We look for ground states and bound states E : R 3 → R 3 to the curl-curl problemwhich originates from nonlinear Maxwell equations. The energy functional associated with this problem is strongly indefinite due to the infinite dimensional kernel of ∇ × (∇ × ·).The growth of the nonlinearity f is controlled by an N -function Φ :We prove the existence of a ground state, i.e. a least energy nontrivial solution, and the existence of infinitely many geometrically distinct bound states. We improve previous results concerning ground states of curl-curl problems. Multiplicity results for our problem have not been studied so far in R 3 and in order to do this we construct a suitable critical point theory. It is applicable to a wide class of strongly indefinite problems, including this one and Schrödinger equations. 2 2010 Mathematics Subject Classification. Primary: 35Q60; Secondary: 35J20, 78A25.In general J ′ is not (sequentially) weak-to-weak * continuous, however we show the weakto-weak * continuity of J ′ for sequences on the topological manifold M. Obviously, the same regularity holds for E ′ and M E .for any (φ, ψ) ∈ V × W.
We look for least energy solutions to the cooperative systems of coupled Schrödinger equations $$\begin{aligned} \left\{ \begin{array}{l} -\Delta u_i + \lambda _i u_i = \partial _iG(u)\quad \mathrm {in} \ {\mathbb {R}}^N, \ N \ge 3,\\ u_i \in H^1({\mathbb {R}}^N), \\ \int _{{\mathbb {R}}^N} |u_i|^2 \, dx \le \rho _i^2 \end{array} \right. i\in \{1,\ldots ,K\} \end{aligned}$$ - Δ u i + λ i u i = ∂ i G ( u ) in R N , N ≥ 3 , u i ∈ H 1 ( R N ) , ∫ R N | u i | 2 d x ≤ ρ i 2 i ∈ { 1 , … , K } with $$G\ge 0$$ G ≥ 0 , where $$\rho _i>0$$ ρ i > 0 is prescribed and $$(\lambda _i, u_i) \in {\mathbb {R}}\times H^1 ({\mathbb {R}}^N)$$ ( λ i , u i ) ∈ R × H 1 ( R N ) is to be determined, $$i\in \{1,\dots ,K\}$$ i ∈ { 1 , ⋯ , K } . Our approach is based on the minimization of the energy functional over a linear combination of the Nehari and Pohožaev constraints intersected with the product of the closed balls in $$L^2({\mathbb {R}}^N)$$ L 2 ( R N ) of radii $$\rho _i$$ ρ i , which allows to provide general growth assumptions about G and to know in advance the sign of the corresponding Lagrange multipliers. We assume that G has at least $$L^2$$ L 2 -critical growth at 0 and admits Sobolev critical growth. The more assumptions we make about G, N, and K, the more can be said about the minimizers of the corresponding energy functional. In particular, if $$K=2$$ K = 2 , $$N\in \{3,4\}$$ N ∈ { 3 , 4 } , and G satisfies further assumptions, then $$u=(u_1,u_2)$$ u = ( u 1 , u 2 ) is normalized, i.e., $$\int _{{\mathbb {R}}^N} |u_i|^2 \, dx=\rho _i^2$$ ∫ R N | u i | 2 d x = ρ i 2 for $$i\in \{1,2\}$$ i ∈ { 1 , 2 } .
We look for least energy solutions to the cooperative systems of coupled Schrödinger equations with G ≥ 0, where ρ i > 0 is prescribed andOur approach is based on the minimization of the energy functional over a linear combination of the Nehari and Pohožaev constraints intersected with the product of the closed balls in L 2 (R N ) of radii ρ i , which allows to provide general growth assumptions on G and to know in advance the sign of the corresponding Lagrange multipliers. We assume that G has at least L 2 -critical growth at 0 and Sobolev subcritical growth at infinity. The more assumptions we make on G, N , and K, the more can be said about the minimizers of the energy functional. In particular, if K = 2, N ∈ {3, 4}, and G satisfies further assumptions, then u = (u 1 , u 2 ) is normalized, i.e., R N |u i | 2 dx = ρ 2 i for i ∈ {1, 2}. J(u)
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