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We are interested in the existence of normalized solutions to the problem (- Δ) m u + μ | y | 2 m u + λ u = g (u) , x = (y , z) ∈ R K × R N - K , ∫ R N | u | 2 d x = ρ > 0 , in the so-called at least mass critical regime. We utilize recently introduced variational techniques involving the minimization on the L 2 -ball. Moreover, we find also a solution to the related curl–curl problem ∇ × ∇ × U + λ U = f (U) , x ∈ R N , ∫ R N | U | 2 d x = ρ , which arises from the system of Maxwell equations and is of great importance in nonlinear optics. [ABSTRACT FROM AUTHOR]

We look for solutions to the Schrödinger equation - Δ u + λ u = g (u) in R N coupled with the mass constraint ∫ R N | u | 2 d x = ρ 2 , with N ≥ 2 . The behaviour of g at the origin is allowed to be strongly sublinear, i.e., lim s → 0 g (s) / s = - ∞ , which includes the case g (s) = α s ln s 2 + μ | s | p - 2 s with α > 0 and μ ∈ R , 2 < p ≤ 2 ∗ properly chosen. We consider a family of approximating problems that can be set in H 1 (R N) and the corresponding least-energy solutions, then we prove that such a family of solutions converges to a least-energy one to the original problem. Additionally, under certain assumptions about g that allow us to work in a suitable subspace of H 1 (R N) , we prove the existence of infinitely, many solutions. [ABSTRACT FROM AUTHOR]

We look for least energy solutions to the cooperative systems of coupled Schrödinger equations - 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 ≥ 0 , where ρ i > 0 is prescribed and (λ i , u i) ∈ R × H 1 (R N) is to be determined, 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 (R N) of radii 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 -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 , N ∈ { 3 , 4 } , and G satisfies further assumptions, then u = (u 1 , u 2) is normalized, i.e., ∫ R N | u i | 2 d x = ρ i 2 for i ∈ { 1 , 2 } . [ABSTRACT FROM AUTHOR]

We look for ground states and bound states E : R 3 → R 3 to the curl–curl problem ∇ × (∇ × E) = f (x , E) in R 3 , which 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 : R → [ 0 , ∞) such that lim s → 0 Φ (s) / s 6 = lim s → + ∞ Φ (s) / s 6 = 0 . We prove the existence of a ground state, that is, 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. [ABSTRACT FROM AUTHOR]