Abstract. We consider the stationary nonlinear magnetic Choquard equationwhere A is a real valued vector potential, V is a real valued scalar potential, N ≥ 3, α ∈ (0, N ) and 2 − (α/N ) < p < (2N − α)/(N − 2). We assume that both A and V are compatible with the action of some group G of linear isometries of R N . We establish the existence of multiple complex valued solutions to this equation which satisfy the symmetry conditionwhere τ : G → S 1 is a given group homomorphism into the unit complex numbers.MSC2010: 35Q55, 35Q40, 35J20, 35B06.
Abstract. We consider the problemwhere Ω is an exterior domain inUnder symmetry assumptions on Ω and W, which allow finite symmetries, and some assumptions on the decay of W at infinity, we establish the existence of a positive solution and multiple sign changing solutions to this problem, having small energy.
We establish the existence of a positive fully nontrivial solution (u, v) to the weakly coupled elliptic systemwhere N ≥ 4, 2 * := 2N N−2 is the critical Sobolev exponent, α, β ∈ (1, 2], α + β = 2 * , µ 1 , µ 2 > 0, and λ < 0. We show that these solutions exhibit phase separation as λ → −∞, and we give a precise description of their limit domains.If µ 1 = µ 2 and α = β, we prove that the system has infinitely many fully nontrivial solutions, which are not conformally equivalent.Pitaevskii equations. This type of systems arises, e.g., in the Hartree-Fock theory for double condensates, that is, Bose-Einstein condensates of two different hyperfine states which overlap in space; see [9]. The sign of µ i reflects the interaction of the particles within each single state. If µ i is positive, this interaction is attractive. The sign of λ, on the other hand, reflects the interaction of particles in different states. This interaction is attractive if λ > 0 and it is repulsive if λ < 0. If the condensates Date: September 29, 2018. M. Clapp was partially supported by CONACYT grant 237661 (Mexico) and UNAM-DGAPA-PAPIIT grant IN104315 (Mexico). A. Pistoia was partially supported by Fondi di Ateneo "Sapienza" Universitá di Roma (Italy).1 2 * 2 for all λ > 0 if N ≥ 5 and for a wide range of λ > 0 if N = 4; see [5,6]. Peng, Peng and Wang [20] studied the system for µ 1 = µ 2 = 1, λ = 1 2 * and different values of α and β, and they obtained uniqueness and nondegeneracy results for positive synchronized solutions. Guo, Li and Wei studied the critical system (1.1) in dimension N = 3 for λ < 0 and they established the existence of positive solutions with k peaks for k sufficiently large in [12]. In [10,11] Gladiali, Grossi and Troestler obtained radial and nonradial solutions to some critical systems using bifurcation methods.Here we focus our attention to the competitive case, i.e., to λ < 0. In this case, the system (1.1) does not have a least energy fully nontrivial solution; see Proposition 2.2 below. This behavior showcases the lack of compactness of the variational functional, which comes from the fact that system is invariant under
We consider the problem -Delta u = vertical bar u vertical bar(2*-2) u in Omega, u = 0 on partial derivative Omega, where Omega is a bounded smooth domain in R-N, N >= 3, and 2* = 2N/N- 2 is the critical Sobolev exponent. We assume that Omega is annular shaped, i.e. there are constants R-2 > R-1 > 0 such that {x epsilon R-N : R-1 < vertical bar x vertical bar < R-2} subset of Omega and 0 is not an element of Omega. We also assume that Omega is invariant under a group Gamma of orthogonal transformations of R-N without fixed points. We establish the existence of multiple sign changing solutions if, either Gamma is arbitrary and R-1/R-2 is small enough, or R-1/R-2 is arbitrary and the minimal Gamma-orbit of Omega is large enough. We believe this is the first existence result for sign changing solutions in domains with holes of arbitrary size. The proof takes advantage of the invariance of this problem under the group of Mobius transformations
Let X be a Hilbert space and , # C 1 (X, R) be strongly indefinite. Assume in addition that a compact Lie group G acts orthogonally on X and that , is invariant. In order to find critical points of , we develop the limit relative category of Fournier et al. in the equivariant context. We use this to prove two generalizations of the symmetric mountain pass theorem and a linking theorem. In the case of the mountain pass theorem the mountain range is allowed to lie in a subspace of infinite codimension. Also other conditions of the classical symmetric mountain pass theorem for G=ZÂ2 (due to Ambrosetti and Rabinowitz) can be weakened considerably. For example, we are able to deal with infinite-dimensional fixed point spaces. The proofs consist of a direct reduction to a relative Borsuk Ulam type theorem. This provides a new proof even for the classical mountain pass theorem. The abstract critical point theorems are applied to an elliptic system with Dirichlet boundary conditions. We only need a weak version of the usual superquadraticity condition. The linking theorem can be applied to asymptotically linear Hamiltonian systems which are symmetric with respect to a (generalized) symplectic group action.
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