An Existence Result for a Problem with Critical Growth and Lack of Strict Convexity

Abstract

Abstract.We prove the existence of a nontrivial solution for a quasilinear elliptic equation involving a nonlinearity having critical growth and a convex principal part, which is not required to be strictly convex. Mathematics Subject Classification (2000). 35J65, 58E05.

A Variant of Clark's Theorem and Its Applications for Nonsmooth Functionals without the Palais--Smale Condition

“…In section 5, using Theorem 1.1, we prove that this equation has infinitely many solutions. In a sense, our result improves the previous results on quasi-linear elliptic equations in [11,22].…”

Minimax solutions for a problem with sign changing nonlinearity and lack of strict convexity

“…On the other hand, only a small literature is available when dealing with equations with a non strictly convex principal part. In this framework, in [7] the author applies non smooth variational methods in presence of subcritical, positive, nonlinearities; while using similar techniques a nonlinearity with criti-cal growth was considered in [9]. The aim of this paper is to extend to the setting of non strictly convex functionals some of the results contained in [2] (existence of a positive solution for λ < λ 1 ) and [8] (existence of a nontrivial solution for any λ.)…”

“…As shown in [7,9], it may happen that Palais Smale sequences, even if bounded in H 1 0 (Ω)-norm, do not admit any subsequence which converges strongly in this norm. And there is no way to prevent the interaction between the area where Ψ loses strict convexity and the values of ∇u.…”

“…Remark 1.3. Arguing as in section 2 of [9] we can deduce the following properties for Ψ, up to modifying the constant µ :…”