We consider the equation

where Q_ε takes the value 1 on each ball B_ε(y), , and the value −1 elsewhere. We establish the existence of a least energy solution for each and show that their H¹ and L^p norms concentrate locally at points of as .

We find a normalized solution u=(u1,…,uK) to the system of K coupled nonlinear Schrödinger equations (Formula presented.) where ρ=(ρ1,…,ρK)∈(0,∞)K is prescribed, (λ,u)∈RK×H1(R3)K are the unknown and 4≤p<6. In the case of two equations we show the existence of multiple solutions provided that the coupling is sufficiently large. We also show that for negative coupling there are no ground state solutions. The main novelty in our approach is that we use the Cwikel-Lieb-Rozenblum theorem in order to estimate the Morse index of a solution as well as a Liouville-type result in an exterior domain.

In this paper, we consider a competitive weakly coupled elliptic system in which each species is attracted to a small region in and repelled from its complement. In this setting, we establish the existence of infinitely many solutions and of a nonnegative least energy solution. We show that, as the regions of attraction shrink, least energy solutions of the system concentrate. We study this behavior and characterize their limit profile. In particular, we show that if each component of a least energy solution is attracted to a different region, then the components decouple in the limit, whereas if all the components are attracted to the same region, they remain coupled.

We consider a weakly coupled singularly perturbed variational elliptic system in a bounded smooth domain with Dirichlet boundary conditions. We show that, in the competitive regime, the number of fully nontrivial solutions with nonnegative components increases with the number of equations. Our proofs use a combination of four key elements: a convenient variational approach, the asymptotic behavior of solutions (concentration), the Lusternik-Schnirelman theory, and new estimates on the category of suitable configuration spaces.

We establish the existence of positive normalized (in the $L^2$ sense) solutions to non-variational weakly coupled elliptic systems of $\ell$ equations. We consider couplings of both cooperative and competitive type. We show the problem can be formulated as an operator equation on the product of $\ell$ $L^2$-spheres and apply a degree-theoretical argument on this product to obtain existence.

We establish the existence of a nonnegative fully nontrivial solution to a non-variational weakly coupled competitive elliptic system. We show that this kind of solutions belong to a topological manifold of Nehari-type, and apply a degree-theoretical argument on this manifold to derive existence.

We study the elliptic system

where Ω is a bounded domain in RN , N ≥ 3, κ1, κ2 ∈ R, µ1, µ2, λ > 0, α, β > 1, and α + β = p ≤ 2∗ := 2N /(N − 2). For p ∈ (2, 2∗) we establish the existence of a ground state and of a prescribed number of fully nontrivial solutions to this system for λ sufficiently large. If p = 2∗ and κ1, κ2 > 0 we establish the existence of a ground state for λ sufficiently large if, either N ≥ 5, or N = 4 and neither κ1 nor κ2 are Dirichlet eigenvalues of −∆ in Ω.

Let Omega subset of R-3 be a Lipschitz domain and let S-curl(Omega) be the largest constant such that integral(R3) vertical bar del x u vertical bar(2) dx >= S-curl(Omega) inf (w is an element of W06(curl;R3)del xw=0) (integral(R3) vertical bar u + w vertical bar(6) dx)(1/3) for any u in W-0(6) (curl; Omega) subset of W-0(6) (curl; R-3), where W-0(6) (curl; Omega) is the closure of C-0(infinity) (Omega, R-3) in {u is an element of L-6(Omega, R-3) : del x u is an element of L-2(Omega, R-3)} with respect to the norm (vertical bar u vertical bar (2)(6) + vertical bar del x u vertical bar(2)(2))(1/2). We show that S-curl(Omega) is strictly larger than the classical Sobolev constant S in R-3. Moreover, S-curl(Omega) is independent of Omega and is attained by a ground state solution to the curl-curl problem del x (del x u) = vertical bar u vertical bar(4)u if Omega = R-3. With the aid of these results we also investigate ground states of the Brezis-Nirenberg-type problem for the curl-curl operator in a bounded domain Omega del x (del x u) + lambda u = vertical bar u vertical bar(4)u in Omega, with the so-called metallic boundary condition nu x u = 0 on partial derivative Omega, where nu is the exterior normal to partial derivative Omega.

We study an optimal $M$-partition problem for the Yamabe equation on the round sphere, in the presence of some particular symmetries. We show that there is a correspondence between solutions to this problem and least energy sign-changing symmetric solutions to the Yamabe equation on the sphere with precisely $M$ nodal domains.

The existence of an optimal partition is established through the study of the limit profiles of least energy solutions to a weakly coupled competitive elliptic system on the sphere.

We look for ground states and bound states  $E:\mathbb{R}^3\to\mathbb{R}^3$ to the  curl-curl problem $$\nabla\times(\nabla\times E)= f(x,E) \qquad\textnormal{ in } \mathbb{R}^3$$ which originates from nonlinear Maxwell equations.

The energy functional associated with this problem is strongly indefinite due to the infinite dimensional kernel of $\nabla\times(\nabla\times \cdot)$. The growth of the nonlinearity $f$ is controlled by an $N$-function $\Phi:\mathbb{R}\to [0,\infty)$ such that $\displaystyle\lim_{s\to 0}\Phi(s)/s^6=\lim_{s\to+\infty}\Phi(s)/s^6=0$. 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 $\mathbb{R}^3$ 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 Schr\"odinger equations.