Nonlinear Maxwell equations a variational approach Version of May 16, 2018

Transcription

1 Nonlinear Maxwell equations a variational approach Version of May 16, 018 Course at the Karlsruher Institut für Technologie Jarosław Mederski Institute of Mathematics of the Polish Academy of Sciences We are interested in the propagation of electromagnetic waves is described by the Maxwell equations for the electric field E, the electric displacement field D, the magnetic field H, and the magnetic induction B. These are time-dependent vector fields in a domain R 3. Given the current intensity J and the scalar charge density ρ, the Maxwell equations in differential form are as follows: t B + E = 0 (Faraday s Law) H = J + t D (Ampere s Law) div (D) = ρ div (B) = 0 (Gauss Electric Law) (Gauss Magnetic Law). These fields are related by constitutive equations determined by the material. The relation between the electric displacement field and the electric field is given by D = εe + P NL (x, E) where ε = ε(x) R 3 3 is the (linear) permittivity tensor of the material, and P NL is the nonlinear part of the polarization. The relation between magnetic field and magnetic induction is B = µh M where µ = µ(x) R 3 3 denotes the magnetic permeability tensor and M the magnetization of the material. In a linear medium one has P NL = 0 leading to the linear Maxwell equations. Suppose there are no currents, charges nor magnetization, i.e. J = 0, ρ = 0, M = 0. Then multiplying Faraday s law with µ 1, taking the curl and using the constitutive relations and Ampere s law leads to the nonlinear electromagnetic wave equation of the form (0.1) (µ(x) 1 E) + ε(x) t E + t P NL (x, E) = 0 for the electric field E. Solving this one obtains D = εe + P NL (x, E) by the constitutive relation and B by time integrating Faraday s law. Finally H = µ 1 B is also determined by the constitutive relation. Equation (0.1) is particularly challenging and in the literature there are several simplifications relying on approximation of the nonlinear electromagnetic wave equation. The most prominent one is the scalar or vector nonlinear Schrödinger equation. In order to justify this approximation one assumes that the term (div (E)) in ( E) = (div (E)) E is 1

2 negligible and can be dropped, and that one can use the so-called slowly varying envelope approximation. However, this approach may produce non-physical solutions and our goal is to find exact solutions of the Maxwell equations and develop analytical tools which allow to look for time-harmonic fields E of the form E(x, t) = u(x) cos(ωt) for x and t R with frequency ω > 0. Suppose that the nonlinear polarization is of the form P NL (x, E) = χ ( x, u(x) ) E i.e. the scalar susceptibility χ depends only on the intensity of E. Then (0.1) reduces to the curl-curl equation which is the main subject of our research (0.) ( µ(x) 1 u ) V (x)u = f(x, u) in, where f(x, u) := χ ( x, u ) u and V (x) = ω ε(x) R 3 3. Probably the most common type of nonlinearity in the physics and engineering literature is the Kerr nonlinearity f(x, u) = χ (3) (x) u u. Other examples for f that appear in applications are nonlinearities with saturation like f(x, u) = χ (3) (x) u 1+ u u. The problem (0.) has a variational nature, i.e. weak solutions correspond to critical points of the functional associated with (0.). The problem is strongly indefinite and during the lectures we will develop variational tools which allow to find ground state and bound state solutions. Plan of the lectures Nonlinear Dirichlet problem on a bounded domain. Mountain Pass Theorem and Nehari manifold approach. General conditions imposed on the nonlinear term. Mountain Pass Theorem vs. Nehari manifold approach. Functional and variational setting for the curl-curl equation on a bounded domain. The role of the cylindrical symmetry in the curl-curl problems. Generalized Nehari manifold approach for strongly indefinite problems. Critical point theory I. Generalized Nehari manifold approach for strongly indefinite problems. Critical point theory II. Ground state solutions and the multiplicity of bound state solutions. Curl-curl equations in R 3 and the lack of compactness. Recent results and the list of open problems.

3 1 Nonlinear Dirichlet problem on a bounded domain. Mountain Pass Theorem and Nehari manifold approach Let R N (N 3) be a Lipschitz domain. Our aim is to find a function u : R such that (DP) u + λu = u p u for x, u = 0 on, where u(x) = N i=1 x i u(x), λ R and p >. Suppose that u C ( ) satisfies the above equation and let ϕ C 0 (). Then, in view of Green s theorem we obtain 0 = ( u + λu u p u)ϕ dx = = u ϕ dx + λuϕ u p uϕ dx, u ν ϕ dσ + u ϕ dx + λuϕ u p uϕ dx where u (x) = u(x)ν(x), ν(x) is the exterior normal, σ is the surface measure. ν Definition 1. Function u is a weak solution to (DP) if (1.1) u ϕ dx + λuϕ u p uϕ dx = 0 for any ϕ C 0 (). We introduce the Hilbert space H 1 () = {u L () : u L ()} equipped with the inner product u, v = u v dx + uv dx. Then u = u, u. Let H0() 1 be the closure of C0 () in H 1 (). If is smooth and u H 1 () C( ), then u H0() 1 if and only if u = 0. In general, we define the trace operator T : H 1 () L ( ) such that T (u) = u for u C 1 ( ) [11, 1]. Then we show that H0() 1 = {u H 1 () T (u) = 0}. 3

4 In view of the Sobolev embedding (1.) H 1 0() L q () for 1 q = N N. Let < p. Let us consider a functional J : H0() 1 R give by (1.3) J(u) = 1 u dx + λ u dx 1 u p dx. p Then (1.4) J (u)(v) = lim t 0 J(u + tv) J(u) t and J C 1. = u v dx + λ uv dx u p uv dx Problem. Applying the Hölder inequality and the Lebesgue s dominated convergence theorem, show formula (1.4). Show the continuity of the Gateaux derivative and infer that J C 1. Observe that (1.1) holds if and only if J (u)(ϕ) = 0 for any ϕ C 0 (). Since C 0 () is dense in H 1 0() we conclude that solutions u H 1 0() of (DP) correspond to critical points of the functional J, i.e. J (u) = Liner case eigenvalue problem The following problem u H0() 1 u + λu = 0 for x, u = 0 on, has a (weak) solution if and only if λ σ( ) = {λ i : i = 1,, 3,...}, for some 0 < λ 1 < λ λ 3... λ n. Let e i be a unit eigenvector with the corresponding eigenvalue λ i, i.e. e i = 1 and e i = λ i e i. Each eigenvalue is of finite multiplicity and we can choose the vectors e i such that {e i } i 1 is the orthonormal basis in H0(), 1 hence span{e i : i = 1,, 3...} = H0() 1 and e i, e j = δ i,j. Suppose that λ k λ < λ k+1 for some k 0, where λ 0 = 0. Let (1.5) (1.6) X := span{e 1, e,..., e k } X + := span{e k+1, e k+,...}. If k = 0 or λ > 0, then we set X = {0} and X + = H 1 0(). 4

5 Proposition 3. There is a constant c > 0 such that (1.7) u dx + λ u dx c u for any u X +, (1.8) u dx + λ u dx 0 for any u X. Corollary 4. If λ > λ 1, then there are constants c 1 c > 0 such that c 1 u u dx + λ u dx c u for u H0() General variational approach and the Montain Pass Theorem In order to solve a variational problem like (DP) we need to recognize the geometry of J which enable us to find a Palais-Smale sequence. Suppose that X is a Banach space and J : X R is of class C 1. We say that (u n ) X is a Palais-Smale sequence of J at level c if J (u n ) 0 and J(u n ) c. If any Palais-Smale sequence (u n ) contains a convergent subsequence, the we say that the Palais-Smale condition is satisfied. The usual 3-steps approach (G-C-S) in variational problems: Step (Geometry): Find the geometry, which provides a bounded Palais-Smale sequence. Step (Compactness): Find a convergent subsequence in some topology (e.g. weak or strong). Step (Solution): Show that the (e.g. weak or strong) limit point is a solution. Theorem 5 (Ambrosetti, Rabinowitz [1,3]). Suppose that X is a Banach space and J : X R is of class C 1 such that for some r > 0 and u 0 X with u 0 > r. inf J(u) > J(0) J(u 0) u =r a) Then there is a Palais-Smale sequence (u n ) at the mountain pass level c, i.e. J (u n ) 0 and J(u n ) c, where c is the mountain pass level c := inf sup J(γ(t)) > 0, γ Γ t [0,1] Γ = { γ C([0, 1], X) : γ(0) = 0 and γ(1) = v }. b) If J satisfies the Palais-Smale condition, then c is a critical value, i.e. there is u X such that J (u) = 0 and J(u) = c. 5

6 1.3 Mountain Pass geometry, Palais-Smale condition and solutions to (DP) Suppose that λ > λ 1 and < p <. Let q be the usual Lebesgue norm in L q (). In view of Corollary 4 J(u) c u 1 p u p p c u C u p = u (c C u p ). Then we find r > 0 such that Moreover for u 0 we get ( 1 J(tu) = t as t. u dx + λ inf J(u) > 0. u =r ) ( 1 ) u dx t p u p dx p Theorem 6 (Rellich). Any bounded sequence in H 1 0() contains a convergent subsequence in L q () for 1 q <. Lemma 7. J satisfies the Palais-Smale condition. Proof. Let (u n ) be a Palais-Smale sequence at level c. Observe that c + o(1) u n = J(u n ) 1 ( 1 p J (u n )(u n ) = p) 1 u + λ u dx ( 1 1 p) c u n, thus u n is bounded in H0(). 1 In view of Rellich theorem there is a subsequence u nk convergent to u 0 H0() 1 in L p () and in L (). Moreover J (u nk )(u nk u 0 ) = u nk, u nk u 0 + (1 λ) u nk (u nk u 0 ) dx u nk p u n (u nk u 0 ) dx. By the Hölder inequality the last two integrals converges to 0. In addition, u nk, u nk u 0 = u nk u 0 + u 0, u nk u 0 and by the Banach-Alaoglu theorem we may assume that u nk weakly sequentially compact in a reflexive space). Hence u 0 (bounded sequence in o(1) = J (u nk )(u nk u 0 ) = u nk u 0 + u 0, u nk u 0 = u nk u 0 + o(1) and we obtain that u nk u 0 in H 1 0(). 6

7 Theorem 8. If λ > λ 1 and < p <, then (DP) has a (weak) solution at the mountain pass level. Proof. We apply Theorem 5 and Lemma 7. Problem 9. Consider J : H0() 1 R such that J(u) = 1 u dx + λ u dx 1 p (u) p + dx, where (u) + (x) = max{u(x), 0}. Show that J has a critical point u 0, such that u(x) 0 for x. Show that u solves (DP). Problem 10. Consider the following (defocusing) problem u + λu = u p u for x, u = 0 on, with λ > λ 1. Show that J : X R is strictly convex and coercive, i.e. J(u) as u. Then J attains the unique global minimum, which is the trivial solution. Nonlinear Dirichlet problem on a bounded domain. Nehari manifold approach We look for solutions to (DP) with λ > λ 1 and < p <, which minimize J among all nontrivial solutions. Observe that, by Proposition 3 ( u λ := u dx + λ defines an equivalent norm in H 1 0(). We introduce the Nehari manifold Lemma 11. m := inf N J > 0. ) 1/ u dx N = {u H 1 0() \ {0} : J (u)(u) = 0}. Proof. If there is u n N such that lim sup n J(u n ) 0, then ( 1 lim sup n p) 1 ( u n + λ u n dx = lim sup J(u n ) 1 ) n p J (u n )(u n ) 7 0.

8 Hence u n 0. On the other hand, by the Sobolev embedding c u n u n + λ u n dx = u n p p C 1 u n p, and we get a contradiction u n p c /C 1 > 0. Proposition 1. If J(u 0 ) = inf N J for some u 0 N, then J (u 0 ) = 0. Moreover N is a manifold of class C 1 diffeomorphic to the unit sphere S = {u H0() 1 : u = 1}. Proof. Let G(u) = J (u)(u) and take any v H 1 0() and let f(s, t) = G(t(u 0 + sv)). Note that t f(0, 1) = G (u)(u) < 0 and by the implicit function theorem there exists δ > 0 and a C 1 function t : ( δ, δ) R such that t(0) = 1 and f(s, t(s)) = 0 for s ( δ, δ). Then γ(s) := t(s)(u+sv) N defines a differentiable curve passing through u. Observe that J(γ(s)) attains a minimum at s = 0, hence Therefore J (u 0 ) = 0. Now observe that G (u)(u) = (1 p) 0 = s J(γ(0)) = J (u 0 )(t (0)u 0 + v) = J (u 0 )(v). u p dx < 0 for u N, hence by the implicit function theorem N is a C 1 -manifold of codimension 1. In fact, for any u 0 we find the unique t = t(u) > 0 such that J(t(u)u) = max t>0 J(tu). Then t(u)u N and t is of class C 1. On the other hand, N u u u to t S. S defines the inverse Theorem 13. There is a ground state solution u 0 of (DP), i.e. u 0 N is such that J (u 0 ) = 0 and J(u 0 ) = inf N J = inf { J(u) : u 0 and J (u) = 0 }. Proof. Take any minimizing sequence (u n ) such that J(u n ) m > 0 and u n N. Observe that J is coercive on N, since ( 1 J(u n ) p) 1 u + λ u dx 8 ( 1 1 p) c u n.

9 Therefore (u n ) is bounded and we may assume that u n u 0 in H 1 0() and u n u 0 in L p () and in L (). Then Since u n N, we get and c = lim n J(u n ) J(u 0 ). u n + λ u n dx = u 0 + λ u 0 dx and u 0 0. Note that we find t 1 such that tu 0 + λ tu 0 dx = u n p dx, u 0 p dx tu 0 p dx, that is tu 0 N. On the other hand ( 1 m J(tu 0 ) = t 1 ) ( 1 u 0 + λ u 0 dx t lim inf p n p) 1 = t lim inf n J(u n) = m Thus t = 1, u 0 N and by Lemma 1 we obtain that J (u 0 ) = 0. u n + λ u n dx 3 Mountain Pass Theorem vs. Nehari manifold approach Theorem 14. Let X be a Banach space, J : X R is a functional of class C 1 and Suppose that for some r > 0, and u 0 X with u 0 > r. N := {u X \ {0} : J (u) = 0}. inf J(u) > J(0) u =r a) Then there is a Palais-Smale sequence (u n ) at the following mountain pass level c = inf sup J(γ(t)) > 0 γ Γ t [0,1] Γ = {γ C([0, 1], X) γ(0) = 0, γ(1) > r and J(γ(1)) < 0}. b) If J satisfies the Palais-Smale condition, then c is a critical value, i.e. there is u X such that J (u) = 0 and J(u) = c. c) Suppose that for any u N we have J(tu) as t. If c is attained by a critical point and J(u) J(tu) for any t 0 and u N, then c = inf N J. 9

10 Proof. c) Since c is attained by a critical point u 0, we get c = J(u 0 ) inf N J. Take any ε > 0 and u N such that J(u) inf N J + ε. We find t 0 > r/ u such that J(t 0 u) < 0. Consider a path γ(t) := tt 0 u for t [0, 1] and observe that γ Γ and Since ε is arbitrary, we infer c = inf N J. c J(γ(t)) J(u) inf N J + ε. Problem 15. Show that in problem (DP) one has c = m, where c is the mountain pass level given in Theorem General nonlinearity Let us consider the following problem u + λu = f(x, u) for x, (DP) u = 0 on, with λ > λ 1, N 3, where f satisfies the following assumptions (F1) f : R R is measurable and continuous in u R for a.e. x R N, and there are c > 0 and < p < such that f(x, u) c(1 + u p 1 ) for all u R, x. (F) f(x, u) = o( u ) uniformly in x as u 0. We observe that the following functional associated with (DP) is given by J(u) = 1 u dx + λ u dx F (x, u) dx and has the mountain pass geometry as in Theorem 5. Hence we may find a Palais-Smale sequence. In order to show the boundedness of the sequence the following condition has been introduced by Ambrosetti and Rabinowitz [1]. (AR) There is γ > such that f(x, u)u γf (x, u) > 0 for u 0. Similarly as in Lemma 7 we estimate J(u n ) 1 γ J (u n )(u n ) and show the boundedness and show the Palais-Smale condition is satisfied. 10

11 Theorem 16. There is a weak solution to (DP) provided that (F1)-(F) and (AR) are satisfied. In order to use the Nehari approach we assume the following conditions: (F3) F (x, u)/ u uniformly in x as u, where F is the primitive of f with respect to u. (F4) u f(x, u)/ u is strictly increasing on (, 0) and (0, ). Note that f(x, u) = u ln(1 + u p ) satisfies (F1)-(F4), but (AR) does not hold. On the other hand, (AR) implies only (F3) and one can easy provide examples which satisfy (AR) and do not satisfy the monotonicity condition (F4). Since J need not to be of class C, so N need not to be of class C 1 and we cannot directly minimize on N. We apply approach due to Szulkin and Weth [] based on the observation that the Nehari manifold N is homemorphic to the unit sphere S where we may apply a critical point theory. 3. Nehari manifold in the abstract setting Let X be a Hilbert space with the norm. We consider a functional J : X R of the following form J(u) = 1 u I(u), where I : X R is of C 1 class. In this case the Nehari manifold is given by N := {u X \ {0} : J (u)(u) = 0} = {u X \ {0} : u = I (u)(u)}. Now we formulate the main result of this section. Theorem 17 ( [8]). Suppose that the following conditions hold: (J1) there is r > 0 such that a := inf u =r J(u) > J(0) = 0; (J) there is q such that I(t n u n )/t q n for any t n and u n u 0 as n ; (J3) for t (0, ) \ {1} and u N (J4) J is coercive on N. t 1 I (u)(u) I(tu) + I(u) < 0; 11

12 Then inf N J > 0 and there exists a bounded minimizing sequence for J on N, i.e. there is a sequence (u n ) N such that J(u n ) inf N J and J (u n ) 0. Observe that condition (J) implies that for any u 0 there is t > 0 such that J (tu) < 0, hence taking into account also (J1) we easily check that J has the classical mountain pass geometry [1, 3] and we are able to find a Palais-Smale sequence. However we do not know whether it is a bounded sequence and contained in N. In order to get the boundedness we assume the coercivity in (J4), which is, in applications, a weaker requirement than the classical Ambrosetti-Rabinowitz condition; see e.g. []. Remark 18. a) In order to get (J3) it is sufficient to check (3.1) (1 t) (ti (u)(u) I (tu)(u)) > 0 for any t (0, ) \ {1} and Indeed, let us consider t (0, ) \ {1}, u N and u such that I (u)(u) > 0. (3.) ϕ(t) = t 1 I (u)(u) I(tu) + I(u). Then I (u)(u) = u > 0, ϕ(1) = 0, ϕ (t) = ti (u)(u) I (tu)(u) > 0 for t < 1 and ϕ (t) < 0 for t > 1. Therefore ϕ(t) < ϕ(1) = 0. b) Observe that (J3) is equivalent to the following condition: u N is the unique maximum point of (0, + ) t J(tu) R. Indeed, note that for u N (3.3) J(tu) = J(u) + ( J(tu) J(u) t 1 J (u)(u) ) = J(u) + ϕ(t) < J(u) if and only if ϕ(t) < 0. Proof of Theorem 17. For a given u 0 we consider a map ϕ : [0, + ) R defined by ϕ(t) = J(tu) J(u) for t [0, + ). Note that from (3.3), ϕ(t) is given by (3.) provided that u N. In view of (J1)-(J) we obtain ϕ(0) = J(u) < ϕ( r ) and ϕ(t) as u t. Therefore there is a maximum point t(u) > 0 of ϕ which is a critical point of ϕ, i.e. J (t(u)u)(u) = 0 and t(u)u N. In view of Remark 18 b) we infer that for any u 0 there is an unique critical point t(u) > 0 of ϕ, i.e. t(u)u N. Let ˆm : X \ {0} N be a map given by ˆm(u) = t(u)u for u 0. We are going to show that ˆm is continuous. Take u n u 0 0 and denote t n = t(u n ) for n 0, so that ˆm(u n ) = t n u n. Observe that if t n then by (3.3) and (J) o(1) = J(u n )/t q n J( ˆm(u n ))/t q n = 1 u n t q n 1 I(t n u n )/t q n

13 and we get a contradiction. Therefore we may assume that t n t 0 0. Again by (3.3) J(t(u 0 )u 0 ) J(t 0 u 0 ) = lim n J(t n u n ) lim n J(t(u 0 )u n ) = J(t(u 0 )u 0 ) we get t 0 = t(u 0 ), which completes the proof of continuity of ˆm. Thus m = ˆm S, where S := {u X : u = 1} is the unit sphere in X, is a homeomorphism. Indeed, the inverse function is given by m 1 (v) = v/ v for v N. Therefore c := inf (J m)(u) = inf J(u) inf J u S u N u N ( r u u ) a > 0. Now, arguing as in [][Proposition.9], we show that J ˆm is of C 1 -class. w, z X, w 0 we have (J ˆm) (w)z = ˆm(w) w J ( ˆm(w))z. Moreover for Indeed, let w X \ {0} and u = ˆm(w) N. Therefore t(w) = u w and choose δ > 0 so small that w t := w + tz X \ {0} u and u = w. Let z X w for t < δ. Let u t = ˆm(w t ). Therefore u t = s t w t for t < δ and some s t > 0, s 0 = u w. The function v ˆm(v) is continuous, thus ( δ, δ) t s t R is also continuous. From the mean value theorem, there is τ t (0, 1) such that (J ˆm)(w t ) (J ˆm)(w) = J(u t ) J(u) = J(s t w t ) J(s 0 w) Since s t s 0 as t 0, we get In a similar way we have J(s t w t ) J(s t w) = J (s t [w + τ t (w t w)])s t (w t w). (J ˆm)(w t ) (J ˆm)(w) s 0 J (u)tz + o(t) as t 0. (J ˆm)(w t ) (J ˆm)(w) s 0 J (u)tz + o(t) as t 0 Therefore (J ˆm) (J ˆm)(w t ) (J ˆm)(w) (w)z = lim = s 0 J (u)z. t 0 t We see that (J ˆm) (w)z is linear and countinuous in z, and it is continouous in w, thus J ˆm is of C 1 class. In view of the Ekeland variational principle [3][Theorem 8.5] we find a minimizing 13

14 sequence (v n ) S for J m such that (J m) (v n ) 0. Then take u n = m(v n ) N and observe that J (u n )(v n ) = 0 and (J m) (v n )(z) = u n J (u n )(z) = u n J (u n )(z + tv n ). for any z T vn S and t R, where T vn S stands for the tangent space S at v n. Therefore (J m) (v n ) = sup (J m) (v n )(z) = u n J (u n ). z T vn S 1, z =1 Since u n N, we have u n η for some η > 0. The coercivity of J implies that sup n u n <. Hence (u n ) is a bounded minimizing sequence for J on N such that J (u n ) Ground state solutions Theorem 19. There is a weak solution u 0 to (DP) such that J(u 0 ) = inf N J provided that (F1)-(F4) are satisfied. Proof. We easy check conditions (J1)-(J4) and in view of Theorem 17 we obtain a bounded Palais-Smale sequence. It contains a convergent subsequence due to Rellich theorem. 4 Maxwell equations We look for a vector field u : R 3 satisfying the following equation (MP) ( µ(x) 1 u ) V (x)u = f(x, u) in derived from (0.1) in the time-harmonic case. Observe that the nonlinearity is a gradient: f(x, u) = u F (x, u) with F (x, u) = 1ψ( x, u ) where ψ(x, s) = s χ(x, r)dr. Solutions of 0 (MP) are critical points of the functional (4.1) J(u) = 1 µ(x) 1 u, u dx 1 V (x)u, u dx F (x, u) dx defined on an appropriate subspace X of H 0 (curl; ) such that F (x, u) and V (x)u, u are integrable. The precise definition of the domain of J will be given in the next subsection. Let us mention already at this point a major difficulty when dealing with this equation. If u = φ is a gradient then u = 0, hence the differential operator in (MP) has an infinite-dimensional kernel and J has no longer the mountain pass geometry. 14

15 The above mentioned difficulty that the curl operator has an infinite-dimensional kernel is of course also present in the variational approach. One of the consequences is that the functional is strongly indefinite, i.e. Morse indices of critical points will be infinite. Another consequence is that the Palais-Smale condition does not hold. And a third difficulty is that the derivative J : X X is not weak-to-weak continuous even when the growth of F is subcritical. Thus even if J has a linking geometry in the spirit of Benci and Rabinowitz [3], the problem cannot be treated by standard variational methods for strongly indefinite functionals as in [3, 4, 14]. Probably the most common type of nonlinearity in the physics and engineering literature is the Kerr nonlinearity (4.) f(x, u) = χ (3) (x) u u. Other examples for f that appear in applications are nonlinearities with saturation like (4.3) f(x, u) = χ (3) u (x) 1 + u u, or cubic-quintic nonlinearities like (4.4) f(x, u) = χ (3) (x) u u χ (5) (x) u 4 u. We refer the reader to [17, 19, 0] and the references therein for these and further examples. When has a boundary then boundary conditions depend of course on the material characteristics of the complement R 3 \. In this lecture we shall only consider the case of being surrounded by a perfectly conducting medium which leads to the so-called metallic boundary condition (BC) ν u = 0 on where ν : R 3 is the exterior normal. References [1] A. Ambrosetti, P. Rabinowitz: Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), [] M.Badiale, E. Serra: Semilinear Elliptic Equations for Beginners: Existence Results via the Variational Approach, Springer

16 [3] V. Benci, P. H. Rabinowitz: Critical point theorems for indefinite functionals, Invent. Math. 5 (1979), no. 3, [4] T. Bartsch, Y. Ding: Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Mathematische Nachrichten 79 (006), no. 1, [5] T. Bartsch, T. Dohnal, M. Plum, W. Reichel: Ground States of a Nonlinear Curl-Curl Problem in Cylindrically Symmetric Media, Nonlin. Diff. Equ. Appl. 3:5 (016), no. 5, 34 pp. [6] T. Bartsch, J. Mederski: Ground and bound state solutions of semilinear time-harmonic Maxwell equations in a bounded domain, Arch. Rational Mech. Anal., Vol. 15 (1), (015), [7] T. Bartsch, J. Mederski: Nonlinear time-harmonic Maxwell equations in an anisotropic bounded domain, J. Funct. Anal. 7 (017), no. 10, [8] B. Bieganowski, J. Mederski: Nonlinear Schrödinger equations with sum of periodic and vanishig potentials and sign-changning nonlinearities, Comm. Pure Appl. Anal 17 (1), (018) [9] M. Costabel: A remark on the regularity of solutions of Maxwell s equations on Lipschitz domains, Math. Methods Appl. Sci. 1, (1990), [10] W. Dörfler, A. Lechleiter, M. Plum, G. Schneider, C. Wieners: Photonic Crystals: Mathematical Analysis and Numerical Approximation, Springer Basel 01. [11] D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Springer- Verlag, Berlin, 001. [1] A. Kirsch, F. Hettlich: The Mathematical Theory of Time-Harmonic Maxwell s Equations: Expansion-, Integral-, and Variational Methods, Springer 015. [13] J. Mederski: Ground states of time-harmonic semilinear Maxwell equations in R 3 with vanishing permittivity, Arch. Rational Mech. Anal. 18 (), (015), [14] J. Mederski: Ground states of a system of nonlinear Schrödinger equations with periodic potentials, Comm. Partial Differential Equations 41 (9), (016), [15] J. Mederski: The Brezis-Nirenberg problem for the curl-curl operator, J. Funct. Anal. 74 (5), (018), [16] P. Monk: Finite Element Methods for Maxwell s Equations, Oxford University Press 003. [17] W. Nie: Optical Nonlinearity: Phenomena, applications, and materials, Adv. Mater. 5, (1993), [18] P. Rabinowitz: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, Vol. 65, Amer. Math. Soc., Providence, Rhode Island [19] C. A. Stuart: Self-trapping of an electromagnetic field and bifurcation from the essential spectrum, Arch. Rational Mech. Anal. 113 (1991), no. 1,

17 [0] C. A. Stuart: Guidance Properties of Nonlinear Planar Waveguides, Arch. Rational Mech. Anal. 15 (1993), no. 1, [1] M. Struwe: Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer 008. [] A. Szulkin, T. Weth: Ground state solutions for some indefinite variational problems, J. Funct. Anal. 57 (009), no. 1, [3] M. Willem: Minimax Theorems, Birkhäuser Verlag