfying the Palais-Smale condition. The nonlinearity of our problem does not satisfy the Ambrosetti-Rabinowitz condition.

Solution for a Class of Quasilinear Elliptic Equation with p-Laplacian

“…Further conditions will be s listed below. Similar problems as (1) have been considered by many authors, see [1,2,4,5,6] and the references therein. Some authors considered the similar problems on the Nehari manifold, see [7,8,9].…”

A priori estimates for super-linear elliptic equation: the Neumann boundary value problem

“…If f (x, u) := |u| p−1 u + g(x, u) and 1 < p < N N −2 , where g is not necessarily odd in u and satisfying suitable subcritical conditions, Ramos-Tavares-Zou used the Morse index [20] to improve the celebrated multiplicity result of Bahri-Lions [2]. When the Palais-Smale; or the Cerami compactness conditions for the energy functional do not seem to follow readily, the proof of existence of solutions is essentially reduced to deriving L ∞ -bounds which in [19,25] has been obtained where the authors adapted the approach of Bahri-Lions to prove the following theorem Theorem 1.1. [3] Assume that f satisfies…”

Ambrosetti-Rabinowitz theorems revisited

“…There exist C > 0 and 1 < q ≤ p such that |f (x, s)| ≥ C|s| q , C > 0, for x ∈ Ω and |s| large,(with q = θ − 1). When m = 1, under the Dirichlet boundary condition, very few existence results have been established when f satisfying (ii) and (i) is relaxed to (SSL)(see for example [9,28,30]). Nevertheless, (SSL) is also violated by many nonlinearities as for example f (s) ∼ as or f (s) ∼ as ln(|s|) at infinity(where a is a positive constant).…”