Neumann problem: p u = juj N() : p 2 u; in ; u n = ; on : Robin problem: R() : Steklov problem: S() : p u = juj p 2 u; in ; p 2 u jruj n + jujp 2 u =

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1 EIGENVALUE PROBLEMS FOR THE p-laplacian AN L^E DEPARTMENT OF MATHEMATICS, UNIVERSITY OF UTAH 55 SOUTH 4 EAST SALT LAKE CITY, UT 842, USA Abstract. We study nonlinear eigenvalue problems for the p-laplace operator subject to dierent kinds of boundary conditions on a bounded domain. Using the Ljusternik-Schnirelman principle, we show the existence of a nondecreasing sequence of nonnegative eigenvalues. We prove the simplicity and isolation of the principal eigenvalue and give a characterization for the second eigenvalue.. Introduction Eigenvalue problems for the p Laplace operator subject to zero Dirichlet boundary conditions on a bounded domain have been studied extensively during the past two decades and many interesting results have been obtained. The investigations principally have relied on variational methods and deduce the existence of a principal eigenvalue as a consequence of minimization results of appropriate functionals. This principal eigenvalue then is the smallest of all possible eigenvalues and its existence proof is very much the same for all possible types of boundary conditions. The study of higher eigenvalues, on the other hand, introduces complications which depend upon the boundary conditions in a signicant way, and thus the existence proofs may dier signicantly, as well. On the other hand, there is a large class of commonly studied eigenvalue problems which allow for a unied treatment. It is such a class of problems which is being studied here. We consider, among others, the following eigenvalue problems: Dirichlet problem: p u = juj D() : p 2 u; in ; u = ; on : No-ux problem: P () : 8 < : R p u = juj p 2 u; in ; u = constant ; on ; u jrujp 2 nds = : 99 Mathematics Subject Classication. 35B45,35J6, 35J7. Key words and phrases. Nonlinear eigenvalue problems, Ljusternik-Schnirelman Principle, p-laplacian, variational methods.

2 Neumann problem: p u = juj N() : p 2 u; in ; u n = ; on : Robin problem: R() : Steklov problem: S() : p u = juj p 2 u; in ; p 2 u jruj n + jujp 2 u = ; on : jruj p u = juj p 2 u; in ; n = juj p 2 u; on : p 2 u Here is a bounded domain in R N, p u := div(jruj p 2 ru) is the p- Laplacian operator with p >, and n u denotes the outer normal derivative of u with respect to. We note that, when N = and = (a; b), P () becomes the periodic boundary value problem 8 < : (ju j p 2 u ) = juj p 2 u; in (a; b); u(a) = u(b); u (a) = u (b): The parameter (which may be a function) in R() is in [; ). We observe that the Dirichlet and Neumann problems correspond to the cases = and =, respectively. Besides being of mathematical interest, the study of the p-laplacian operator is also of interest in the theory of non-newtonian uids both for the case p 2 (dilatant uids) and < p < 2 (pseudo-plastic uids), see [4]. It is also of geometrical interest for p 2, some of which is discussed in [32]. Many results have been obtained on the structure of the spectrum of the Dirichlet problem D(). It is shown in [7] that there exists a nondecreasing sequence of positive eigenvalues f n g tending to as n!. Moreover, the rst eigenvalue is simple and isolated, see [2, 23]. Recently, in [3], a characterization of the second eigenvalue of D() was also given. The existence of such a sequence of eigenvalues can be proved using the theory of Ljusternik - Schnirelman (e.g. see [7, 6, 35]). For that reason we call this sequence the L-S sequence f n g. To establish the simplicity of the rst eigenvalue, one shows that any (nontrivial) eigenfunction associated to does not change sign and that any two rst eigenfunctions are constant multiples of each other. It follows from the proof of the simplicity of that any eigenfunction associated to an eigenvalue 6= has to change sign. This fact together with the closedness of the spectrum give the isolation of. It will also be shown that the eigenvalue 2 of the L-S sequence is in fact the least of all eigenvalues which exceed the rst eigenvalue. The spectrum of the Dirichlet problem D() has, among others, the following properties: 2

3 (i) There exists a nondecreasing sequence of nonnegative eigenvalues obtained by the Ljusternik-Schnirelman principle (L-S sequence) f n g tending to as n! (Garca Azorero and Peral Alonso [7]). (ii) The rst eigenvalue is simple and only eigenfunctions associated to do not change sign (Anane [2] and Lindqvist [23]). (iii) The set of eigenvalues is closed (Anane and Tousli [3]). (iv) The rst eigenvalue is isolated (Lindqvist [23]). (v) The eigenvalue 2 is the second eigenvalue ([3]), i.e., 2 = inff : is an eigenvalue of D() and > g: Comparatively, the spectra of N(), P (), R(), S() have been investigated little and it is natural to pose the problem of analyzing the structures of the spectra of N(); P (); R(); S() and compare these to that of D(): We remark that in the case N = fairly complete information on the spectra of the Dirichlet, Neumann, and periodic boundary value problems is available (see, e.g., [25], [4]). The purpose of this paper is to study this problem and show, among other things, that properties (i)-(v) of the spectrum of the Dirichlet problem also hold for P (); N(); R() and S(). By choosing appropriate function spaces, we will see later in section 2 that we can unify all problems as one single abstract problem. And if we denote by f D n g; f P n g; f N n g the corresponding L-S sequences of eigenvalues of D(), P (), N() (respectively) then (.) D n P n N n ; for all n: The paper is organized as follows: We rst present, for the sake of completeness the Ljusternik-Schnirelman principle and applications to our setting. We then establish the existence of L-S sequences for the Dirichlet, Periodic, Neumann, Robin, and Steklov problems. This is followed by a discussion of global boundedness and C ; smoothness of eigenfunctions. In Section 5 we establish the promised properties of the spectra of the Dirichlet, Periodic, Neumann, Robin and Steklov problems. The nal section consists of some appendices containing useful results which are needed in the development. 2. The Ljusternik-Schnirelman principle and its applications 2.. The Ljusternik-Schnirelman principle in Banach spaces. We recall here a version of the Ljusternik-Schnirelman principle which was discussed by F. Browder [7] and E. eidler [34], [35] (section 44.5, remark 44.23). We then shall apply the principle (theorem 2.) to establish the existence of a sequence of eigenvalues for eigenvalue problems in closed subspaces of W ;p (). 3

4 Let X be a real reexive Banach space and F; G be two functionals on X. Consider the following eigenvalue problem (2.) F (u) = G (u); u 2 S G ; 2 R; where S G is the level S G = fu 2 X : G(u) = g. We assume that: (H) F; G : X!R are even functionals and that F; G 2 C (X; R) with F () = G() =. (H2) F is strongly continuous (i.e. u n * u in X implies F (u n )!F (u)) and hf (u); ui = ; u 2 cos G implies F (u) =, where cos G is the closed convex hull of S G. (H3) G is continuous, bounded and satises condition (S ), i.e. as n!, u n * u; G (u n ) * v; hg (u n ); u n i!hv; ui implies u n!u: (H4) The level set S G is bounded and u 6= implies hg (u); ui > ; lim G(tu) = +; inf t!+ u2s G hg (u); ui > : It is known that (u; ) solves (2.) if and only if u is a critical point of F with respect to S G (see eidler [35], proposition 43.2). For any positive integer n; denote by A n the class of all compact, symmetric subsets K of S G such that F (u) > on K and (K) n, where (K) denotes the genus of K, i.e., (K) := inffk 2 N : 9h : K!R k n fg such that h is continuous and odd g. We dene: suph2an inf a n = u2h F (u); A n 6= ;: (2.2) ; A n = ;: (2.3) Also let = supfn 2 N : an > g; if a > ; ; if a = : We now state the Ljusternik-Schnirelman principle. Theorem 2.. Under assumptions (H)-(H4), the following assertions hold: () If a n >, then (2.) possesses a pair u n of eigenvectors and an eigenvalue n 6= ; furthermore F (u n ) = a n. (2) If =, (2.) has innitely many pairs u of eigenvectors corresponding to nonzero eigenvalues. (3) > a a 2 ::: and a n! as n!. (4) If = and F (u) = ; u 2 cos G implies hf (u); ui =, then there exists an innite sequence f n g of distinct eigenvalues of (2.) such that n! as n!. (5) Assume that F (u) = ; u 2 cos G implies u =. Then = and there exists a sequence of eigenpairs f(u n ; n )g of (2.) such that u n * ; n! as n! and n 6= for all n. Proof. We refer to [7] or [34] for the proof. 4

5 2.2. Applications of the L-S principle in W ;p () and its subspaces. Let be a bounded domain in R N with C boundary. Let X be a closed subspace of W ;p () such that W ;p () X W ;p () with the norm k k induced by the norm in W ;p (). Dene on X the functionals (2.4) (2.5) F (u) = G(u) = a(x)ju(x)j p dx + (jru(x)j p + ju(x)j p )dx + b(s)ju(s)j p ds; (s)ju(s)j p ds; where a 2 L () and b; 2 L () such that a; b; a.e. (We refer to [2] where the the surface integral on and the spaces L p () are discussed.) As before we dene S G = fu 2 X : G(u) = g. It is easy to see that F and G are C functionals. Let A = p F ; B = p G ; where (2.6) (2.7) hau; vi = hbu; vi = + ajuj p 2 uvdx + bjuj p 2 uvds; (jruj p 2 ru rv + juj p 2 uv)dx juj p 2 uvds; u; v 2 X: (2.8) Then (2.) becomes Au = Bu; where G(u) =. Thus for any v 2 X, ajuj p 2 uvdx + (jruj p 2 ru rv + juj p 2 uv)dx + bjuj p 2 uvds = juj p 2 uvds We claim that F; G satisfy hypotheses (H), (H2), (H3), and (H4) mentioned in 2.. It follows straightforwardly from (2.6), (2.7) that (H) and (H4) hold. Proposition 2.2. Let F be dened in (2.4), then F satises (H2). Proof. It suces to show that A is strongly continuous. Let u n * u in X, we need to show that Au n!au in X. 5 :

6 For any v 2 X, by Holder's inequality in the space L p () and Sobolev's embedding theorem, it follows that jhau n Au; vij a(ju n j p 2 u n juj p 2 u)vdx + b(ju n j p 2 u n juj p 2 u)vds kak kju n j p 2 u n juj p 2 uk p p kvk p + kbk kju n j p 2 u n juj p 2 uk L p p ()kvk L p () C kak kju n j p 2 u n juj p 2 uk p p kvk + C 2 kbk kju n j p 2 u n juj p 2 uk p kvk; L p () where we have denoted (and we shall continue to do so) by k k, k k p the norms in W ;p (), and L p (), respectively. We next show ju n j p 2 u n!juj p 2 u in L p p (). To see this, let wn = ju n j p 2 u n and w = juj p 2 u. Since u n * u in W ;p (), u n!u in L p (), it follows w n (x)! w(x); a.e. in and jw n j p p p dx! jwj p dx: We conclude from lemma A. that w n! w in L p=(p ) (). Using the compact embedding W ;p (),! L p () and arguing as above, we obtain that ju n j p 2 u n!juj p 2 u in L p p (). Therefore Aun!Au in X. In order to verify (H3) we need the following lemma which uses a calculation from chapter 6 of [2]. Lemma 2.3. Let B be dened in (2.7), then for any u; v 2 X one has hbu Bv; u vi (kuk p kvk p )(kuk kvk): Furthermore, hbu Bv; u vi = if and only if u = v a.e. in. Proof. Straightforward computations give us = hbu Bv; u vi + + [jruj p + jrvj p jruj p 2 ru rv jrvj p 2 rv ru]dx (juj p + jvj p juj p 2 uv jvj p 2 vu)dx (juj p + jvj p juj p 2 uv jvj p 2 vu)ds: 6

7 Since we have (juj p + jvj p juj p 2 uv jvj p 2 vu)ds (juj p + jvj p juj p jvj jvj p juj)ds = hbu Bv; u vi + (juj p jvj p )(juj jvj)ds ; [jruj p + jrvj p jruj p 2 ru rv jrvj p 2 rv ru]dx = kuk p + kvk p (juj p + jvj p juj p 2 uv jvj p 2 vu)dx (jruj p 2 ru rv + juj p 2 uv)dx (jrvj p 2 rv ru + jvj p 2 vu)dx: Using Holder's inequality, we obtain (2.9) (jruj p 2 ru rv + juj p 2 uv)dx Applying the inequality + p jruj p p p juj p p (a + b) (c + d) a c + b d ; jrvj p p jvj p p : which holds for any 2 (; ) and for any a > ; b > ; c > ; d >, with we conclude that Hence, c = a = jruj p dx; b = jrvj p dx; d = juj p dx; jvj p dx; = p p ; (jruj p 2 ru rv + juj p 2 uv)dx kuk p kvk: (jrvj p 2 rv ru + jvj p 2 vu)dx kvk p kuk: 7

8 Therefore, hbu Bv; u vi kuk p + kvk p kuk p kvk kvk p kuk : (kuk p kvk p )(kuk kvk) Now let u; v be such that hbu Bv; u vi =. Then we have hbu Bv; u vi = (kuk p kvk p )(kuk kvk) = : It follows that kuk = kvk and that the equality holds in (2.9). As equality in Holder's inequality is characterized, we obtain from (2.9) that u = kv a.e. in, for some constant k. Therefore, k = and u = v a.e. in. Proposition 2.4. Let G be dened in (2:5) then G satises (H3). Proof. As B = G =p, it suces to show this for B. Using Sobolev's embedding theorem, Holder's inequality and following the arguments used in the proof of proposition 2.2 one can easily see that B is continuous and bounded. It remains to be shown that B satises condition (S ). That means if fu n g is a sequence in X such that u n * u; Bu n * v; and hbu n ; u n i!hv; ui for some v 2 X and u 2 X, then it follows that u n!u in X. By Sobolev's compact embedding theorem we have u n!u in L p (). Since X is a reexive Banach space, by the Lindenstrauss-Asplund-Troyanski theorem (see [3]) one can nd an equivalent norm such that X with this norm is locally uniformly convex. In such a space weak convergence and norm convergence imply (strong) convergence. Thus to show u n!u in X, we only need to show ku n k!kuk. To this end, we rst observe that lim hbu n Bu; u n ui = lim (hbu n; u n i hbu n ; ui hbu; u n ui) = : n! n! On the other hand, it follows from lemma 2.3 that hbu n Bu; u n ui (ku n k p kuk p )(ku n k kuk): Hence ku n k!kuk as n!. And therefore B satises condition (S ). We now can apply theorem 2. to conclude the following. Theorem 2.5 (Existence of L-S sequence). Let X be a closed subspace of W ;p () such that W ;p () X and let F; G be the two functionals dened in (2.4), (2.5). Then there exists a nonincreasing sequence of nonnegative eigenvalues f n g obtained from the Ljusternik-Schnirelman principle such that n! as n!, where (2.) n = sup H2A n inf F (u); u2h and each n is an eigenvalue of F (u) = G (u). 8

9 Proof. The existence of such a sequence f n g follows from theorem 2.-(5). To verify (2.) we observe, using (2.4), (2.5), (2.6) and (2.7), that n = n G(u n ) = n hbu n ; u n i = hau n ; u n i = F (u n ) = a n : Combining this with (2.2) we obtain (2.). 3. Eigenvalue problems for the p-laplacian We shall establish the existence of a sequence of eigenvalues using the principle given in the previous section. We rst notice that by choosing the function spaces appropriately, the Dirichlet problem D(), the No-ux problem P (), and the Neumann problem N() yield the same formula for weak solutions. 3.. Weak solutions. Denition 3.. (3.) (i) Let X be either W ;p (), W ;p () R or W ;p (). Then a pair (u; ) 2 X R is a weak solution of D(); P (); N(), respectively, provided that jruj p 2 ru rvdx = juj p 2 uvdx; for any v 2 X: (ii) A pair (u; ) 2 W ;p () R is a weak solution of the Robin problem R() provided that (3.2) jruj p 2 ru rvdx + juj p 2 uvds = juj p 2 uvdx; for any v 2 W ;p (). (iii) A pair (u; ) 2 W ;p ()R is a weak solution of the Steklov problem S() provided that (3.3) jruj p 2 ru rvdx + juj p 2 uvdx = juj p 2 uvds; for any v 2 W ;p (). In all cases, such a pair (u; ), with u nontrivial, is called an eigenpair, is an eigenvalue and u is called an eigenfunction. It follows from (3.), (3.2) and (3.3) that all eigenvalues are nonnegative (by choosing v = u). It will be shown that if is of class C ; ; then eigenfunctions of (3.), (3.2), (3.3) belong to C ; (). Hence ru exists on and the boundary conditions of the problems P (), N(), R(), and S() make sense. The following lemma assures that if an eigenfunction u is smooth enough, then u solves the corresponding partial dierential equation. 9

10 Lemma 3.2. Let (u; ) be an eigenpair, i.e., a weak solution, of (3.) (with X = W ;p (), W ;p () R or W ;p ()), or (3.2), or (3.3) such that u is in W 2;p (), then (u; ) solves D(), P (), N(), R(), and S(), respectively. Proof. Let us verify this for the No-ux problem P () and the Steklov problem S(). The verication for the others proceeds in a similar way. Let X = W ;p () R and (u; ) 2 W ;p () R R + be an eigenpair of (3.) with u 2 W 2;p (). Since u =constant on, to show (u; ) solves P () it remains to show (3.4) and (3.5) ( p u)vdx = jruj juj p 2 uvdx; 8v 2 C p 2 u ds = : n We recall the rst formula of Green (see [28]) ( p u)vdx + jruj p 2 ru rvdx = (); p 2 u jruj n vds; which holds for a C boundary and for any u 2 W 2;p (), v 2 W ;p (). Applying Green's rst identity to (3.), we obtain ( p u)vdx + p 2 u jruj n vds = juj p 2 uvdx; 8v 2 W ;p () R: Thus (3.4) follows immediately, i.e., p u = juj p 2 u in. Consequently, jruj p 2 u n We obtain (3.5), since v =constant on. vds = ; 8v 2 W ;p () R: Now let (u; ) 2 W 2;p () R + be an eigenpair of (3.3). Using again Green's rst identity, it follows from (3.3) that ( p u)vdx + p 2 u jruj n vds + juj p 2 uvdx = for any v 2 W ;p (). Thus, taking any v in C ( p u + juj p 2 u)vdx = ; () we have juj p 2 uvds; which implies p u = juj p 2 u in. Furthermore, since the range of the trace mapping W ;p (),! L p () is dense in L p (), we have p 2 u jruj n vds = juj p 2 uvds; 8v 2 L p (): p 2 u Therefore, jruj n = jujp 2 u on.

11 3.2. Existence results. Let F and G be dened in (2.4) and (2.5). We will show that by choosing an appropriate subspace X of W ;p () and appropriate functions of a; b; we can apply theorem 2.5 to the Dirichlet, the No-ux, the Neumann, the Robin, and the Steklov problems. Theorem 3.3 (Existence of L-S sequences for D(); P (); N()). Let F and G be dened in (2.4) and (2.5) with a ; b and. Let X be W ;p (); W ;p () R, or W ;p (), then there exists a nondecreasing sequence of nonnegative eigenvalues f X n g of (3.) obtained by using the Ljusternik-Schnirelman principle such that X n =! X n as n!, where each X n is an eigenvalue of the corresponding equation F (u) = G (u) dened in (2.). Proof. With a ; b and, F and G become F (u) = G(u) = juj p dx; And thus F (u) = G (u) is equivalent to or juj p 2 uvdx = jruj p 2 ru rvdx = (jruj p + juj p )dx: (jruj p 2 ru rv + juj p 2 uv)dx; 8v 2 X; juj p 2 uvdx; 8v 2 X: Comparing the last equation to (3.) and applying theorem 2.5 we obtain the result. As mentioned above, if X = W ;p (), W ;p () R, or W ;p () we have f D n g, fp n g, and fn n g are the corresponding L-S sequences of eigenvalues of D(), P (), and N(), respectively. Since S W ;p S () W ;p ()R S W ;p () where S X = fu 2 X : G(u) = g, it follows from (2.) that D n P n N n. Thus D n P n N n, for all n: This proves inequality (.) mentioned in section. Theorem 3.4 (Existence of L-S sequences for R()). Let X be W ;p () and F, G be dened in (2.4), (2.5) with a(x), b(x) and (x) >. Then there exists a nondecreasing sequence of nonnegative eigenvalues f n g of (3.2) obtained by using the Ljusternik- Schnirelman principle such that n = n! as n!, where each n is an eigenvalue of the corresponding equation F (u) = G (u) dened in (2.).

12 Proof. With a(x) ; b(x) and (x), F and G become F (u) = G(u) = juj p dx; Thus F (u) = G (u) is equivalent to juj p 2 uvdx = for any v 2 W ;p (); or jruj p 2 ru rvdx + (jruj p + juj p )dx + juj p ds: (jruj p 2 ru rv + juj p 2 uv)dx + juj p 2 uvds = juj p 2 uvds ; juj p 2 uvdx; for any v 2 W ;p (). Comparing the last equation to (3.2) and applying theorem 2.5 we obtain the result. Theorem 3.5 (Existence of L-S sequences for S()). Let X be W ;p () and F; G be dened in (2.4), (2.5) with a ; b and. Then there exists a nondecreasing sequence of nonnegative eigenvalues f n g of (3.3) obtained using the Ljusternik-Schnirelman principle such that n = n! as n!, where each n is an eigenvalue of the corresponding equation F (u) = G (u) dened in (2.). Proof. With a ; b and, F and G become F (u) = G(u) = juj p ds; And thus F (u) = G (u) is equivalent to juj p 2 uvds = (jruj p + juj p )dx: (jruj p 2 ru rv + juj p 2 uv)dx; 8v 2 W ;p (); or (jruj p 2 ru rv + juj p 2 uv)dx = juj p 2 uvds; 8v 2 W ;p (): Comparing the last equation to (3.3) and applying theorem 2.5 we obtain the result Remarks. We notice that theorem 2.5 assures the existence of an L-S sequence of eigenvalues for any closed subspace X of W ;p () and any functionals F, G dened in (2.4), (2.5). It follows that we can study 2

13 (3.6) eigenvalue problems with mixed boundary conditions, i.e., any combination of the Dirichlet, the No-ux, the Neumann, the Robin conditions. To give an example, let us consider the following Dirichlet- No-ux-Neumann problem: DP N() : 8 >< >: R 2 jruj p u = juj p 2 u; in ; u = ; on ; u = constant, on 2 ; ds = ; p 2 u n u n = ; on 3 ; where, 2, 3 are disjoint open connected subsets of such that [ 2 [ 3 =. Let X := fu 2 W ;p () : uj = ; uj 2 = constantg. Then X is a closed subspace of W ;p (). We say a pair (u; ) 2 X R is a weak solution of (3.6) if jruj p 2 rurvdx = juj p 2 uvdx; 8v 2 X: Then we apply theorem 2.5 to obtain a nondecreasing sequence of nonnegative eigenvalues f DP k N g of (3.6) such that N k DP k N D k for each k, where D k and N k are the k-th L-S eigenvalues of the Dirichlet problem and the Neumann problem, respectively. Furthermore, if an eigenfunction u of (3.6) is in W 2;p (), then u solves (3.6). To see this, we use Green's rst identity and follow the arguments used in the proof of lemma 3.2. In the next section, we will show that eigenfunctions are in C ; () provided that is regular enough. However, in order to have the result stated in lemma 3.2 we require that eigenfunctions are in W 2;p (). When N = 2, the authors in [9] showed that solutions to p u = in, p 6= 2, in general, do not have any better regularity than C ;. To our knowledge, higher degrees of regularity of eigenfunctions are unknown. 4. Regularity results on eigenfunctions In this section we shall prove boundedness of eigenfunctions and use this fact to obtain C ; () and C ; () smoothness of (weak) eigenfunctions of the nonlinear eigenvalue problems D(), P (), N(), R(), and S(). 4.. Boundedness for eigenfunctions. Let be a bounded domain in R N with C boundary and < p <. To obtain the regularity of eigenfunctions in and on the boundary we need to show that such eigenfunctions are in L (). If p > N the answer follows from Sobolev's embedding theorem (see theorem A.5). The following theorems extend lemma 3.2 of Drabek - Kufner - Nicolosi [3], which asserts that any nonnegative eigenfunction of the Dirichlet problem (.) is in L (). 3

14 Theorem 4. (Boundedness for solutions of D(); P (); N()). Let X be W ;p (), W ;p () R or W ;p () and let (u; ) 2 X R + be an eigensolution of the weak form (3.). Then u 2 L (). Proof. By Sobolev's embedding theorem it suces to consider the case p N. In this proof, we use the Moser iteration technique (see for example [3]). Let us assume rst that u. For M > dene v M (x) = minfu(x); Mg. Letting f(x) = x, if x M and f(x) = M, if x > M; it follows from theorem B.3 that v M 2 X \ L (): For k > dene ' = v kp+ M, then r' = (kp + )rvkp M. It follows that ' 2 X \ L (): Using ' as a test function in (3.), one obtains (kp+) jruj p 2 rurv M v kp M dx = juj p 2 uv kp+ M dx juj (k+)p dx; or (kp + ) (k + ) p and then (kp + ) (k + ) p (jrv k+ thus jrv k+ M jp dx = M jp + jv k+ M jp )dx (k + kvm k+ kp )p (kp + ) + u (k+)p dx; (kp + ) + (k + ) p u (k+)p dx; kuk (k+)p (k+)p : However, by Sobolev's embedding theorem, there is a constant c > such that kv k+ M k p c kv k+ M k; here we take p = Np N p ; if p < N and p = 2p, if p = N. Thus kv M k (k+)p kv k+ c M k+ k=(k+) p (k + )p (kp + ) + p(k+) kuk(k+)p : Using calculus, we can nd a constant c 2 > such that (k + )p p (kp + ) + p k+ c2 ; for any k >. Thus p k+ k+ kv M k (k+)p c c2 kuk (k+)p : Letting M!, Fatou's lemma implies p k+ (4.) kuk (k+)p c k+ c2 kuk (k+)p : Choosing k such that (k + )p = p, then (4.) becomes kuk (k +)p c k + c 4 p k + 2 kuk p :

15 Next, we choose k 2 such that (k 2 + )p = (k + )p, then taking k 2 = k in (4.), we have kuk (k2 +)p c k 2 + By induction we obtain c p k2 + kuk (kn+)p c 2 kuk (k2 +)p = c p p p kn+ c kn+ p k 2 + k2 + c 2 kuk (kn +)p ; 2 kuk (k +)p : where the sequence fk n g is chosen such that (k n +)p = (k n +)p, k =. n. It is easy to see that k n + = Hence P ni= k kuk (kn+)p c i + c P ni= p ki + 2 kuk p : As p p <, there is C > such that for any n = ; 2; ::: kuk (kn+)p Ckuk p ; where r n = (k n + )p! as n!. We will indirectly show that u 2 L (). Suppose u 62 L (), then there exists " > and a set A of positive measure in such that ju(x)j > Ckuk p + " = K, for all x 2 A. Then lim inf n! kuk r n lim inf n! A K rn =rn = lim inf n! KjAj=rn = K > Ckuk p ; which contradicts what has been established above. If u (as an eigenfunction of (3.) ) changes sign, we consider u +. By lemma B.2, u + 2 X. Dene for each M >, v M (x) = minfu + (x); Mg. Taking again ' = v kp+ M as a test function in X, we obtain (kp + ) jruj p 2 ru rv M v kp M dx = juj p 2 uv kp+ M dx; which implies (kp + ) jru + j p 2 ru + rv M v kp M dx = ju + j p 2 u + v kp+ M dx: Proceeding the same way as above we conclude that u + 2 L (). Similarly we have u 2 L (). Therefore u = u + + u is in L (). Corollary 4.2 (Global boundedness for R() solutions). Let (u; ) be an eigensolution of the weak form (3.2). Then u 2 L (). Proof. Let u be an eigenfunction of (3.2). We assume rst that u. Let v M = minfu; Mg and ' = v kp+ M. Theorem B.3 implies that v M; ' 2 5

16 W ;p () \ L () and v M j = minfuj ; Mg. Since >, we have (kp + ) (kp + ) jruj p 2 ru rv M v kp M dx jruj p 2 ru rv M v kp M dx + = juj p 2 uv kp+ M ds juj p 2 uv kp+ M dx: Then we use the argument used in the proof of theorem 4. to conclude that u 2 L (). If u changes sign, one can easily show that both u + and u are in L (). Therefore u 2 L (). Theorem 4.3 (Global boundedness for S() solutions). Let (u; ) be an eigensolution of the weak form (3.3). Then u 2 L (). Proof. Arguing as in theorem 4., we can assume that < p N and u. For M > dene v M (x) = minfu(x); Mg. For k > dene ' = v kp+ M, then r' = (kp + )rv M v kp M. It follows that ' 2 W ;p () \ L (). Using ' as a test function we have (kp+) jruj p 2 rurv M v kp M dx+ juj p 2 uv kp+ M dx = juj p 2 uv kp+ M ds; which implies (kp + ) (k + ) p jrvm k+ jp dx + juj p 2 uv kp+ M dx u (k+)p ds: Letting M!, using Fatou's lemma we obtain Since (kp+) (k+) p (kp + ) (k + ) p jru k+ j p dx + juj (k+)p dx < for any k >, we conclude that u (k+)p ds: (kp + ) (k + ) p (jru k+ j p + ju k+ j p )dx u (k+)p dx: By Sobolev's embedding theorem, there exists c > such that here we take q = (N )p N p kuk L (k+)q () c k+ ku k+ k L q () c ku k+ k;, if p < N and q = 2p if p = N. Thus (k + )p (kp + ) p(k+) kukl (k+)p () : Using the iteration method in the proof of theorem 4. we obtain that u 2 L (). Hence for any k >, u (k+)p (kp + ) dx (k + ) p jru k+ j p dx + u (k+)p dx u (k+)p dx: 6

17 Thus kuk (k+)p R u(k+)p dx (k+)p = (jj) Ckuk L () ; (k+)p kuk L () where jj = () is the boundary measure of. Letting k! we conclude that u 2 L () Regularity results on eigenfunctions. Let be a bounded domain in R N, < p <. Consider the degenerate elliptic equation (4.2) p u(x) = f(x; u(x)); in ; where f : R!R is a Caratheodory function, i.e. x 7! f(x; u) is measurable on for all u 2 R, u 7! f(x; u) is continuous for a.e. x 2. A function u 2 W ;p () is called a weak solution of (4.2) if loc jruj p 2 ru r'dx = f(x; u)'dx; 8' 2 C (): The following result was established by DiBenedetto [2] and Tolksdorf [29]. Theorem 4.4. Let u be a weak solution of (4.2) and let g(x) = f(x; u(x)), a.e. x 2. If g belongs to L q () with q > p N, then u is a C ; () p function for some >. In particular, the result holds if g 2 L (). Combining theorem 4., corollary 4.2, theorem 4.3 and theorem 4.4 we obtain Theorem 4.5. If u 2 W ;p () is an eigenfunction of (3.), (3.2) or (3.3), then u is in C ; (). Proof. We have shown in theorems 4., 4.3 and corollary 4.2 that any eigenfunction of (3.), (3.2) or (3.3) is in L (). If we dene g(x) = ju(x)j p 2 u(x) in, then g is also in L (). Therefore, it follows from theorem 4.4 that any eigenfunction of (3.), (3.2) or (3.3) is in C ; (). We shall also need, as an important tool in our development, a Harnack type inequality due to Trudinger [3] (theorem., p.724 and corollary., p.725) given in the following theorem. Theorem 4.6 (Harnack inequality). Let u 2 W ;p () be a weak solution of (4.2). Suppose that for all M < and for all (x; s) 2 ( M; M) the condition jf(x; s)j b (x)jsj p + b 2 (x) holds, where b ; b 2 are nonnegative functions in L () depending only on M. 7

18 Then if u(x) < M in a cube K(3r) := K x (3r), there exists a constant C such that max u(x) C min u(x); K(r) K(r) where K x (r) denotes a cube in R N of edgelength r and center x whose edges are parallel to the coordinate axes. Corollary 4.7. If u 2 W ;p () is a nonnegative eigenfunction of (3.), (3.2) or (3.3), then u is strictly positive in the whole domain. Proof. Let u be a nonnegative eigenfunction, then u is in C ; () \ L () and u 6. Suppose u(x ) = for some x 2. By theorem 4.6 u is identically zero on any cube in containing x and thus by connectedness we obtain u in, which is a contradiction. Therefore u is strictly positive in. Having proved that any (weak) eigenfunction of either D(), P (), N(), R() or S() is in L (), we now can use boundary regularity results for solutions of degenerate elliptic equations in Lieberman [22] to obtain that u is in C ; (). We state the results as follows: Theorem 4.8. Let be a bounded domain in R N with C ; boundary, <. Let u be a bounded weak solution of the problem (4.3) p u(x) = g(x); a.e. in ; u = ; on ; with kuk M. If g is in L () and is in C ; () with kgk K and kk C ; () L, then there exists a positive constant = (; N; p; M; K) such that u is in C ; () and kuk C ; ( ) C(; N; p; M; K; L; ): Theorem 4.9. Let be a bounded domain in R N with C ; boundary, <. Let u be a bounded weak solution of the problem p u(x) = g(x); a.e. in ; (4.4) p 2 u jruj n = (x; u); on ; with kuk M. If g is in L () with kgk K and satises the condition j(x; z) (y; w)j L[jx yj + jz wj ]; j(x; z)j L; for all (x; z) and (y,w) in [ M; M]. Then there exists a positive constant = (; N; p; M; K) such that u is in C ; () and kuk C ; ( ) C(; N; p; M; K; L; ): We recall that a weak solution u in W ;p () of (4.3) satises jruj p 2 ru r'dx = u 2 W ;p (); 8 g'dx; 8' 2 W ;p ();

19 while a weak solution u in W ;p () of (4.4) satises jruj p 2 ru r'dx = g'dx + (x; u)'dx; 8' 2 W ;p (): We observe that if or (x; z) = jzj p 2 z then satises the hypotheses of theorems 4.8 and 4.9 for any < minfp ; g. Therefore if is of class C ;, then eigenfunctions of (3.), (3.2) or (3.3) are in C ; (). 5. On the spectrum of the p-laplacian In this section we will study the spectra of the Dirichlet, No-ux, Neumann, Robin, and Steklov problems. In fact, as mentioned in Section, we will show for all the above problems the following: The rst eigenvalue is simple and only eigenfunctions associated to do not change sign. The set of eigenvalues is closed. The rst eigenvalue is isolated. The eigenvalue 2 is the second eigenvalue, i.e., 2 = inff : is an eigenvalue and > g: Here and 2 are the rst two eigenvalues of the L-S sequence established (using the Ljusternik-Schnirelman principle) in section 3 (theorems 3.3, 3.4, and 3.5). In what follows we assume that is a bounded domain in R N with C ; boundary, > ; and < p <. 5.. Simplicity of the rst eigenvalue. We will show that the rst element of the L-S sequence of eigenvalues is simple and only eigenfunctions associated to do not change sign. We recall from the regularity results of Section 4 that eigenfunctions are in C ; () if is of class C (theorem 4.5) and are in C ; () if is of class C ; (theorems 4.8 and 4.9). Let us recall the abstract problem which we have discussed in Section 2. We have established in theorem 2.5 that there exists a nonincreasing sequence of nonnegative values f n g tending to as n! such that n = sup H2An inf u2h F (u) and f n g are eigenvalues of F (u) = G (u), where F and G are dened in (2.4), (2.5) and A n is dened in (2.2) The Dirichlet, No-ux, Neumann problems. We rst notice that D > P = N = which agrees with inequality (.), a consequence of theorem 3.3. Here f D n g, f P n g, and f N n g denote the corresponding L-S sequences of eigenvalues of D(), P (), and N(), respectively. For simplicity we will write instead of D; P or N when there is no ambiguity. It is easy to see from the characterization of in (2.) that + = = inff (jruj p + juj p )dx : u 2 X and 9 juj p dx = g:

20 Thus (5.) = inf u2xnfg R jrujp dx R jujp dx ; where X is W ;p (); W ;p () R or W ;p (). It follows immediately that is the smallest eigenvalue. Theorem 5.. Given X = W ;p (); W ;p () R or W ;p (). Then the rst eigenvalue is simple. Moreover, all rst eigenfunctions do not change sign. Proof. If X = W ;p (), the result is due to Anane [2] and Lindqvist [23] and their technique of proof will be used again in the proof of theorem 5.4 to show the simplicity of the rst eigenvalue of the Robin problem. In case X = W ;p () R or W ;p () (No-ux or Neumann problem), by choosing v in (3.) we have = which is the smallest eigenvalue. And all rst eigenfunctions (eigenfunctions associated with = ) have zero gradients and thus are nonzero constant functions. Thus the eigenspace is simple. Next, let us show that any eigenfunction associated to an eigenvalue > has to change sign. Proposition 5.2. Let (u; ) be an eigenpair of (3.) with >. Then u has to change sign in. Proof. Again when X = W ;p () the result is proved in Anane [2] and Lindqvist [23]. Now let X = W ;p () R or X = W ;p (), as (u; ) satises (3.) for any v 2 X, by choosing v one obtains: Therefore, u has to change sign. juj p 2 u = : The Robin problem. It follows from (2.) and theorem 3.4 that the rst eigenvalue can be characterized as = inf f jruj p dx + juj p ds : juj p dx = g: u2w ;p () Lemma 5.3. Let u be an eigenfunction associated with, then either u > or u < in. Proof. We notice that if u is a rst eigenfunction, so is juj. By the Harnack inequality (theorem 4.6), either juj > in the whole domain or juj. To see this, let assume juj(x ) = for some x 2. Then theorem 4.6 implies that juj is identically zero in a ball centered at x. Covering by such balls we conclude that u in, which is a contradiction. Thus, juj must be positive in. By the continuity of u, either u or u is positive in the whole domain. 2

21 Theorem 5.4. The principal eigenvalue is simple, i.e., if u and v are two eigenfunctions associated with, then there exists c such that u = cv. Proof. By lemma 5.3 we can assume u and v are positive in. In this proof we use the technique that Lindqvist [23] used to prove the simplicity of the rst eigenvalue of the Dirichlet problem. Let = (u + ")p (v + ") p (u + ") p and = (v + ")p (u + ") p (v + ") p ; where " is a positive parameter. Then r = + (p ) v + " p ru p u + " v + " p rv: u + " Since u and v are bounded (corollary 4.2), r is in L p () and thus is in W ;p (). By symmetry, the gradient of the test-function in the corresponding equation for v has a similar expression with u and v interchanged. Set u " = u + " and v " = v + ": Inserting these test functions into their respective equations obtained from (3.2) and adding these equations, we obtain u p = = + p p + " u p " vp (u p v" p " vp ")dx + (p ) v" p v" p u " u p u p " u " p jru " j p + jru " j p 2 ru " rv " + p u" vp v p " (u p " v p ")ds (u p " v p ")(jr ln u " j p jr ln v " j p )dx = L " + v p "jr ln u " j p 2 r ln u " (r ln v " r ln u " )dx u p " jr ln v "j p 2 r ln v " (r ln u " r ln v " )dx u p (u p " vp ")ds vp u p " v" p u p u p " vp v p " (u p " v p ")dx: 2 + (p ) u" v " p jrv " j p dx p # jrv " j p 2 rv " ru " v "

22 Taking x = r ln u " ; y = r ln v " and vice versa, it follows from inequality (A.3) in lemma A.2 that L " = p p : (u p " vp ")(jr ln u " j p jr ln v " j p )dx v p "jr ln u " j p 2 r ln u " (r ln v " r ln u " )dx u p " jr ln v "j p 2 r ln v " (r ln u " r ln v " )dx By the Dominated Convergence Theorem, which also holds in L p (); it is apparent that u p (5.2) lim "! vp (u p + u p " v" p " v")dx p = : and (5.3) lim "! + u p u p " vp (u p v" p " v")ds p = : (We have from theorem 4.8 that u and v are in C ; ().) Let us consider the case p 2. According to inequality (A.) in lemma A.2 we have C(p) v" p + u p jv " ru e u " rv " j p dx " L " u p u p " for every " >. lemma we obtain vp (u p v" p " v")dx p u p u p " vp (u p v" p " v")ds; p Recalling (5.2), (5.3), letting "! +, and using Fatou's lim v "ru e u " rv " = a.e. in ; "! + and thus vru = urv a.e. in : u We obtain immediately that r v =, i.e., there is a constant k such that u = kv a.e. in. By continuity, u = kv at every point in. This proves the result for the case p 2. The case < p < 2 is very similar. Applying inequality (A.2) in lemma A.2 we obtain C(p) L " (u " v " ) p (u p " + v") p jv "ru e u " rv " j 2 dx jv " ru e + u " rv " j2 p u p u p " vp (u p v" p " v")dx p 22 u p u p " vp (u p v" p " v")ds; p

23 for every " >. Using (5.2) and (5.3), we obtain that u = kv for some constant k. Proposition 5.5. Let v be an eigenfunction associated to 6=. Then v changes sign in. Proof. Suppose that v does not change sign in, then by theorem 4.6 we can assume that v > in. Let u be an eigenfunction associated to. Making similar computations as in the proof of theorem 5.4 we conclude that u p u p u p " = L " : vp v p " Letting "! + we obtain (u p " vp ")dx ( ) u p " (u p v p )dx : vp v p " Hence if we take ku instead of u we obtain for any k > that ( ) (k p u p v p )dx ; (u p " vp ")ds: R R which yields a contradiction if we choose k p > vp dx= up dx. Therefore, v changes sign in The Steklov problem. Arguing as for the Dirichlet and Robin problems, one sees that the rst eigenvalue of S() can be characterized as = inf u2w ;p () (jruj p + juj p )dx : juj p ds = Lemma 5.6. If u is an eigenfunction associated with, then either u > or u < in. Proof. We have that ju j is also a minimizer. It follows from the Harnack inequality (theorem 4.6) and theorem 4.9 that ju j > on and ju j is in C ; (). Thus if there is x 2 such that u (x ) =, by the Hopf lemma (see [33], theorem 5) we obtain ju j n (x ) <. But the boundary condition jruj p 2 u n = jujp 2 u imposes that ju j n (x ) =. This contradiction implies that ju j > in, which proves the lemma. Theorem 5.7. The principal eigenvalue is simple, i.e., if u and v are two eigenfunctions associated with, then there exists a constant c such that u = cv. Proof. The proof of this theorem is due to Martnez and Rossi [24] in which they use the technique developed in [2, 23] (see theorem 5.4). However, in order to carry the arguments made in [24], it requires that u, and v are bounded eigenfunctions. For the sake of completion we include the proof here. 23 :

24 By lemma 5.6 we can assume u and v are positive in. We take = (u p v p )=u p and 2 = (v p u p )=v p as test functions to obtain u p v p u p v p jruj p 2 ru r dx = u p juj p 2 u ds u p u p v p juj p 2 u dx: and v p u p jrvj p 2 rv r v p Adding both equations we get (5.4) u p v p = jruj p 2 ru r Using the fact that u p v p r u p u p the rst term of (5.4) becomes jruj p dx p jr ln uj p u p dx p dx = u p v p u p jvj p 2 v v p v p u p jvj p 2 v v p dx + jrvj p 2 rv r = ru p vp vp rv + (p ) up u p ru; v p u p jrujp 2 rv rudx + (p ) v p jr ln uj p 2 r ln u r ln vdx + (p ) dx: v p u p v p ds dx: v p u p jrujp dx = jr ln uj p v p dx: We have an analogous expression for the second term of equation (5.4) and thus (5.4) becomes = p p (u p v p )(jr ln uj p jr ln vj p )dx v p jr ln uj p 2 r ln u (r ln v r ln u)dx u p jr ln vj p 2 r ln v (r ln u r ln v)dx: For p 2, letting fx; yg be fr ln u; r ln vg and applying inequality (A.) in lemma A.2 we obtain C(p)jr ln u r ln vj p (u p + v p )dx: Hence, = jr ln u r ln vj: This implies u = kv. For p < 2 we use inequality (A.2) in lemma A.2 to obtain the same result. 24

25 Proposition 5.8. Let u be an eigenfunction associated to 6= then u changes sign on, i.e., the sets fx 2 : u(x) > g and fx 2 : u(x) < g have positive boundary measure. Proof. Suppose that u does not change sign in, then we can assume that u > in due to the Harnack inequality (theorem 4.6). Let u be an eigenfunction associated to. Making similar computations as in [24] we conclude that ( ) (u p up )ds C jr ln u r ln u j p (u p + up )dx: Hence if we take ku instead of u we obtain for any k > that (u p kp u p )ds ; R which yields a contradiction if we choose k p < upds=r up ds. Therefore u changes sign in. Suppose that u does not change sign on. We then can assume u on. Using u + as a test function in (3.3) we conclude jruj p 2 ruru + dx + juj p 2 uu + dx = : Since u changes sign in, the left hand side is strictly positive. This contradiction implies that u changes sign on Closedness of the set of eigenvalues. We will show that the spectra of the Dirichlet, the No-ux, the Neumann, the Robin, and the Steklov problems are closed. Precisely, we will prove that the sets of all numbers that satisfy (3.), (3.2) or (3.3), respectively, are closed. We rst show the closedness of the sets of eigenvalues of the Dirichlet, the No-ux, the Neumann, and the Robin problems. Theorem 5.9. The sets of eigenvalues of D(), P (), N(), and R() (equations (3.) and (3.2)) are closed. Proof. Let X be either W ;p (), W ;p () R or W ;p (). Let f(u n ; n )g be a sequence of eigenpairs of (3.) or (3.2) such that n! for some. Without loss of generality we can assume ku n k = and thus fu n g has a weakly convergent subsequence, i.e., we may assume that u n * u in X. By lemma 2.3, hb(u n ) B(u); u n ui (ku n k p kuk p )(ku n k kuk): However, as the (u n ; n )'s are eigenpairs, the left hand side equals hb(u n ) B(u); u n ui = ( n + )hau n ; u n ui + hbu; u n ui which tends to, as n!. 25

26 Since hb(u n ) B(u); u n ui! we conclude that ku n k! kuk as n!. It follows that u n! u in X, since W ;p () is locally uniformly convex (with respect to an equivalent norm, see the proof of proposition 2.4 or [3]). To show that is an eigenvalue of (3.) or (3.2) and u is an associated eigenfunction we need to show for any v 2 X as n!, (5.5) (5.6) (5.7) jru n j p 2 ru n rvdx! ju n j p 2 u n vdx! ju n j p 2 u n vdx! jruj p 2 ru rvdx; juj p 2 uvdx; juj p 2 uvdx: Let w n = jru n j p 2 ru n and w = jruj p 2 ru. Then as u n!u in W ;p () w n (x)! w(x); a.e. in and jw n j p p p dx! jwj p dx: It follows from lemma A. that w n! w in L p=(p ) (). Thus, by Holder's inequality we obtain (5.5). Similarly, we have (5.6) and (5.7). We now show the closedness of the spectrum of the Steklov problem. Theorem 5.. The set of eigenvalues of (3.3) is closed. Proof. Let f(u n ; n )g be a sequence of eigenvalues of (3.3) such that n! for some. Without loss of generality we can assume ku n k = and thus fu n g has a weakly convergent subsequence, i.e., we may assume that u n * u in W ;p (). We recall that each (u n ; n ) satises jru n j p 2 ru n rvdx + ju n j p 2 u n vdx = n ju n j p 2 u n vds; or hb(u n ); vi = n for all v 2 W ;p (). By lemma 2.3, we have ju n j p 2 u n vds; hb(u n ) B(u); u n ui (ku n k p kuk p )(ku n k kuk): However, the left hand side equals hb(u n ) B(u); u n ui = n ju n j p 2 u n (u n u)dx + hbu; u n ui; which tends to as n!. The rest of the proof follows from the argument that we used in the proof of theorem 5.9. Therefore, (u; ) is an eigenpair of (3.3) Isolation of the rst eigenvalue. 26

27 5.3.. The Dirichlet, No-ux, Neumann, and Robin problems. Let us recall (see appendix C) that if u is a continuous function on then the set (u) = fx 2 : u(x) = g is called the zero set of u and any component! of n (u) is called a nodal domain of u. Given, an eigenvalue of either (3.) or (3.2), and u an associated eigenfunction to, we dene: N(u) = the number of components of n (u); N() = supfn(u) : u is an associated eigenfunction to g: We will show that N() is nite. Theorem 5.. Let (u; ) be an eigenpair of (3.) or (3.2) and let! be a nodal domain of u. Then there exist two constants c and r independent of!; u and such that the Lebesgue measure j!j [( + )c p )] r =: C > : Therefore N() jj=c. (In the case X = W ;p () R we assume further that is connected so that theorem C.3 holds.) Proof. Let X be either W ;p (), W ;p () R or W ;p (). We will prove the theorem for (u; ) satisfying (5.8) jruj p 2 ru rvdx + juj p 2 uvds = juj p 2 uvdx; 8v 2 X; with a given. Thus, if we take = we obtain the result for the Dirichlet, the No-ux, and the Neumann problems and if we take >, X = W ;p () we obtain the result for the Robin problem. We rst notice that the regularity results from section 4 assure that u is in C(). Let u = u! be the restriction of u on!. Then by theorem C.3 we have u 2 X. Furthermore, ru = ru!. Taking the test function v in (5.8) to be u and using lemmas C.3 and C.4, we obtain jruj p dx + juj p ds = juj p dx =! juj p dx: Adding R jujp dx to both sides and using Holder's inequality, we conclude Thus (jruj p + juj p )dx + juj p dx = ( + ) (jruj p + juj p )dx ( + )j!j p p ( = ( + )j!j p p (!! juj p dx: juj p dx) p p Here we choose p = Np N p, if p < N and p = 2p, if p N. 27 juj p dx) p p :

28 By Sobolev's embedding theorem one has which implies kuk L p () ckuk = c (jruj p + juj p )dx kuk p L p () c p ( + )j!j p p kuk p L p () : p ; Since u 6=, we conclude that j!j (( + )c p ) r, where r = N=p, if p < N and r = 2, if p N. Corollary 5.2. Let (u; ) be an eigenpair of (3.) or (3.2) and let + = fx 2 : u(x) > g such that j + j >. Then there exist two constants c and r independent of u and such that j + j [( + )c p )] r = > : The result also holds for = fx 2 : u(x) < g; if j j >. In fact, the corollary is still valid if is only of class C (so that the embedding theorem A.5 can be applied) instead of class C ;. Proof. By lemma B.2 we have u + is in X (the lemma does not require any regularity on the boundary ). Replacing u in the proof of theorem 5. by u + and using the same argument, we obtain the corollary. We are in the position to prove the isolation of the rst eigenvalue. Theorem 5.3. The rst eigenvalue of (3.) or (3.2) is isolated. Proof. Let X be W ;p (); W ;p () R or W ;p (). Suppose is not isolated. Then by theorem 5.9 there exists a sequence of eigenpairs f(u n ; n )g such that as n!, u n!u in X and n!, where u is an eigenfunction corresponding to. We can assume that ku n k = kuk = for any n and that u > in. Dene for each n n = fx 2 : u n (x) < g and + n = fx 2 : u n (x) > g: By corollary 5.2, there exists a > such that j n j a > for any n, i.e., the measure j n j is uniformly bounded from below. Since u is continuous and positive on, there exists " > such that j " j > jj a=4, where " = fx 2 : u(x) > "g. By Egoro's theorem there is a measurable subset E of " such that jej > j " j a=4 and u n converges uniformly to u on E. Thus there exists n " such that ju n (x) u(x)j < "=2, for any x 2 E and any n n ". In particular E + n ". Thus j n " j < jj jej < jj j " j + a=4 < jj jj + a=2 = a=2. We have arrived at a contradiction. Therefore is isolated. 28

29 The Steklov problem. Given, an eigenvalue of (3.3) and u an eigenfunction associated to, theorem 4.9 implies that the eigenfunction u is in C ; (). Thus we may dene: (u) = fx 2 : u(x) = g; N(u) = the number of components of n (u); N() = supfn(u) : u is an associated eigenfunction to g: We will show again that N() is nite. Theorem 5.4. Let (u; ) be a (weak) eigenpair of S() and let! be a component of n (u). Then there exists a constant C independent of!; u and such that j \!j C ; where = (N )=(p ), if < p < N and = 2, if p N. Here jaj denotes the boundary measure of a measurable subset A of. Consequently, N() jj =C. Proof. Let u = u!, then by theorem C.3 we have u 2 W ;p (). Taking u as a test function in (3.3) we obtain that jruj p 2 ru ru + juj p 2 uudx = juj p 2 uuds: Hence, using Holder's inequality in L p () and lemma C.4 we have kuk p W ;p () juj p ds = j \!j = : If < p < N, we choose = (N )=(N p) and = (N )=(p ). Then we use Sobolev's embedding theorem (theorem A.5-(iii)) to conclude that there exists a constant C such that kuk p L p () Ckukp W ;p () : If p N we choose = = 2 and we argue as before using the embedding W ;p (),! L p (). Corollary 5.5. Let u be an eigenfunction associated to 6=, then there exists a constant C such that j + j C and j j C ; where + = \ fu > g; = \ fu < g. We can now establish the isolation of. Theorem 5.6. The principal eigenvalue of S() is isolated. That is, there exists a > such that is the unique eigenvalue in [; a]. 29

30 Proof. Suppose is not isolated. Then by theorem 5., there exists a sequence of eigenpairs f(u n ; n )g such that as n!, u n!u in W ;p () and n!, where u is an eigenfunction associated to. We can assume ku n k = kuk = for any n and u > in. Dene for each n n = fx 2 : u n (x) < g and + n = fx 2 : u n (x) > g: Then, by corollary 5.5, there is a > such that j n j a > for any n, i.e., the measure j n j is uniformly bounded from below. Since u is continuous and positive on there exists " > such that j " j > jj a=4, where " = fx 2 : u(x) > "g. By Egoro's theorem, there is a subset E of " such that jej > j " j a=4 and u n converges uniformly to u on E. Thus there exists n " such that ju n (x) u(x)j < "=2 for any x 2 E and any n n e. In particular E + n ". Thus j n " j < jj jej < jj j " j + a=4 < jj jj + a=2 = a=2. We have arrived at a contradiction. Therefore is isolated On the second eigenvalue. In this subsection we will show that the eigenvalue 2 of the L-S sequence of eigenvalues whose existence was established in theorems 3.3, 3.4, and 3.5 is actually the smallest eigenvalue of the spectrum that is greater than the principal eigenvalue. This work is motivated by the result in [3] in which Anane and Tsouli consider the Dirichlet problem. We begin by proving an interesting property on the number of nodal domains of a given eigenvalue of the Dirichlet, the Neumann or the Robin problems. Proposition 5.7. For any eigenvalue of (3.) or (3.2), we have N () : Here N() is the maximal number of nodal domains associated with (see theorem 5.) and N () is the N()-th eigenvalue taken from the L-S sequence of theorem 3.3 or theorem 3.4. Proof. Let r = N(), then there is an eigenfunction u 6= associated to such that N(u) = r. Let! ;! 2 ; :::;! r be the r-components of n (u). For i = ; 2; :::; r we dene v i (x) = 8 < : u(x) [ R! i juj p dx] =p ; if x 2! i; ; if x 2 n! i : By theorem C.3 we have that v i 2 X (=W ;p (); W ;p () R, or W ;p ()), for i = ; 2; :::; r. Let X r denote the subspace of X spanned by fv ; v 2 ; :::; v r g. Since the v i 's are linearly independent, we have that 3