Abstract
We discuss and prove existence of multiple solutions for critical elliptic systems in potential form on compact Riemannian manifolds.
Highlights
Let (M, g) be a smooth, compact Riemannian n-manifold, n ≥ 3
Mps(R) is the vector space of p × p real matrices S = (Sij) which are such that Sij = Sji for all i, j
Let (M, g) be a smooth, compact Riemannian manifold of dimension n ≥ 3, let p ≥ 1 be a natural number, and let A be a smooth map from M to Mps (R) such that the operator ∆pg + A is coercive on H12,p (M )
Summary
Let (M, g) be a smooth, compact Riemannian n-manifold, n ≥ 3. Following the very nice construction in Clapp–Weth [10], we prove the existence of such a map with k = n + 1, see Section 7, as soon as we can prove the existence of a two-parameters family of test functions Ux,ε, x ∈ Ω and ε > 0, Ux,ε depending continuously on x, such that (i) IA,g(WUx,ε) < K−n/n uniformly in x as ε → 0, (ii) Supp Ux,ε ⊂ Bx(ε) for all x and all ε > 0, where WUx,ε is as in (1.7), Ω is an open subset of M , Supp Ux,ε stands for the support of Ux,ε, and Bx(ε) is the ball in M of radius ε centered at x With such a test function reduction, which extends classical existence conditions of Aubin’s type [4], Theorem 1.1 provides several examples of systems like (1.1)–(1.2) with multiplicity of solutions. We mention the reference Vetois [44] for closely related developments
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