New asymptotically quadratic conditions for Hamiltonian elliptic systems

This paper is concerned with the following Hamiltonian elliptic system − Δ u + V ( x ) u = W v ( x , u , v ) , x ∈ R N , − Δ v + V ( x ) v = W u ( x , u , v ) , x ∈ R N , $$ \left\{ \begin{array}{ll} -\Delta u+V(x)u=W_{v}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},\\ -\Delta v+V(x)v=W_{u}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},\\ \end{array} \right. $$ where z = (u, v) : ℝ N → ℝ2, V(x) and W(x, z) are 1-periodic in x. By making use of variational approach for strongly indefinite problems, we obtain a new existence result of nontrivial solution under new conditions that the nonlinearity W ( x , z ) := 1 2 V ∞ ( x ) | A z | 2 + F ( x , z ) $ W(x,z):=\frac{1}{2}V_{\infty}(x)|Az|^2+F(x, z) $ is general asymptotically quadratic, where V ∞(x) ∈ (ℝ N , ℝ) is 1-periodic in x and infℝ N V ∞(x) > minℝ N V(x), and A is a symmetric non-negative definite matrix.

Multiple Solutions to a Nonlinear Curl–Curl Problem in $${\mathbb {R}}^3$$

We look for ground states and bound states $$E:{\mathbb {R}}^3\rightarrow {\mathbb {R}}^3$$E:R3→R3 to the curl–curl problem $$\begin{aligned} \nabla \times (\nabla \times E)= f(x,E) \qquad \text { in } {\mathbb {R}}^3, \end{aligned}$$∇×(∇×E)=f(x,E)inR3,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 $$\Phi :{\mathbb {R}}\rightarrow [0,\infty )$$Φ:R→[0,∞) such that $$\displaystyle \lim _{s\rightarrow 0}\Phi (s)/s^6=\lim _{s\rightarrow +\infty }\Phi (s)/s^6=0$$lims→0Φ(s)/s6=lims→+∞Φ(s)/s6=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 $${\mathbb {R}}^3$$R3 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.

An Abstract Linking Theorem Applied to Indefinite Problems Via Spectral Properties

An abstract linking result for Cerami sequences is proved without the Cerami condition. It is applied directly in order to prove the existence of critical points for a class of indefinite problems in infinite-dimensional Hilbert Spaces. The applications are given to Schrödinger equations. Here spectral properties inherited by the potential features are exploited in order to establish a linking structure, and hence hypotheses of monotonicity on the nonlinearities are discarded.