Ground States of Time-Harmonic Semilinear Maxwell Equations in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^3}$$\end{document} with Vanishing Permittivity

We investigate the existence of solutions E:R3→R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${E:\mathbb{R}^3 \to \mathbb{R}^3}$$\end{document} of the time-harmonic semilinear Maxwell equation ∇×(∇×E)+V(x)E=∂EF(x,E)inR3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \times (\nabla \times E) + V(x) E = \partial_E F(x, E) \quad {\rm in} \mathbb{R}^3$$\end{document}where V:R3→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V:\mathbb{R}^3 \to \mathbb{R}}$$\end{document}, V(x)≦0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V(x) \leqq 0}$$\end{document} almost everywhere on R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^3}$$\end{document}, ∇×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\nabla \times}$$\end{document} denotes the curl operator in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^3}$$\end{document} and F:R3×R3→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F:\mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}}$$\end{document} is a nonlinear function in E. In particular we find a ground state solution provided that suitable growth conditions on F are imposed and the L3/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^{3/2}}$$\end{document} -norm of V is less than the best Sobolev constant. In applications, F is responsible for the nonlinear polarization and V(x)=-μω2ε(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V(x) = -\mu\omega^2 \varepsilon(x)}$$\end{document} where μ > 0 is the magnetic permeability, ω is the frequency of the time-harmonic electric field R{E(x)eiωt}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{R}\{E(x){\rm e}^{i\omega t}\}}$$\end{document} and ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon}$$\end{document} is the linear part of the permittivity in an inhomogeneous medium.


Introduction
We study the propagation of electromagnetic waves (E, B) in the absence of charges, currents and magnetization. The constitutive relations between the electric displacement field D and the electric field E as well as between the magnetic induction H and the magnetic field B are given by D = εE + P N L and H = 1 μ B, (1.1) where ε is the (linear) permittivity of an inhomogeneous material, and P N L stands for the nonlinear polarization which depends nonlinearly on the electric field E. In inhomogeneous media, ε and P N L depend on the position x ∈ R 3 and we assume that the magnetic permeability is constant μ > 0. As usual, the Maxwell equations ∇ × H = ∂ t D, div (D) = 0, ∂ t B + ∇ × E = 0, div (B) = 0, (1.2) together with the constitutive relations (1.1) lead to the equation (see Saleh and Teich [22]) In the time-harmonic case the fields E and P N L are of the form E(x, t) = {E(x) e iωt }, P N L (x, t) = {P(x)e iωt }, where E(x), P(x) ∈ R 3 and we arrive at the time-harmonic Maxwell equation where V (x) = −μω 2 ε(x) 0 and f (x, E) = μω 2 P(x, E). Here E : R 3 → R 3 is a vector field and V : R 3 → R. In a Kerr-like medium the strong electric field E of high intensity causes the refractive index to vary quadratically with the field and then the polarization has the form P N L = α(x) |E| 2 E, where |E| 2 stands for the time average of the intensity of E, hence P(x, E) = 1 2 α(x)|E| 2 E (see Nie [18] and Stuart [23]). In applications, for low intensity |E| the Kerr effect is often considered to be linear, P N L is negligible and therefore we may assume that P N L decays rapidly as |E| → 0. In order to model these nonlinear phenomena we consider nonlinearities of the form f (x, E) = (x) min{|E| p−2 , |E| q−2 }E, 2 < p q, (1.4) where ∈ L ∞ (R 3 ) is positive, periodic and bounded away from 0. Case p = 4 corresponds to the Kerr effect for the strong field E. In fact, we will able to deal with general nonlinearities of the form f (x, E) = ∂ E F(x, E), where F : R 3 ×R 3 → R.
Some other examples of nonlinearities in physical models can be found for example in Stuart [23] (see also Section 2). We look for weak solutions to (1.3) in a certain D(curl, p, q) space, where p and q are provided by the growth of f ; see Section 3 for details. Note that a solution E of (1.3) determines P N L and D by the first constitutive relation in (1.1) whereas B and H are obtained from ∇ × E by time-integration. We will show that if E ∈ D(curl, p, q) solves (1.3), then the total electromagnetic energy is finite. We do not know whether the fields E, D, B and H are localized, that is decay to zero as |x| → ∞, however D(curl, p, q) lies in the sum of Lebesgue spaces L p,q := L p (R 3 , R 3 ) + L q (R 3 , R 3 ) and therefore it does not contain the usual nontrivial travelling waves E propagating in a given direction z ∈ R 3 such that E(x) = E(x + z) for all x ∈ R 3 . The finiteness of the electromagnetic energy and the localization problem attract a strong attention in the study of self-guided beams of light in a nonlinear medium; see for example [23,24].
We restrict our considerations to optical metamaterials having permittivity ε close to zero, that is the so-called epsilon-near-zero (ENZ) media (see for example [2,12,15] and references therein). The ENZ materials exhibit strong nonlinear effects, for example the Kerr effect, governed by the polarization P N L and the propagation of time-harmonic electric field waves is described by (1.3). Our principal aim is to investigate the existence and the nonexistence of solutions to (1.3) under appropriate assumptions imposed on V and F. In particular, the closeness to zero of ε will be expressed in terms of L 3 2 -norm of V (see Section 2). Moreover ground state solutions which have the least possible energy among all nontrivial solutions will be of our major interest owing to their physical importance. It is worth mentioning that usually naturally occurring materials have the permittivity positive and bounded away from zero, that is V (x) = −μω 2 ε(x) is negative and bounded away from 0. However it is not clear in which space one should seek weak solutions of this problem with such V and a nonlinearity of the form (1.4), and whether any variational method can be used. We will show, in fact, that (1.3) does not admit classical solutions in case of constant and negative V ; see Corollary 2.5.
Recall that semilinear equations involving the the curl-curl operator ∇ ×∇ ×(·) in R 3 have been recently studied by Benci and Fortunato in [7]. They introduce a model for a unified field theory for classical electrodynamics which is based on a semilinear perturbation of the Maxwell equations. In the magnetostatic case, in which the electric field vanishes and the magnetic field is independent of time, they are lead to an equation of the form for the gauge potential A related to the magnetic field H = ∇ × A. Here F(A) = 1 2 W (|A| 2 ) is nonlinear in A. We emphasize that proof of the existence of solutions to (1.6) in [7] contains a gap and the techniques from [7] do not seem to be sufficient. Indeed, in order to deal with the lack of compactness issue they restrict the space of divergence-free vector fields to the radially symmetric ones, which becomes the null space. Finally in [1] Azzollini et al. use the cylindrical symmetry of the equation to find solutions of (1.6) of the form A field of this form is divergence-free and hence standard methods of nonlinear analysis apply. In D'Aprile and Siciliano [14] one finds another kind of cylindrical solution to the equation again using symmetry arguments and the scaling properties of (1.6). Observe that (1.3) cannot be treated neither by the Palais principle of symmetric criticality [19] nor by the rescaling arguments due to the presence of nonsymmetric and vanishing V , that is V ∈ L 3 2 (R 3 ). We would like to emphasize that we are also able to deal with functions F(x, E) that depend on x and are not radial in E. Therefore, we point out that the existence of ground states solutions of (1.3) with V = 0 will shed a new light on Equation (1.6) and on a new formulation of the Maxwell equations due to Born and Infeld [7,10].
Problem (1.3) has a variational structure and (weak) solutions correspond to critical points of the energy functional defined on a space D(curl, p, q) which will be introduced in Section 3. One difficulty from a mathematical point of view is that the curl-curl operator ∇ × ∇ × (·) has an infinite-dimensional kernel, namely all gradient vector fields. Moreover the functional E is unbounded from above and from below and its critical points have infinite Morse index. In addition to these problems related to the strongly indefinite geometry of E, we also have to deal with the lack of compactness issues. Namely functional E is not (sequentially) weak-to-weak * continuous, that is the weak convergence E n E in D(curl, p, q) does not imply that E (E n ) E (E) in D(curl, p, q) * (see the discussion preceding Corollary 5.3). Therefore we do not know whether a weak limit of a bounded Palais-Smale sequence is a critical point. Moreover the lack of the sufficient regularity of E makes this problem difficult to treat with the available variational methods for indefinite problems for example [4,6,21].
In order to find solutions to (1.3) we use a generalization of the Nehari manifold technique for strongly indefinite functionals obtained recently by Bartsch and the author in [5] (see also Szulkin and Weth [32,33]). Namely we introduce a Nehari-Pankov manifold (cf. [20]) which is homeomorphic with a sphere in the subspace of D(curl, p, q) consisting of divergence-free vector fields. This allows us to find a minimizing sequence on the sphere and hence on the Nehari-Pankov manifold. However in [5] we are in a position to find a limit point of the sequence being a critical point because the space of divergence-free vector fields on a bounded domain is compactly embedded into certain L p spaces and a variant of the Palais-Smale condition is satisfied. Since (1.3) is modelled in R 3 , the minimizing sequences are no longer compact. Therefore the critical point theory developed in [5,Section 4] is insufficient to find a solution to (1.3). Moreover the lack of the weak-to-weak * continuity of E makes this problem impossible to treat by a concentration-compactness argument in the spirit of Lions [16,17] in D(curl, p, q). Our approach is based on a new careful analysis of a bounded sequence (E n ) of the Nehari-Pankov manifold (Theorem 2.2) with a possibly infinite splitting (2.7) of the limit This result enables us to get the the weak-to-weak * continuity of E on the Nehari-Pankov manifold. Moreover, in the spirit of the global compactness result of Struwe [30,31] or Coti Zelati and Rabinowitz [13], we are able to find a finite splitting of the ground state level lim n→∞ E(E n ) with respect to a minimizing sequence (E n ) of the Nehari-Pankov manifold (Theorem 2.3). Finally, comparisons of energy levels will imply the existence of solutions to (1.3) (Theorem 2.1). The paper is organized as follows. In Section 2 we formulate our hypotheses on V and F, and we state our main results concerning the existence and the nonexistence of solutions and ground state solutions. In Section 3 we introduce the variational setting, in particular the spaces on which E will be defined. Moreover we provide the Helmholtz decomposition of a vector field E into the divergence-free component u and the curl-free component ∇w, that allows us to treat E as a functional J of two variables (u, w) [see (3.4) and Proposition 3.3]. Next, in Section 4 we introduce the Nehari-Pankov manifold on which we minimize J in order to find a ground state. In Section 5 we provide an analysis of bounded sequences in D(R 3 , R 3 ) and we obtain a splitting of a bounded sequence of the Nehari-Pankov manifold in Theorem 2.2. We investigate Palais-Smale sequences in Section 6 and we prove Theorem 2.3. Finally in Section 7 we prove Theorem 2.1 which states the existence of solutions and ground state solutions of (1.3) and we obtain a variational identity in Theorem 2.4 implying a nonexistence result Corollary 2.5.

Main Results
We impose on V : R 3 → R the following condition.
is the classical best Sobolev constant.
Here and in the sequel | · | q denotes the L q -norm. Now we collect assumptions on the nonlinearity F(x, u).
is uniformly strictly convex with respect to u ∈ R 3 , that is for any compact (F3) There are 2 < p < 6 < q and constants c 1 , c 2 > 0 such that for all x, u ∈ R 3 . (F4) For any x ∈ R 3 and u ∈ R 3 , u = 0 If in addition F(x, u) = F(x, v) then the strict inequality holds.
The periodicity arises in the study of dielectric materials, for example in photonic crystals and we assume it in (F1). The convexity condition (F2) is rather harmless (see examples below). Also, observe that condition (F4) is reminiscent of the Ambrosetti-Rabinowitz condition. The growth condition (F3) describes a supercritical behavior |u| q of F for |u| small and subcritical behavior |u| p for large |u|. Note that 6 = 2 * is the critical Sobolev exponent. This kind of growth has been considered for Schrödinger equations in the zero-mass case for example by Berestycki and Lions [9] or Benci et al. [8]. Moreover, similarly to [1,7,14] in the study of (1.6), condition (F3) requires to work in L p,q in order to ensure that the nonlinear term of energy functional (1.7) is finite; see Section 3 for details. The technical condition (F5) is a variant of the monotonicity condition for vector fields (see for example Szulkin and Weth [32]) and will be needed to set up the Nehari-Pankov manifold (cf. conditions (F1)-(F7) in [5]).
Our model examples are of the form Since M contains all nontrivial critical points of E, then a ground state solution is a nontrivial solution with the least possible energy E. Moreover we show that any E ∈ M admits the Helmholtz decomposition E = u + ∇w with u = 0 and div (u) = 0.
We provide a careful analysis of bounded sequences in M which plays a crucial role in proof of Theorem 2.1. Namely, setting (2.4) we get the following result.
As a consequence of Theorem 2.2 we get the sequentially weak-to-weak * continuity of E in M ∪ {0} (cf. Corollary 5.3). Moreover, in the spirit of the global compactness result of Struwe [30,31] or Coti Zelati and Rabinowitz [13], we obtain a finite splitting of energy levels with respect to a Palais-Smale sequence in M.

8)
where E 0 is the energy functional given by (1.7) under assumption V = 0.
Observe that if 0 < c < inf M 0 J 0 then N = 0, J (Ē 0 ) = c andĒ 0 is a nontrivial critical point of J . In this way the comparison of energy levels will imply the existence of nontrivial solutions.
Finally we provide a variational identity for an autonomous version of (1.3) and we get a corollary justifying to some extent the optimality of growth condition (F3).
Observe that for any 2 < p q the following growth condition is satisfied by nonlinearities given by (2.1), (2.2) and implies the first inequality in (F3). Now we formulate nonexistence results as a consequence of Theorem 2.4.
In particular, for the Kerr nonlinearity, that is p = q = 4 and f (x, E) = |E| 2 E there exist no classical solutions to (1.3) for constant V 0. Therefore example (1.4) with p = 4 and q > 6 incorporates the Kerr effect only for strong fields E in order to solve (1.3).

Variational Setting
Let 1 < p q and Recall that in L p,q we can introduce an equivalent norm and by [3, Proposition 2.5] the infimum in | · | p,q,1 is attained. Below we recall some properties of L p,q given for example in [3, Corollary 2.19, Proposition 2.21].
We show that the natural space for the energy functional E is where div E has to be understood in the distributional sense.
It is clear that W is linearly isometric to The following Helmholtz's decomposition holds.
∇W is a closed subspace of L p,q and cl U ∩ ∇W = {0} in L p,q . Moreover if p 6 q, then and the norms · D and · curl, p,q are equivalent on U.
Proof. Since W is a complete space, then clearly ∇W is a closed subspace of L p,q .
can be written as is the Newton potential of div (ϕ). Since ϕ has compact support, then for any u ∈ U. By the Sobolev embedding we have that U is continuously embedded in L 6 (R 3 , R 3 ) and by (3.1) also in L p,q . Therefore the norms · D and · curl, p,q are equivalent on U and by the density argument we get the decomposition (3.2).
Let us assume that (F1), (F3) and (V) hold. We introduce a norm in U × W by the formula and consider a functional J : U × W → R given by The next Lemma 3.4(a) and [3,Corollary 3.7] imply that E : U ⊕ ∇W → R and J : U × W → R are well defined and of class C 1 with for any (u, w), (φ, ψ) ∈ U × W. Thus we get the following observation.
and the electromagnetic energy (1.5) is finite for all t.
Proof. The first equivalence follows from Lemma 3.2 and the above discussion. Let E = u +∇w be a critical point of E and ϕ ∈ C ∞ 0 (R 3 , R 3 ). We find a decomposition ϕ = ϕ 1 + ∇ϕ 2 with ϕ 1 ∈ U, ϕ 2 ∈ W according to (3.3) and observe that Clearly if E = u +∇w ∈ U ⊕∇W is a weak solution, then by the density argument we have E (E) = 0. Now observe that At the end of this section we collect some helpful inequalities.
Observe that by the Hölder inequality (b) Note that by (F3) and by Lemma 3.1(a) Then it is enough to observe the following inequalities
In view of Lemma 3.1(a), for any ε > 0 there is n 0 ∈ N and δ > 0 such that for any with | | < δ the following inequality holds for any n n 0 . Thus ( f (x, E n − E +t E), E ) n is uniformly integrable. Moreover for any ε > 0 there is n 0 ∈ N and ⊂ R 3 with | | < +∞ such that for for any n n 0 Proof of Lemma 4.2. We show that (up to a subsequence) E n (x) → E(x) almost everywhere on R 3 . Since I (E n ) → I (E) by lower semicontinuity we have . Thus E n → E almost everywhere on R 3 . Assume that F is uniformly strictly convex in u ∈ R 3 (see (F2)). Then for any 0 < r R m := inf Observe that by the convexity of F in u ∈ R 3 0 lim sup Therefore setting and thus μ( n ) → 0 as n → ∞. Since 0 < r R are arbitrary chosen, we deduce E n → E almost everywhere on R 3 .

In view of Lemma 4.3 we obtain
By Lemma 3.4(b) we get |E n − E| p,q → 0. Now we are able to apply the critical point theory on the Nehari-Pankov manifold developed in [5,Section 4]. Namely we get the following result. Moreover the following condition holds.
(A4) There exists r > 0 such that inf u D =r J (u, 0) > 0. We prove the following condition.

Indeed, in view of Lemma 3.4(c) and by (F3) for any
(B2) I(t n (u n , w n ))/t 2 n → ∞ if t n → ∞ and u n → u for some u = 0 as n → ∞.

Observe that by Lemma 3.4 (b)
F(x, t n (u n + ∇w n )) dx c 1 min{|t n u n + ∇w n | p p,q , |t n u n + ∇w n | q p,q } c 1 t 2 n min{t p−2 n |u n + ∇w n /t n | p p,q , t q−2 n |u n + ∇w n /t n | q p,q }.
If lim inf n→∞ |u n + ∇w n /t n | p,q = 0 as n → ∞, then passing to a subsequence we get |u + ∇(w n /t n )| p,q → 0.
Hence we get a contradiction to the assumption u = 0. Therefore |u n +∇w n /t n | p,q is bounded away from 0 and I(t n (u n , w n ))/t 2 n → ∞ as n → ∞. Finally the arguments provided in proof of Proposition 4.1 show that: (B3) t 2 −1 2 I (u, w), (u, w) + t I (u, w), (0, ψ) + I(u, w) − I(tu, tw + ψ) < 0 for any t 0, u ∈ U and w, ψ ∈ W such that (tu, tw + ψ) = (u, w). Since there is no compact embedding of U into L p,q , the critical point theory provided in [5,Section 4] is not sufficient to show that c = inf N J is achieved by a critical point of J . Therefore in the next Section 5 we provide an analysis of bounded sequences in D(R 3 , R 3 ) and of bounded sequences of the Nehari-Pankov manifold.

Analysis of Bounded Sequences
We need further properties of I.

t (u)u, w(t (u)u)).
Proof. (a) Let u ∈ L p,q . Since W w → I(u, w) ∈ R is continuous, strictly convex and coercive, then there exists a unique w(u) ∈ W such that (5.1) holds. We show that the map w : L p,q → W is continuous. Let u n → u in L p,q . Since 0 I(u n , w(u n )) I(u n , 0) (5.2) we obtain that w(u n ) is bounded and we may assume that w(u n ) w 0 for some w 0 ∈ W. Observe that by the (sequentially) lower semi-continuity of I we get I(u, w(u)) I(u, w 0 ) lim inf n→∞ I(u n , w(u n )) lim inf n→∞ I(u n , w(u)) = I(u, w(u)).

In view of Proposition 4.4(a) we get m(u) = (t (u)u, w(t (u)u).
Below we analyse a bounded sequence (u n ) in D(R 3 , R 3 ) and provide a possibly infinite splitting of lim n→∞ I(u n , w(u n )).
such that x 0 n = 0 and the following conditions hold: loc and almost everywhere in R 3 for any 0 i < N + 1, ∇w(ū 0 ) and ∇w 0 (u n )(· + x i n ) ∇w 0 (ū i ) in L p,q for any 1 i < N + 1, (f) ∇w(u n ) → ∇w(ū 0 ) and ∇w 0 (u n )(· + x i n ) → ∇w 0 (ū i ) in L p,q loc and almost everywhere in R 3 for any 1 i < N + 1, If N < ∞ then we takeū i = 0 for i > N . If N = ∞ then the above conditions hold for any i 0. Observe that we may assume that (x i n ) n i ⊂ Z 3 . Hence the local convergence in (c) follows directly from (5.3). Moreover the boundedness of (w(u n )) n∈N and (w 0 (u n )) n∈N in W implies that we may assume Observe that (a), (e)-(h) are a consequence of the following claims and the almost everywhere convergence in (c) and (f) follows from the local convergence in L p,q (see [3,Prop. 2.8]). (B(0, 1), R 3 ). Therefore, up to a subsequence, u n (· + x i+1 n ) →ū i+1 in L 2 (B(0, 1), R 3 ) and then Indeed, observe that Lemma 3.4(b), the weak lower semicontinuity of I 0 and conditions (b), (5.7) imply that 2 ) ) I 0 (u n , w 0 (u n )) for any k ∈ N. By Lemma 5.1(b) we obtain that (I 0 (u n , w 0 (u n ))) n∈N is bounded. Therefore, up to a subsequence, (5.8) holds.
Note that similarly as in Lemma 3.4(a) we obtain then we get (5.9).

Claim 4.
lim sup Let v 0 := ∇w(ū 0 ) and for i 1 Note that for given 0 j n Let ε > 0 and observe that by (5.8) there is n 0 1 such that Then for sufficiently large n n i=0 as n → ∞. Moreover similarly we show that as n → ∞. Therefore from (5.11), (5.9) we obtain (5.10).

Claim 6. (g) holds.
From (c) and ( f ) we know that for any i 0 and thus lim n→∞ I(u n , w(u n )) = I(ū 0 , w(ū 0 )) + lim for n ∈ N, 0 j < N + 1. Again by Lemma 4.3 and then lim Similarly we show for any 0 and then Thus we obtain In general J is not (sequentially) weak-to-weak * continuous. Indeed, take for example F(x, u) = 1 p ((1 + |u| q ) p q − 1), and observe that ∇w n ∇w in L p,q does not imply However we show the weak-to-weak * continuity of J for sequences on the Nehari-Pankov manifold N . Obviously the same regularity holds for E and M.
Proof. Observe that by Propositions 4.1, 4.4(a) and Lemma 5.1(c) we get w n = w(u n ). In view of Lemma 5.2(c) and (f) we may assume that u n +∇w n → u 0 +∇w 0 almost everywhere on R 3 . Observe that for In view of the Vitaly convergence theorem we obtain

Analysis of Palais-Smale Sequences in N
The following lemma implies that any Palais-Smale sequence of J in N is bounded.
Proof. Suppose that (u n , w n ) ∈ N , (u n , w n ) → ∞ as n → ∞ and J (u n , w n ) M for some constant M > 0. Let u n := u n (u n , w n ) .
In view of Lemma 5.2(c) we may assume thatū n ū 0 in U andū n →ū 0 almost everywhere in R 3 . Moreover there is a sequence (x n ) n∈N ⊂ R 3 such that If lim inf n→∞ |∇w n | p,q = 0 then, up to a subsequence, |∇w n | p,q → 0, and for sufficiently large n we get If lim inf n→∞ |∇w n | p,q > 0 then there is C 2 ∈ (0, 1) such that for sufficiently large n Therefore, passing to a subsequence if necessary, for any s 0. The obtained contradiction shows that (6.1) holds. Then we may assume that (x n ) ⊂ Z 3 and lim inf n→∞ B(0,r ) |ū n (x + x n )| 2 dx > 0 for some r > 1, henceū n (· + x n ) →ū 0 in L 2 loc (R N ) for someū 0 = 0. Take any bounded ⊂ R 3 of positive measure such that Observe that for any x ∈ |u n (x + x n )| = |ū n (x + x n )| · (u n , w n ) → ∞ and by Fatou's lemma as n → ∞. Since norms | · | p,q and | · | p are equivalent on L p ( , R 3 ) (see [3,Corollary 2.15]), then the periodicity of F in x, Lemma 3.4(b) and (6.4) imply fore some constant C 3 > 0. Thus by (6.5) we get as n → ∞ and the obtained contradiction completes proof.
Lemma 6.2. If E ∈ L p,q and x n ∈ R 3 is such that |x n | → +∞ as n → +∞, then Proof. Observe that for any R > 0. Therefore and we get the conclusion by taking R → +∞. Lemma 6.3. Let J 0 : U × W → R be the functional given by F(x, u + ∇w) dx. (6.6) for (u, w) ∈ U × W. Let (u n , w n ) ∈ N be a (P S) c -sequence for some c > 0. Then there is N 0 and there are sequences such that x 0 n = 0 and, up to a sequence, Proof.
Step 1. Construction of (ū i ,w i ), (x i n ) n i and proof of (6.7). Since (u n , w n ) ∈ N then by Propositions 4.1, 4.4(a) and Lemma 5.1 m(u n ) = (u n , w n ) and w n = w(u n ).
In view of Lemma 6.1 (u n , w n ) is bounded in U × W. Thus we may assume that Step 2. J 0 (ū i ,w i ) = 0 for 1 i < N + 1. From (b) and (e) of Lemma 5.2 and arguing as in Corollary 5.3 we obtain for any (φ, ψ) ∈ U × W. On the other hand |V (x + y n )||φ + ∇ψ| 2 dx 1 2 and by Lemma 6.2 we get Step for any t 0. Thus Note that by (5.8) (ū i + ∇w i ) i 1 is bounded and if, up to a subsequenceū i → 0 in L p,q , then which contradicts (6.13). Therefore Step 4. N < ∞ and proof of (6.8), (6.9) and (6.11).
Observe that for some constant C 1 > 0 and for any k 1 where the last inequalities follows from the fact that B(x i n , n−2 and taking into account Step 3 we obtain thatū i = 0 for finitely many i 1. Thus N < ∞ and (6.8), (6.9), (6.11) follow from Step 2, Step 3 and Lemma 5.2(g).
Step 6. Proof of (6.12). Since N < ∞ and Lemma 5.2(h) holds, then we need to prove the following convergence (6.14) Note that Proof of Theorem 2.3. Proof follows directly from Lemma 6.3 by decomposing E n = u n +∇w n , where (u n , w n ) ∈ N and by takingĒ i =ū i +∇w i for 0 i N .

Proofs of Theorems 2.1 and 2.4
Now we are ready to prove the existence and nonexistence results. Note that any critical point (ū,w) of J 0 such thatū = 0 belongs to N 0 and hence J 0 (ū,w) c 0 > 0.
Since ∇ϕ n (x) = 0 for |x| < n 2 , then by the Lebesgue dominated theorem we get which completes the proof.
Proof of Corollary 2.5. Suppose that V = 0 and E = u+∇w is a classical solution to (7.3) with u = 0 and p > 6 or q < 6. Then by (2.11) Therefore R 3 F(E) dx = 0 and E = 0 almost everywhere on R 3 . Thus u = 0 and we obtain a contradiction. If V (x) = V 0 < 0 is constant and E = u + ∇w is a classical solution to (1.3) with u = 0 and q 6, then by Theorem 2.4 and, similarly to the above, we get a contradiction.