Existence results for a supercritical Neumann problem with a convex–concave non-linearity

Abstract

We shall consider the following semi-linear problem with a Neumann boundary conditionwhere B 1 is the unit ball in R N , N ≥ 2, a, b are nonnegative radial functions, and p, q are distinct numbers greater than or equal to 2. We shall assume no growth condition on p and q. Our plan is to use a new variational principle that allows one to deal with problems with supercritical Sobolev non-linearities. Indeed, we first find a critical point of the Euler-Lagrange functional associated with this equation over a suitable closed and convex set. Then we shall use this new variational principle to deduce that the restricted critical point of the Euler-Lagrange functional is an actual critical point. Keywords Semi-linear elliptic problems • Calculus of variations • Variational principles Mathematics Subject Classification 35J15 • 58E30 Abbas Moameni is pleased to acknowledge the support of the National Sciences and Engineering Research Council of Canada. Leila Salimi is supported by a Grant from IPM (No. 96470045).

Existence of radial weak solutions to Steklov problem involving Leray–Lions type operator

“…We say that the point u ∈ X is a critical point of I K , if DΦ(u) ∈ Ψ K (u) or equivalently, it satisfies the following inequality Notice that a global minimum point is a critical point. Definition 2.3 (Point-Wise Invariance Condition) The triple (Ψ, Φ, K) satisfies the point-wise invariance condition at a point u ∈ X if there exist a convex Gâteaux differentiable function G ∶ X → ℝ and a point v ∈ K such that Now, we bring a variational principle verified in [10] which is the main tool of this paper. Theorem 2.4 Let X be a reflexive Banach space and K be a convex and weakly closed convex subset of X.…”

Super-critical Neumann problems on unbounded domains

“…When t 2 * the situation is more delicate and additional conditions on D must be imposed. In this regard, there have been extensive new studies on super-critical Neumann problems, see [5,12,18,21,22] and references therein, where the domain is bounded and radial (this is by no means an exhaustive list). Motivated by the results proved in [5], Cowan and Moameni [6] considered the problem (3) where the nonlinearity is super-critical by assuming D is a bounded domain with certain symmetry conditions well beyond but including the radial domains.…”

“…For the convenience of the reader, by choosing the functions Φ, Ψ and the convex set K in lines with problems (1) and (2), we shall provide a proof to a particular case of theorem 1.5 applicable to these problems (please see theorem 2.2 and proposition 3.1). We would like to remark that theorem 1.5 has been successfully used to prove existence for several super-critical problems such as the well-known Ambrosetti-Brezis-Cerami concave-convex problem [14], and super-critical Neumann problems on radial and non-radial domains [6,18].…”