Multiple Solutions for Parametric Neumann Problems with Indefinite and Unbounded Potential

Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 8, Number 2, pp. 281 293 (2013) http://campus.mst.edu/adsa Multiple Solutions for Parametric Neumann Problems with Indefinite and Unbounded Potential Nikolaos S. Papageorgiou and George Smyrlis National Technical University of Athens Department of Mathematics Zografou Campus, Athens 157 80, Greece npapg@math.ntua.gr gsmyrlis@math.ntua.gr Abstract We consider a parametric Neumann problem with an indefinite and unbounded potential. Using a combination of critical point theory with truncation techniques and with Morse theory, we produce four nontrivial smooth weak solutions: one positive, one negative and two nodal (sign changing). AMS Subject Classifications: 35J20, 35J60, 58E05. Keywords: Indefinite and unbounded potential, truncations-perturbations, mountain pass theorem, Morse theory. 1 Introduction Let R N (N 3) be a bounded domain with a C 2 -boundary. We study the following Neumann problem u(z) + β(z)u(z) = λu(z) f(z, u(z)) in, u n = 0, on. (P) λ Here β L q () (q > N) and is in general indefinite (i.e., it is sign-changing) and unbounded from below. Also λ R is a parameter, f(z, x) is a Carathéodory perturbation and n( ) is the outward unit normal on. Moreover, we assume that for a.a. z f(z, ) is superlinear near ±. The problem (P) λ was studied in the past under Dirichlet boundary conditions and only in the special case where β 0, f(z, x) = f(x), z, x R. We refer to Received October 28, 2012; Accepted December 3, 2012 Communicated by Eugenia N. Petropoulou

282 N. S. Papageorgiou and G. Smyrlis the works [3, 4, 11], where it is proved that for λ > λ 2 = (the second eigenvalue of the Dirichlet ), the problem (P) λ has at least three nontrivial smooth weak solutions. No sign information is provided for the third solution. In this paper, we extend the aforementioned works to Neumann boundary value problems with an indefinite and unbounded potential β( ) and we produce four nontrivial smooth weak solutions: one positive, one negative and two nodal (sign-changing). Our approach uses variational methods based on the critical point theory, truncation techniques and Morse theory (critical groups). 2 Preliminaries Let X be a Banach space, X its topological dual and let, denote the duality brackets for the pair (X, X). Let ϕ C 1 (X). We say that ϕ satisfies the Palais Smale condition (the PS-condition for short), if the following holds: Every sequence {x n } n 1 X such that {ϕ(x n )} n 1 is bounded and ϕ (x n ) 0 in X, admits a strongly convergent subsequence. This compactness-type condition leads to a deformation theorem from which we derive minimax characterizations of certain critical values of ϕ, for example the mountain pass theorem. Next, let us recall some basic definitions and facts from Morse theory. Given ϕ C 1 (X) and c R, we introduce the following sets: ϕ c = {x X : ϕ(x) c}, K ϕ = {x X : ϕ (x) = 0}. Let Y 1, Y 2 be two topological spaces such that Y 2 Y 1 X. For every integer k 0, by H k (Y 1, Y 2 ) we denote the kth-relative singular homology group for the topological pair (Y 1, Y 2 ). Recall that for k < 0, H k (Y 1, Y 2 ) 0. The critical groups of ϕ at an isolated x K ϕ are defined by C k (ϕ, x) = H k (ϕ c U, ϕ c U \ {x}) for all k 0, where c = ϕ(x) and U is a neighborhood of x such that K ϕ ϕ c U = {x}. The excision property of singular homology theory implies that the above definition of critical groups, is independent of the particular neighborhood U. If x is a local minimizer of ϕ, then C k (ϕ, x) = δ k,0 Z, for all k 0. Suppose that ϕ satisfies the PS-condition and inf ϕ(k ϕ ) >. Let also c < inf ϕ(k ϕ ). The critical groups of ϕ at infinity are defined by C k (ϕ, ) = H k (X, ϕ c ) for all k 0. The Second Deformation Theorem implies that the above definition of critical groups at infinity, is independent of the particular level c < inf ϕ(k ϕ ) used. If ϕ is bounded from below, then C k (ϕ, ) = δ k,0 Z, for all k 0.

Multiple Solutions for Parametric Neumann Problems 283 Suppose that K ϕ is finite. We set M(x) = k 0( 1) k rankc k (ϕ, x), x K ϕ. A particular case of the Morse relation says that M(x) = ( 1) k rankc k (ϕ, ). (2.1) x K ϕ k 0 Suppose X = H is a Hilbert space, x H, U is a neighborhood of x and ϕ C 2 (U). If x K ϕ, then the Morse index of x, denoted by µ(x), is defined to be the supremum of the dimensions of the vector subspaces of H on which ϕ (x) is negative definite. We say that x K ϕ is nondegenerate, if ϕ (x) is invertible. If x K ϕ is nondegenerate and has Morse index µ = µ(x), then C k (ϕ, x) = δ k,µ Z, for all k 0. (2.2) Here δ k,µ is the Kronecker symbol. Suppose that X = Y V. We say that ϕ C 1 (X) has a local linking at 0, if there exists r > 0 such that ϕ(x) 0 (resp. 0) for all x Y (resp. for all x V ) with x r. From [12, Proposition 2.3], we have the following. Proposition 2.1. If H is a Hilbert space, ϕ C 2 (H), 0 is an isolated critical point of ϕ, ϕ (0) is a Fredholm operator and ϕ has a local linking at 0 with respect to the direct sum decomposition H = Y V with dim Y = µ = the Morse index of 0, then C k (ϕ, 0) = δ k,µ Z, for all k 0. In the study of problem (P) λ, we will use the Sobolev space (H 1 (), ) and the ordered Banach space C 1 (). The order cone of C 1 () and its interior are given by C + = { u C 1 () : u 0 }, intc + = {u C + : u > 0 }, respectively. We know that intc +. If h 1, h 2 H 1 () such that h 1 (z) h 2 (z), a.e. on, then we write h 1 h 2 and we define the order interval [h 1 h 2 ] = {u H 1 () : h 1 u h 2 }. If h 1, h 2 C 1 () with h 2 h 1 intc +, then we write h 1 h 2 and we define the order interval (h 1 h 2 ) = { u C 1 () : h 1 u h 2 }. The latter is an open subset of the Banach space (C 1 (), C 1 () ).

284 N. S. Papageorgiou and G. Smyrlis Finally, let g(z, x), z, x R be a Carathéodory function and u 1, u 2 : R be two measurable functions such that u 1 (z) u 2 (z). a.e. in. The truncation of g at the pair (u 1, u 2 ) is defined by g(z, u 1 (z)) if x < u 1 (z) ĝ(z, x) = g(z, x) if u 1 (z) x u 2 (z) g(z, u 2 (z)) if u 2 (z) < x. Then ĝ(z, x), z, x R is also a Carathéodory function. 3 Regularity and Sign of the Solutions We impose the following hypotheses on β( ), f(, ). H β : β L q () for some q > N, β + L (). H f : f : R R is a Carathéodory function such that f(z, 0) = 0 a.e. in and (i) f(z, x) α(z) + c x r 1 for a.a. z, all x R, with α L () +, c > 0, 2 < r < 2 = 2N/(N 2). (ii) lim x 0 f(z, x) x = 0 uniformly for a.a. z. Proposition 3.1. Under H β, H f (i), (ii), let u H 1 () be a weak solution of problem (P) λ. Then u C 1 (). If in addition u 0 and u 0 (resp. u 0) on, then u intc + (resp. u intc + ). Proof. Hypotheses H f (i), (ii) imply that We set f(z, x) c 1 ( x + x r 1 ) for a.a. z, all x R, with c 1 > 0. (3.1) f(z, u(z))/u(z) if u(z) 0 h(z) = 0 if u(z) = 0. Due to H f (i), (ii), h is measurable. Moreover (3.1) yields h(z) c 1 (1 + u(z) r 2 ), for a.a. z. (3.2) Since 2 < r < 2, we also have 0 < (r 2)N/2 < 2. By the Sobolev embedding theorem, u H 1 () L 2 () L (r 2)N/2 (), so h L N/2 () (see (3.2)). It follows that γ( ) = λ h( ) β( ) L N/2 (). But now we have u(z) = γ(z)u(z) a.e. in, u n = 0, on. (3.3)

Multiple Solutions for Parametric Neumann Problems 285 Invoking the regularity theory of Wang [15, Lemma 5.1], we get that u L t () for all t 1. This fact combined with (3.2) and with H β gives that γ L q (). To proceed, choose s (N, q) and set p = q/s > 1. Applying Hölder s inequality for the exponents p and p = p/(p 1) we infer that γ( )u( ) L s () (recall that u L t () for all t 1). Then u L s () (see (3.3)). Now the regularity theory of Wang [15, Lemma 5.2] implies that u W 2,s (). But the Sobolev embedding theorem says that for τ = 1 N/s (0, 1), W 2,s () C 1,τ () C 1 (). Thus, u C 1 (). Next, suppose that u 0 and u 0 on. Since u C 1 (), we have that h L () (see (3.2)) and u(z) = (β(z) λ + h(z))u(z) ( β + + λ + h )u(z), for a.a. z. It follows that u intc + (see Vazquez [14]). We argue in a similar way in the case where u 0, u 0. 4 The Spectrum of u + βu In this section we study the spectral properties of the differential operator u u + βu. So, we consider the following linear eigenvalue problem: u u(z) + β(z)u(z) = λu(z) in, = 0 on. (4.1) n We assume that H β holds (see Section 3). By using standard arguments mainly based on the continuous embedding H 1 () L 2N/(N 2) (), we may prove the following. Proposition 4.1. Let σ : H 1 () R be defined by σ(u) = Du 2 2 + β(z)u(z) 2 dz for all u H 1 () (Du is the gradient of u). Then (i) the smallest eigenvalue λ 1 R of (4.1) is given by [ < λ σ(u) 1 = inf : u H 1 (), u 0 u 2 2 (ii) there exist ξ > 0, ξ > λ 1 and c > 0 such that ] ; (4.2) σ(u) + ξ u 2 2 c u 2 for all u H 1 (). (4.3) By using Proposition 4.1 and working in a similar way as in the case of we obtain a sequence of eigenvalues < λ 1 < λ 2 <... < λ m <..., λm + as m. Also there is a corresponding sequence {û m } m 1 of eigenfunctions of

286 N. S. Papageorgiou and G. Smyrlis (4.1) such that û m C 1 (), for all m 1 (see Proposition 3.1) and {û m } m 1 is an orthonormal basis of L 2 () and an orthogonal basis of H 1 (). For each m 2, we have the following variational characterization: [ σ(u) λ m = sup u 2 2 where H m = m ] [ σ(u) : u H m, u 0 = inf : u H u m 1, u 0 2 2 E( λ i ) and H 1 () = H m 1 H m 1. ], (4.4) E( λ i ), Hm 1 = i=1 i m Here E( λ i ) denotes the eigenspace corresponding to the eigenvalue λ i. The infimum in (4.2) is realized on E( λ 1 ) and both the supremum and infimum in (4.4) are realized on E( λ m ). We know that λ 1 is simple (i.e., dim E( λ 1 ) = 1) and is the only eigenvalue with eigenfunctions of constant sign. All the higher eigenvalues have nodal eigenfunctions. In the sequel by û 1 C + \ {0} we denote the L 2 -normalized (i.e., û 1 2 = 1), positive eigenfunction. Then Proposition 3.1 implies that û 1 int C +. The eigenspaces E( λ i ), i 1 all have the so-called Unique Continuation Property (UCP), namely if u E( λ i ) and u vanishes on a set of positive measure, then u 0 (see Garofalo Lin [9] and de Figueiredo Gossez [8]). An easy application of the UCP and of (4.4), reveals that if m 1, then there exists ξ > 0 such that σ(u) λ m+1 u 2 2 ξ u 2, for all u H m. (4.5) 5 Solutions of Constant Sign In addition to hypotheses H f (i), (ii), we also assume that f(z, x) H f : (iii) lim x x = + uniformly for a.a. z (superlinearity). Hypothesis H f (iii) enables us to construct an upper solution u intc + and a lower solution u intc + of the problem (P) λ. In fact, u (resp. u) is an appropriate positive (resp. negative) multiple of û 1 (see Section 4). Proposition 5.1. If hypotheses H(β) and H f (i) (iii) hold and λ > λ 1, then problem (P) λ has at least two nontrivial constant sign smooth solutions u 0 [0 u] int C + and v 0 [u 0] ( int C + ). Sketch of the proof. First we construct u 0 (the construction of v 0 is similar). Let g λ +(z, x) be the truncation of the function (λ + ξ)x f(z, x) at the pair (0, u) (see Section 2), where ξ appears in (4.3).

Multiple Solutions for Parametric Neumann Problems 287 We set G λ +(z, x) = ϕ λ +(u) = 1 2 Du 2 2 + 1 2 x 0 g λ +(z, s)ds and consider ϕ λ + C 1 (H 1 ()) defined by (ξ + β(z))u(z) 2 dz G λ +(z, u(z))dz, for all u H 1 (). The functional ϕ λ + is coercive (this may be shown with the aid of (4.3)) so, it attains its minimum at some point u 0 H 1 (). In particular, u 0 is a critical point of ϕ λ +. By using suitable test functions and taking into account the definition of g λ +, we may show that u 0 is a weak solution of the problem (P) λ such that u 0 [0 u]. Moreover, employing H f (ii) together with the hypothesis that λ > λ 1, we may choose t > 0 sufficiently small such that ϕ λ +(tû 1 ) < 0 = ϕ λ +(0). This fact implies that u 0 0 (recall that u 0 is a global minimizer of ϕ λ +). Now we apply Proposition 3.1. Proposition 5.2. Under hypotheses of Proposition 5.1, the problem (P) λ has a smallest positive smooth solution u and a biggest negative smooth solution v. Sketch of the proof. We establish the existence of the smallest positive solution (the existence of the biggest negative solution is proved in a similar way). Let S λ + = { u H 1 () : u [0 u], u 0, u is a solution of (P) λ }. From Propositions 5.1 and 3.1 we know that S+ λ and S+ λ int C +. We show that S+ λ has a minimal element with respect to the pointwise order ( ). To this end, let C S+ λ be a chain. We can find a decreasing sequence {u n } n 1 C such that inf C = inf u n. Each u n is a weak solution of (P) λ and sup u n u <. n 1 n 1 By using u n as a test function, we may show that {u n } n 1 H 1 () is bounded. By passing to subsequences we may assume that {u n } n 1 weakly converges to some u H 1 (). Exploiting the strong monotonicity of the operator u u combined with the Sobolev embedding theorem, we deduce that u n u in H 1 (). Then passing to the limit as n, we infer that u solves (P) λ and u [0 u]. It remains to prove that u 0. Arguing indirectly, suppose that u = 0. We have u n 0 in H 1 (). We set y n = u n, n 1. As before, by passing to subsequences, u n we may assume that y n y in H 1 () for some y H 1 () with y = 1, y 0 (5.1) and also We set u n (z) 0, y n (z) y(z) a.e. in. (5.2) g n (z) = f(z, u n(z)) u n = f(z, u n(z)) y n (z), n 1, z. u n (z)

288 N. S. Papageorgiou and G. Smyrlis By using (3.1), (5.1), (5.2), combined with hypothesis H f (ii) and Lebesgue s theorem, we may check that g n 0 in L 2 (). (5.3) Now we observe that each y n is a weak solution of the problem (P) λ, where the nonlinearity f(, ) is replaced by the function g n ( ). Then by passing to the limit as n and taking into account (5.1), (5.3) we infer that y is a λ-eigenfunction of the operator u u + βu. Since λ > λ 1, y must be nodal (see Section 4) which contradicts (5.1). Therefore u = inf C S λ +. Since C S λ + is an arbitrary chain, Zorn s lemma guarantees that S λ + has a minimal element u S λ +. But as in [2, Lemma 1], we may show that (S λ +, ) is downward directed, i.e., if u 1, u 2 S λ +, we can find u S λ + such that u u 1, u u 2. Hence, u intc + is the smallest positive solution of (P) λ. 6 Nodal Solutions Under more restrictive hypotheses on the data of the problem (P) λ, we are going to construct two nodal smooth solutions. For λ > λ 1, we consider the corresponding energy functional ϕ λ defined by ϕ λ (u) = 1 2 σ(u) λ 2 u 2 2 + F (z, u(z))dz, u H 1 (), where σ(u) is defined in Proposition 4.1 and F (z, x) = x 0 f(z, s)ds, z, x R. Under hypotheses H β, H f (i),(iii), ϕ λ C 1 (H 1 ()) and it is coercive, thus it satisfies the Palais Smale condition. It is not clear whether if ϕ λ has the Geometry of the mountain pass theorem (GMPT) or not. Nevertheless, we have the following. Proposition 6.1. Under hypotheses H β, H f (i) (iii) and for λ > λ 1, there exists a coercive functional ψ λ C 1 (H 1 ()), such that (i) ϕ λ, ψ λ coincide on [v u ] and K ψλ K ϕλ [v u ]. (ii) u, v are both local minimizers of ψ λ. (iii) ψ λ has the GMPT and there exists y 0 K ψλ \ {u, v } such that C 1 (ψ λ, y 0 ) 0. Sketch of the proof. (i), (ii). Let h λ +, h λ, h λ be the truncations of the function (λ+ξ)x f(z, x) at the pairs (0, u ), (v, 0), (v, u ) respectively (see Section 2), where ξ appears in (4.3). We define the functionals ψ λ ±, ψ λ C 1 (H 1 ()) exactly as the functional ϕ λ +

Multiple Solutions for Parametric Neumann Problems 289 in the proof of Proposition 5.1, replacing g λ + by h λ ±, h λ respectively. The functionals ψ λ ±, ψ λ are all coercive and K ψ λ + K ϕλ [0 u ], K ψ λ K ϕλ [v 0], K ψ λ K ϕλ [v u ]. The extremality of u, v yields K ψ λ + = {0, u } and K ψ λ = {v, 0}. Now minimizing ψ λ ± we obtain that u, v are both local minimizers of ψ λ with respect to the norm C 1 () (note that ψλ coincides with ψ λ ± on ±intc + ). It follows from a result of Brezis Nirenberg [6] that u, v are also local minimizers of ψ λ in H 1 (). (iii) We may assume that K ψ λ is finite (or otherwise we already have an infinity of nodal solutions). Then u, v are isolated critical points and local minimizers of ψ λ. We may also assume that ψ λ (v ) ψ λ (u ). Arguing as in [1, proof of Prop. 29], we may find ρ (0, 1) such that v u > ρ and ψ λ (v ) ψ λ (u ) < inf[ψ λ (u) : u u = ρ]. Hence, ψ λ has the GMPT. Now the conclusion follows from an improved version of the mountain pass theorem (see for example, [7, p.89]). To proceed, we strengthen the hypotheses on the data of (P) λ. Namely we assume H f: f : R R is measurable such that for a.a. z, f(z, 0) = 0, f(z, ) C 1 (R) and (i) f x(z, x) α(z) + c x r 2 for a.a. z, all x R, with α L () +, c > 0, 2 < r < 2. (ii) f x(z, 0) = lim x 0 f(z, x) x (iii) f(z, x) lim x x = 0 uniformly for a.a. z. = + uniformly for a.a. z. (iv) f(z, x)x 0 for a.a. z, all x R. Moreover, we assume that λ > λ 2 (see Section 4). Hypotheses H β, H f(i) imply that ϕ λ C 2 (H 1 ()). This will be crucial in computing the critical groups of ϕ λ. Let m 2 be the unique positive integer such that λ ( λ m, λ m+1 ]. We have. Proposition 6.2. If hypotheses H β, H f(i) (iv) hold, then C k (ϕ λ, 0) = δ k,dm Z, for all m k 0, where d m = dim H m 2, H m = E( λ i ). i=1 Sketch of the proof. For all x H 1 (), ϕ λ(0)(x), x = Dx 2 2 + [β(z) λ]x(z) 2 dz = σ(x) λ x 2 2 (see H f (ii)).

290 N. S. Papageorgiou and G. Smyrlis Since λ ( λ m, λ m+1 ], the variational expressions (4.4) imply that ϕ λ(0)(x), x 0 (resp. 0 ), for all x H m (resp. for all x H m ). Hence, the Morse index of 0 equals to µ = d m 2. If λ ( λ m, λ m+1 ), then 0 is a nondegenerate critical point of ϕ λ and the conclusion follows immediately from (2.2). Suppose that λ = λ m+1. Then by using H f(ii) in conjunction with (4.5), we may find δ > 0 such that for u H m with u C 1 () δ, we have ϕ λ(u) 0. But all the norms on H m are equivalent, so we may also find δ > 0 such that for u H m with u H 1 () δ, inequality ϕ λ (u) 0 still holds. On the other hand, employing (4.4) together with H f(iv), we may show that ϕ λ (u) 0 for all u H m. It turns out that ϕ λ has a local linking at 0 (see Section 2) and the conclusion follows from Proposition 2.1. Now we are ready to establish the existence of the first nodal solution. Proposition 6.3. Suppose that H β, H f(i) (iv) hold, λ > λ 2 and let y 0 be given by Proposition 6.1. Then y 0 (v u ) \ {0} and thus, it is a nodal smooth weak solution of (P) λ. Proof. First we show that y 0 0. Indeed, ψ λ, ϕ λ coincide on (v u ) which is an open neighborhood of 0 with respect to the norm C 1 (). Meanwhile, the linear space C 1 () is H 1 ()-dense in the Hilbert space H 1 (). By a result of Palais [10], we infer that for all k 0, C k (ψ λ, 0) = C k (ψ λ C 1 (), 0) = C k(ϕ λ C 1 (), 0) = C k(ϕ λ, 0) = δ k,dm Z (6.1) (see Proposition 6.2). In particular, C 1 (ψ λ, 0) = 0 (recall that d m 2), whereas C 1 (ψ λ, y 0 ) 0 (see Prop. 6.1(iii)). Thus, y 0 0. Next, we set R = max { u, v }. By using H f(i) combined with the mean value theorem we may find η > 0 such that for a.a. z, the function x ηx f(z, x) is nondecreasing on [ R, R]. This fact implies that for w {y 0 v, u y 0 }, w(z) (β(z) + η)w(z), for a.a. z. Now H β coupled with the Strong Maximum principle of Vazquez [14] yields that y 0 v, u y 0 intc +. In order to derive a second nodal solution, we also need to compute the critical groups of ψ λ at y 0. Namely, we have the following: Proposition 6.4. C k (ψ λ, y 0 ) = δ k,1 Z, for all k 0. Proof. Arguing exactly as in the first part of the proof of Proposition 6.3 with 0 being replaced by y 0, we deduce that C k (ψ λ, y 0 ) = C k (ϕ λ, y 0 ), for all k 0. In particular, C 1 (ϕ λ, y 0 ) = C 1 (ψ λ, y 0 ) 0 (see Proposition 6.1(iii)). It suffices to prove that C k (ϕ λ, y 0 ) = δ k,1 Z, for all k 0. To this end, let σ(ϕ λ(y 0 )) denote the spectrum of

Multiple Solutions for Parametric Neumann Problems 291 ϕ λ(y 0 ) and assume that σ(ϕ λ(y 0 )) [0, + ). This means that the Morse index of y 0 is zero and we have Du 2 2 ηu 2 dz for all u H 1 (), (6.2) where η( ) = λ β( ) f x(, y 0 ( )) L q () (q > N) (see H β, H f(i)). If u kerϕ λ(y 0 ), then u u(z) = η(z)u(z) a.e. in, = 0 on. (6.3) n If η + = 0, then from [ (6.3) it follows that u = 0. If η ] + 0, then from (6.2) we have λ 1 (η) = inf Du 2 2 : ηu 2 dz = 1, u H 1 () 1, which is the principal eigenvalue of with Neumann boundary conditions and weight η. This eigenvalue is simple (see [13, Theorem 2.5]) and so from (6.3) it follows that dim kerϕ λ(y 0 ) 1. Then invoking a result of Bartcsh [5, Proposition 2.5 ] we infer that C k (ϕ λ, y 0 ) = δ k,1 Z, for all k 0, which finishes the proof of Proposition 6.4. Now we are ready to establish the existence of a second nodal solution. Proposition 6.5. Assume that H β and H f(i) (iv) hold. Then for each λ > λ 2, the problem (P) λ has a second nodal solution ŷ y 0. Proof. Regarding the critical groups of ψ λ at its critical points 0, u, v, y 0, we have for all k 0, C k (ψ λ, 0) = δ k,dm Z, C k (ψ λ, y 0 ) = δ k,1 Z (see (6.1) and Proposition 6.4), (6.4) C k (ψ λ, u ) = C k (ψ λ, v ) = δ k,0 Z, (see Proposition 6.1(ii) and Section 2) (6.5) and C k (ψ λ, ) = δ k,0 Z, since ψ λ is coercive (see Section 2). (6.6) Assume that K ψλ = {0, u, v, y 0 }. Then exploiting the Morse relation (2.1) together with (6.4), (6.5) and (6.6) we obtain ( 1) dm +1+1+( 1) = 1 (false!). Hence, we may find ŷ K ψλ \ {0, u, v, y 0 }. Arguing as in the second part of the proof of Proposition 6.3, we may check that ŷ (v u ) and thus, ŷ is nodal. Summarizing the above results we obtain our main existence result stated below: Theorem 6.6. Assume that H β and H f(i) (iv) hold. Then for each λ > λ 2, the problem (P) λ has at least four nontrivial smooth weak solutions: one positive, one negative and two nodal.

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