By Murnaghan F. D.

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**Example text**

29) Since uε →0 in H 1 (Ω) it follows that, in order to deduce our claim, it is enough to apply the Arzel`a–Ascoli theorem, after observing that uε C 1,α ≤C. This uniform estimate follows by a standard bootstrap argument. Indeed, assuming that uε ∈ Lq1 (Ω) for some q1 ≥ 2N/(N − 2), then f (x, uε ) ∈ Lr (Ω), where r = q1 /p. Thus, by Schauder elliptic estimates, uε ∈ W 2,r (Ω) ⊂ Lq2 (Ω), with 1 p 2 = − , q2 q1 N provided that the right hand-side is positive. In the remaining case we have uε ∈ Lq (Ω) for any q > 1.

In the sequel, we are interested to find conditions in order to have Σ ∩ Σ⊥ = {0}, that is, (PSC) to be verified. There are two ways to achieve this purpose, the so-called “compact” as well as the “isometric” cases. We are dealing now with the first one whose original form has been given by Palais [232]. 25 Let G be a compact topological group and the representation π of G over X is continuous, that is, (g, u) → gu is a continuous function G × X into X. Then (PSC) holds. 41) G where dg is the normalized Haar measure on G.

Lions [98] observed that defining the closed convex cone K = {u ∈ H01 (ω × R) : u is nonnegative, y → u(x, y) is nonincreasing for x ∈ ω, y ≥ 0, and y → u(x, y) is nondecreasing for x ∈ ω, y ≤ 0}, (K) the bounded subsets of K are relatively compact in Lp (ω × R) whenever p ∈ (2, 2∗ ). Note that 2 = ∞, if m = 1. The main goal of this section is to give a new approach to treat elliptic (eigenvalue) problems on domains of the type Ω = ω × R. The genesis of our method relies on the Szulkin-type functionals.