Semilinear Robin problems resonant at both zero and infinity

Transcription

1 Forum Math. 2018; 30(1): Research Article Nikolaos S. Papageorgiou and Vicențiu D. Rădulescu* Semilinear Robin problems resonant at both zero and infinity DOI: /forum Received December 27, 2016; revised May 2, 2017 Abstract: We consider a semilinear elliptic problem, driven by the Laplacian with Robin boundary condition. We consider a reaction term which is resonant at ± and at 0. Using variational methods and critical groups, we show that under resonance conditions at ± and at zero the problem has at least two nontrivial smooth solutions. Keywords: Resonance, critical groups, Morse index, nullity, unique continuation property MSC 2010: 35J20, 35J60, 58E05 Communicated by: Christopher D. Sogge 1 Introduction Let R N be a bounded domain with a C 2 -boundary. In this paper, we study the following semilinear Robin problem: u u(z) = f(z, u(z)) in, + β(z)u = 0 on. (1.1) n In this problem, the reaction term f(z, x) is a measurable function from R into R, which is continuously differentiable in the x-variable. The boundary coefficient β( ) belongs to W 1, ( ) and satisfies β(z) 0 for all z. When β 0, we recover the Neumann problem. It is well known that the existence and multiplicity of nontrivial solutions depends on the interaction of the limits f(z, x) lim x ± x and f(z, x) lim x 0 x with the spectrum of the Robin Laplacian. The most difficult and therefore interesting case is when the above limits hit the spectrum. Such problems are called resonant. Resonant Dirichlet problems were studied by Landesman, Robinson and Rumbos [10], Liang and Su [14], Li and Willem [13], de Paiva [6], and Su [21]. Neumann problems were investigated by Gasinski and Papageorgiou [9], Li [11], Li and Li [12], Papageorgiou and Rădulescu [18], and Qian [20]. We mention also the work of Barletta and Papageorgiou [2] on periodic ordinary differential equations. In this paper, combining variational methods with Morse theory (critical groups), we show that under conditions of resonance at both zero and infinity (double resonance), problem (1.1) can have at least two nontrivial solutions. Nikolaos S. Papageorgiou: Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece, npapg@math.ntua.gr *Corresponding author: Vicențiu D. Rădulescu: Faculty of Sciences, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia; and Department of Mathematics, University of Craiova, Street A.I. Cuza 13, Craiova, Romania, vicentiu.radulescu@imar.ro

2 238 N. S. Papageorgiou and V. D. Rădulescu, Robin problems with multiple resonance In the following section, for easy reference, we recall the main mathematical tools which we will use in the sequel. 2 Mathematical background Let X be a Banach space and X its topological dual. By, we denote the duality brackets for the pair (X, X). Given φ C 1 (X, R), we say that φ( ) satisfies the Cerami condition (the C-condition for short), if the following property holds: Every sequence u n } n 1 X such that φ(u n )} n 1 R is bounded and admits a strongly convergent subsequence. (1 + u n )φ (u n ) 0 in X, as n, This compactness-type condition is central in the minimax theory of the critical values of φ. On we employ the (N 1)-dimensional Hausdorff (surface) measure σ( ). From the general theory of Sobolev spaces we know that there exists a unique continuous linear map γ 0 : H 1 () L 2 ( ), known as the trace map, such that γ 0 (u) = u for all u H 1 () C(). We know that γ 0 is compact from H 1 () into L τ ( ) with 1 τ < 2(N 1) N 2 N = 1, 2. We have im γ 0 = H 1 2,2 ( ) and ker γ 0 = H0 1 (). if N > 2, and with 1 τ < if In the sequel, for notational simplicity, we drop the use of the map γ 0. All restrictions of Sobolev functions on, are understood in the sense of traces. Throughout this work, the standing hypothesis on β( ) is the following: (H(β)) β W 1, ( ), β(z) 0 for all z. In what follows, ϑ : H 1 () R is the C 1 -functional defined by ϑ(u) = Du β(z)u 2 dσ for all u H 1 (). Also by we denote the norm of the Sobolev space H 1 (), that is, u = [ u Du 2 2 ]1/2 for all u H 1 (). Let η L (), η(z) 0 for almost all z, η problem: u(z) = λη(z)u(z) in, = 0. We consider the following weighted linear eigenvalue u + β(z)u = 0 on. (2.1) n We say that λ R is an eigenvalue if problem (2.1) has a nontrivial solution u H 1 () which is known as an eigenfunction corresponding to the eigenvalue λ. Using the spectral theorem for compact self-adjoint operators, we can show that problem (2.1) has a sequence λ k (η)} k 1 [0, + ) of eigenvalues such that λ k (η) + as k +. If η 1, then we write λ k (1) = λ k for all k N (see Papageorgiou and Rădulescu [17, 18]). By E( λ k (η)) (for k N) we denote the eigenspace corresponding to the eigenvalue λ k (η). The regularity theory implies that E( λ k (η)) C 1 () for all k N. Moreover, each eigenspace E( λ k (η)) has the so-called unique continuation property (UCP for short), which asserts that if u E( λ k (η)) vanishes on a set of positive measure, then u 0 (see Motreanu, Motreanu

3 N. S. Papageorgiou and V. D. Rădulescu, Robin problems with multiple resonance 239 and Papageorgiou [15, p. 234] and Papageorgiou and Rădulescu [18]). We set H k = k i=1 E( λ i (η)) and H k = H k = E( λ i (η)). i k+1 For every k N, H k is finite-dimensional and we have the orthogonal direct sum decomposition H 1 () = H k H k for all k N. We have the following variational characterizations of the eigenvalues: and for k 2 we have ϑ(u) λ 1 (η) = inf[ ηu 2 dz : u H1 (), u = 0], (2.2) ϑ(u) λ k (η) = inf[ ηu 2 dz : u ϑ(u) = sup[ ηu 2 dz : u H k 1, u H k, u = 0] = 0]. (2.3) In both (2.2) and (2.3), the infimum and the supremum are realized on the corresponding eigenspace E( λ k (η)), k N. We know that λ 1 (η) is simple (that is, dim E( λ 1 (η)) = 1) and from (2.2) it is clear that the elements of E( λ 1 (η)) do not change sign. If by u 1 we denote the L 2 -normalized (that is, u 1 2 = 1) positive eigenfunction corresponding to λ 1 (η) 0, then u 1 (z) > 0 for all z. The aforementioned properties of the eigenvalues and eigenspaces lead to the following easy but useful results (see Gasinski and Papageorgiou [8, 9]). Proposition 2.1. If η(z) η (z) for almost all z and the inequality is strict on a set of positive measure, then λ k (η ) < λ k (η) for all k N. Proposition 2.2. (i) If e L (), e(z) λ k for almost all z and the inequality is strict on a set of positive measure, then ϑ(u) e(z)u 2 dz c u 2 for some c > 0 and all u H k 1. (ii) If e L (), e(z) λ k for almost all z and the inequality is strict on a set of positive measure, then ϑ(u) e(z)u 2 dz c 1 u 2 for some c 1 > 0 and all u H k. Finally, we mention that all nonprincipal eigenvalues λ k (η) (that is, k 2) have nodal (that is, sign-changing) eigenfunctions. Next we recall some basic definitions and facts for critical groups. So let X be a Banach space, φ C 1 (X, R) and c R. We introduce the following sets: φ c = u X : φ(u) c} (the sublevel set at c R), K φ = u X : φ (u) = 0} (the critical set of φ), Kφ c = u K φ : φ(u) = c} (the critical set of φ at the level c R). Let (Y 1, Y 2 ) be a topological pair such that Y 2 Y 1 X. For very k N 0, let H k (Y 1, Y 2 ) denote the kth relative singular homology group for the pair (Y 1, Y 2 ) with integer coefficients. Let u K φ be isolated and c = φ(u) (that is, u K c φ). The critical groups of φ at u are defined by C k (φ, u) = H k (φ c U, φ c U \ u}) for all k N 0, with U being a neighborhood of u such that K φ φ c U = u}. From the excision property of singular homology, we see that the above definition of critical groups is independent of the choice of the neighborhood U. Now suppose that φ C 1 (X, R) satisfies the C-condition and inf φ(k φ ) >. Let c < inf φ(k φ ). Then the critical groups of φ at infinity are defined by C k (φ, ) = H k (X, φ c ) for all k N 0.

4 240 N. S. Papageorgiou and V. D. Rădulescu, Robin problems with multiple resonance The second deformation theorem (see, for example, Gasinski and Papageorgiou [8, p. 628]) implies that the above definition is independent of the choice of the level c R. Assuming that K φ is finite, we have rank C k (φ, ) u K φ rank C k (φ, u) for all k N (2.4) (see, for example, Motreanu, Motreanu and Papageorgiou [15, p. 160]). From (2.4) we infer the following result. Proposition 2.3. If φ C 1 (X, R) satisfies the C-condition, K φ is finite and for some k N 0 we have C k (φ, ) = 0, then there exists u K φ such that C k (φ, u) = 0. A related result is the following (see [15, p. 173]). Proposition 2.4. If φ C 1 (X, R) satisfies the C-condition, K φ is finite and for some k N 0 we have C k (φ, 0) = 0 and C k (φ, ) = 0, then we can find u K φ such that either φ(u) < 0 and C k 1 (φ, u) = 0, or φ(u) > 0 and C k+1 (φ, u) = 0. Suppose X = H is a Hilbert space and φ C 2 (H, R). Let u K φ. The Morse index of u, denoted by m(u), is defined to be the supremum of the dimensions of the vector subspace on which φ (u) is negative definite. The nullity of u, denoted by ν(u), is defined to be the dimension of ker φ (u). We say that u K φ is nondegenerate, if φ (u) is invertible (that is, ν(u) = 0). For a nondegenerate u K φ with Morse index m(u) = m, we have C k (φ, u) = δ k, m Z for all k N 0. Hereafter by δ k, m we denote the Kronecker symbol δ k, m = 1 if k = m, 0 if k = m, k N 0. In dealing with degenerate critical points, the main tool is the so-called shifting theorem (see, for example, Motreanu, Motreanu and Papageorgiou [15, p. 156]). A consequence of this theorem which we will need in the sequel, is the following proposition. Proposition 2.5. If H is a Hilbert space, φ C 2 (H, R) and u K φ has finite Morse index m and nullity ν, then one of the following statements holds: (i) C k (φ, u) = 0 for all k m and all k m + ν; (ii) C k (φ, u) = δ k, m Z for all k N 0 ; (iii) C k (φ, u) = δ k, m+ ν Z for all k N 0. Finally, we fix our notation. Given a measurable function h : R R (for example, a Carathéodory function), we set N h (u)( ) = h(, u( )) for all u H 1 () (the Nemitski or superposition operator corresponding to the function h). By N we denote the Lebesgue measure on R N and, as we already mentioned earlier in this section, by we denote the norm of the Sobolev space H 1 (), that is, u = [ u Du 2 2 ]1/2 for all u H 1 (). By H k (X) (for k N 0 ), we denote the reduced homology groups, that is, H k (X) = H k (X, ) for all k N 0, with X. By A L(H 1 (), H 1 () ) we denote the operator defined by A(u), h = (Du, Dh) R N dz for all u, h H 1 ().

5 N. S. Papageorgiou and V. D. Rădulescu, Robin problems with multiple resonance Pairs of nontrivial solutions In this section, we establish the existence of two nontrivial solutions for problem (1.1). The hypothesis on the boundary term β( ) is (H(β)) from the previous section. Note that the case β 0 corresponds to the Neumann problem. So, our result here subsumes the works on resonant Neumann problems. The hypotheses on the reaction term f(z, x) are the following: (H) f : R R is a measurable function such that for almost all z we have f(z, 0) = 0, f(z, ) C 1 (R) and the following hold: (i) fx(z, x) a(z)(1 + x r 2 ) for almost all z, all x R with a L () and 2 r < 2, where (ii) (the critical Sobolev exponent); there exist m 3 and η L () + such that η(z) 2N 2 if N 3, = N 2 + if N = 1, 2 λ m+1 for almost all z, η λ m+1, λ f(z, x) f(z, x) m lim inf lim sup η(z) uniformly for almost all z, x ± x x ± x and if F(z, x) = x f(z, s) ds, then 0 (iii) lim [f(z, x)x 2F(z, x)] = uniformly for almost all z ; x ± there exist l N and δ > 0 such that l m 2, λ l x 2 f(z, x)x λ l+1 x 2 for almost all z and all x δ; (iv) f(z, x)x f x(z, x)x 2 for almost all z and all x R, f x(z, x) λ m+1 for almost all z and all x R, and for every ρ > 0, there exists ρ with ρ N > 0 such that f x(z, x) < The following function satisfies hypotheses (H): λ m+1 for almost all z ρ and all x ρ. λ m x ( λ l λ m )( 1 2 ln x + x ) if x < 1, f(x) = λ l x if 1 x 1, λ m x + ( λ l λ m )( 1 2 ln x + x ) if x > 1. Remark 3.1. Hypotheses (H(ii)), (H(iii)) imply that we can have resonance at both ± and zero (double resonance). Hypothesis (H(iv)) implies that for almost all z, To see this, note that x f(z, x) x = nondecreasing on (0, + ), nonincreasing on (, 0). f(z, x) ( ) = f x(z, x)x f(z, x) 0 if x > 0, x x 2 0 if x < 0. Let φ : H 1 () R be the energy functional for problem (1.1) defined by φ(u) = 1 2 ϑ(u) F(z, u(z)) dz for all u H 1 (). Evidently, φ C 2 (H 1 ()).

6 242 N. S. Papageorgiou and V. D. Rădulescu, Robin problems with multiple resonance Proposition 3.2. If hypotheses (H(β)), (H) hold, then the functional φ satisfies the C-condition. Proof. Let u n } n 1 H 1 () be a sequence such that φ(u n ) M 1 for some M 1 > 0 and all n N, (3.1) (1 + u n )φ (u n ) 0 in H 1 () as n. (3.2) From (3.2) we have φ (u n ), h ϵ n h for all h H 1 (), with ϵ n u n A(u n), h + β(z)u n h dσ f(z, u n )h dz ϵ n h 1 + u n for all n N. (3.3) In (3.3) we choose h = u n H 1 (). Then On the other hand, from (3.1) we have Adding (3.4) and (3.5), we obtain ϵ n ϑ(u n ) + f(z, u n )u n dz for all n N. (3.4) 2M 1 ϑ(u n ) 2F(z, u n ) dz for all n N. (3.5) M 2 [f(z, u n )u n 2F(z, u n )] dz for some M 2 > 0 and all n N. (3.6) Suppose that u n } n 1 H 1 () is unbounded. By passing to a subsequence if necessary, we may assume that u n. Let y n = u n u n, n N. Then y n = 1 for all n N, and so we may assume that From (3.3) we have for all n N, y n w y in H 1 () and y n y in L 2 () and in L 2 ( ). (3.7) A(y n), h + β(z)y n h dσ Hypotheses (H(i)), (H(ii)) imply that We may assume that N f (u n ) u n h dz f(z, x) c 1 (1 + x ) for almost all z and all x R, with c 1 > 0 N f (u n ) u n } n 1 L2 () N f (u n ) u n Moreover, hypothesis (H(ii)) implies that is bounded. ϵ n h (1 + u n ) u n. (3.8) w ξ in L 2 () as n. (3.9) ξ(z) = η(z)y(z) for almost all z with λ m η(z) η(z) for almost all z (3.10) (see [1]).

7 N. S. Papageorgiou and V. D. Rădulescu, Robin problems with multiple resonance 243 Therefore, if in (3.8) we pass to the limit as n and use (3.7), (3.9) and (3.10), we obtain A(y), h + β(z)yh dσ = η(z)yh dz y(z) = for all h H 1 () η(z)y(z) for almost all z, (see Papageorgiou and Rădulescu [17]). If η λ m (see (3.10)), then using Proposition 2.1, we have λ m ( η) < λ m ( λ m ) = 1 and 1 = λ m+1 ( λ m+1 ) < λ m+1 ( η) y = 0 (see (3.11)). From (3.8) with h = y n H 1 (), we see that Dy n 2 0 y n 0 in H 1 () (see (3.7)), a contradiction to the fact that y n = 1 for all n N. If η(z) = λ m for almost all z (see (3.10)), then from (3.11) we have y E( λ m ). y + β(z)y = 0 on (3.11) n Also, if in (3.8) we choose h = y n y H 1 (), pass to the limit as n and use (3.7) and (3.9), then lim A(y n), y n y = 0 n Dy n 2 Dy 2 y n y (by the Kadec Klee property of Hilbert spaces; see [8, p. 911]) y = 1 The UCP of the eigenspaces implies that y(z) Then hypothesis (H(ii)) implies that and so y E( λ m ) \ 0}. = 0 for almost all z u n (z) + for almost all z. f(z, u n (z))u n (z) 2F(z, u n (z)) for almost all z. (3.12) Using (3.12), hypothesis (H(ii)) and Fatou s lemma, we have [f(z, u n )u n 2F(z, u n )] dz as n. (3.13) Comparing (3.6) and (3.13), we reach a contradiction. So, we have proved that u n } n 1 H 1 () is bounded. We may assume that u n w u in H 1 () and u n u in L 2 () and in L 2 ( ). (3.14) In (3.3) we choose h = u n u H 1 (), pass to the limit as n and use (3.14). Then as desired. lim A(u n), u n u = 0 n Du n 2 Du 2 u n u in H 1 () (again by the Kadec Klee property; see [8, p. 911]) φ satisfies the C-condition,

8 244 N. S. Papageorgiou and V. D. Rădulescu, Robin problems with multiple resonance Let H l = l i=1 E( λ i ) and H l = H l = E( λ 1 ), i l+1 where l comes from hypothesis (H(iii)). We have the following orthogonal direct sum decomposition: So, every u H 1 () admits a unique decomposition with u H l and u H l. H 1 () = H l H l. (3.15) u = u + u (3.16) Proposition 3.3. If hypotheses (H(β)), (H) hold, then C k (φ, 0) = δ k,dl Z for all k N 0 with d l = dim H l < +. Proof. We assume that 0 K φ is isolated. Otherwise, we already have a sequence of distinct nontrivial solutions of (1.1) converging to zero, and so we are done. Let η 0 ( λ l, λ l+1 ) and consider the functional ψ : H 1 () R defined by ψ(u) = 1 2 ϑ(u) η 0 2 u 2 2 for all u H 1 (). Evidently, ψ C 2 (H 1 ()) and using (2.2) and (2.3), we see that ψ H l 0 and ψ H l 0. From Motreanu, Motreanu and Papageorgiou [15, Proposition 6.84 (ii) and Theorem 6.87], we have C dl (ψ, 0) = 0 with d l = dim H l <. Since ψ C 2 (H 1 ()), from Proposition 2.5 we infer that We consider the homotopy h(t, u) defined by C k (ψ, 0) = δ k,dl Z for all k N 0. h(t, u) = (1 t)φ(u) + tψ(u) for all t [0, 1] and all u H 1 (). For t (0, 1] and u C 1 () with u C 1 () δ (δ > 0 is as in hypothesis (H(iii))), we have h u(t, u), v = (1 t) φ (u), v + t ψ (u), v for all v H 1 (). (3.17) Let v = u u H 1 () (see (3.16)). Exploiting the orthogonality of the component subspaces in (3.15), we have φ (u), u u = ϑ( u) ϑ( u) Recall that u C 1 () with u C 1 () δ. So, using hypothesis (H(iii)), we have f(z, u(z))( u u)(z) Using (3.19) in (3.18), we have Also, we have λ l+1 f(z, u)( u u) dz. (3.18) u(z) 2 u(z) 2 for almost all z. (3.19) λ l φ (u), u u γ( u) λ l+1 u 2 2 [γ( u) λ l u 2 2 ] 0 (see (2.2), (2.3)). (3.20) ψ (u), u u = γ( u) η 0 u 2 2 [γ( u) η 0 u 2 2 ] c 2 u 2 for some c 2 > 0 (3.21) (see Proposition 2.2 and recall that η 0 ( λ l, λ l+1 )).

9 N. S. Papageorgiou and V. D. Rădulescu, Robin problems with multiple resonance 245 So, for t (0, 1], from (3.20) and (3.21) we have h u(t, u), u u tc 2 u 2. On the other hand, h(0, ) = φ( ) and 0 K φ is isolated. Therefore, the homotopy h [0,1] C 1 () preserves the isolation of the critical point u = 0. From Corvellec and Hantoute [4, Theorem 5.2] we have as desired. C k (φ C 1 (), 0) = C k(ψ C 1 (), 0) for all k N 0 C k (φ, 0) = C k (ψ, 0) for all k N 0 (since C 1 () is dense in H 1 (); see Palais [16]) C k (φ, 0) = δ k,dl Z for all k N 0 (see (3.17)), Next we compute the critical groups of φ at infinity. For the same integer m as in hypothesis (H(ii)), we consider the following orthogonal direct sum decomposition: H 1 () = H m H m, (3.22) where H m = m i=1 E( λ i ) and H m = Proposition 3.4. If hypotheses (H(β)), (H) hold, then C dm (φ, ) H m = E( λ i ). i m+1 = 0 with d m = dim H m (see (3.22)). Proof. Hypothesis (H(ii)) implies that given any ξ > 0, we can find M 3 = M 3 (ξ) > 0 such that For almost all z and all s Then we have d F(z, s) f(z, s)s 2F(z, s) ds s 2 = f(z, x)x 2F(z, x) ξ for almost all z and all x M 3. (3.23) F(z, v) v 2 = 0, we have Note that hypothesis (H(ii)) implies that d F(z, s) ds s 2 = f(z, s)s2 2F(z, s)s s 4 f(z, s)s 2F(z, s) = s 2. s s 3 ξ s 3 for almost all z and all s M 3 (see (3.23)) F(z, s) x 2 ξ 2 [ 1 v 2 1 x 2 ] for almost all z and all v x M 3. (3.24) λ 2F(z, u) 2F(z, u) m lim inf u ± u 2 lim sup u ± u 2 η(z) (3.25) uniformly for almost all z. So, if in (3.24) we pass to the limit as v + and use (3.25), then λ m 2 F(z, x) x 2 ξ 2 1 x 2 for almost all z and all x M 3 λ m x 2 2F(z, x) ξ for almost all z and all x M 3 λ m x 2 2F(z, x) as x, uniformly for almost all z (3.26) (recall that ξ > 0 is arbitrary).

10 246 N. S. Papageorgiou and V. D. Rădulescu, Robin problems with multiple resonance On the negative semiaxis, we have d F(z, s) f(z, s)s 2F(z, s) ds s 2 = s 2 s (see (3.23) and recall that s < 0), which implies ξ s 2 s for almost all z and all s M 3 F(z, v) F(z, x) v 2 x 2 ξ 2 [ 1 x 2 1 v 2 ] for almost all z and all x v M 3. Letting x and using (3.25), we obtain F(z, v) λ v 2 m 2 ξ 1 2 v 2 for almost all z and all v M 3 2F(z, v) λ m v 2 ξ for almost all z and all v M 3 λ m v 2 2F(z, v) as v, uniformly for almost all z (3.27) (recall that ξ > 0 is arbitrary). From (3.26) and (3.27) it follows that We introduce the two sets λ m x 2 2F(z, x) as x ±, uniformly for almost all z. (3.28) S r = u H 1 () : u = r, ϑ(u) A = u H 1 () : ϑ(u) λ m+1 u 2 2 }. λ m u 2 2 } (r > 0), The set S r is a C 1 -Hilbert manifold with boundary and from Degiovanni and Lancelotti [5, Theorem 3.2]. We have ind S r = ind(h 1 () \ A) = d m, where ind denotes the Fadell Rabinowitz index (see [7]). From Cingolani and Degiovanni [3, Theorem 3.6], we know that the sets S r and A homologically link in dimension d m. Hypotheses (H(i)), (H(ii)) imply that given ϵ > 0, we can find c 3 = c 3 (ϵ) > 0 such that Then for all u A, we have F(z, x) 1 2 (η(z) + ϵ)x2 + c 3 for almost all z and all x R. (3.29) φ(u) = 1 2 ϑ(u) F(z, u) dz Choosing ϵ (0, c), we see that Next we show that 1 2 [ϑ(u) η(z)u 2 dz ϵ u 2 ] c 3 N (see (3.29)) c ϵ 2 u 2 c 3 N for some c > 0 (see Proposition 2.2). inf φ >. A φ(u) as u S r and r +. Arguing by contradiction, we suppose that one can find r n +, u n S rn and M 4 > 0 such that M 4 φ(u n ) for all n N and u n. (3.30)

11 N. S. Papageorgiou and V. D. Rădulescu, Robin problems with multiple resonance 247 Let y n = u n u n, n N. Then y n = 1 for all n N and so we may assume that We have y n w y in H 1 () and y n y in L 2 () and in L 2 ( ). (3.31) ϑ(y n ) If y = 0, from (3.31), (3.32) and hypothesis (H(β)) we have λ m y n 2 2 for all n N. (3.32) Dy n 0 in L 2 (, R N ) y n 0 in H 1 () (see (3.31)), a contradiction to the fact that y n = 1 for all n N. Therefore y = 0, and so u n (z) + for almost all z 0 = y = 0}, 0 N > 0. Then from (3.28) and Fatou s lemma we have [ λ m u 2 n 2F(z, u n )] dz as n. (3.33) On the other hand, from (3.30) and since u n S rn, n N, we have 2M 4 2φ(u n ) [ λ m u 2 n 2F(z, u n )] dz for all n N. (3.34) Comparing (3.33) and (3.34), we reach a contradiction. Therefore, we have φ(u) as u S r and r +. It follows that for r > 0 big, we have sup φ < inf φ. S r A As before, we assume that K φ is finite (otherwise, we already have a whole sequence of distinct solutions, and so we are done). Hence we can have sup φ a < inf φ. S r K φ We consider the triple of sets We have the commutative diagram H dm 1(S r ) S r φ a H 1 () \ A. H dm 1(φ a ) i H dm 1(H 1 () \ A). Since i = 0 (recall that we established earlier that S r and A homologically link in dimension d m ; see Motreanu, Motreanu and Papageorgiou [15, Definition 6.77]), we have H dm 1(φ a ) = 0. But from Motreanu, Motreanu and Papageorgiou [15, Proposition 6.64] we have C dm (φ, ) = H dm (H 1 (), φ a ) = H dm 1(φ a ) = 0. For the multiplicity theorem we need to strengthen hypothesis (H(ii)) a little. So, the new conditions on the reaction term f(z, x) are the following.

12 248 N. S. Papageorgiou and V. D. Rădulescu, Robin problems with multiple resonance (H ) f(z, x) = λ m x + f 0 (z, x) with f 0 : R R is a measurable function such that for almost all z we have f 0 (z, 0) = 0, f 0 (z, ) C 1 (R) and the following holds: (i) (f 0 ) x(z, x) a 0 (z)(1 + x r 2 ) for almost all z and all x R, with a 0 L () +, 2 r < 2 ; (ii) there exists a function η L () + such that 0 η(z) λ m+1 λ m for almost all z, η λ m+1 λ m, f η(z) 0 (z, x) = lim x ± x and if F 0 (z, x) = x 0 f 0(z, s) ds, then uniformly for almost all z, (iii) lim [f 0(z, x)x 2F 0 (z, x)] = uniformly for almost all z ; x ± there exist l N and δ > 0 such that l m 2, ( λ l λ m )x 2 f 0 (z, x) ( λ l+1 λ m )x 2 for almost all z and all x δ; (iv) f 0 (z, x)x (f 0 ) x(z, x)x 2 for almost all z and all x R, (f 0 ) x(z, x) λ m+1 λ m for almost all z and all x R, and for every ρ > 0 there exists ρ with ρ N > 0 such that ( f 0 ) x(z, x) < λ m+1 for almost all z ρ and all x ρ. Theorem 3.5. If hypotheses (H(β)), (H ) hold, then problem (1.1) admits at least two nontrivial solutions u 0, u C 1 (). Proof. From Proposition 3.4 we know that C dm (φ, ) u 0 K φ such that C dm (φ, u 0 ) u 0 On the other hand, from Proposition 3.3 we have = 0. So, according to Proposition 2.3 we can find = 0. (3.35) C k (φ, 0) = δ k,dl Z for all k N 0. (3.36) Since l m 2 (see hypothesis (H (iii)), we have d l = d m. Then from (3.35) and (3.36) we infer that = 0. Note that hypotheses (H (i)) (H (iii)) imply that f(z, x) c 4 x for almost all z and all x R, with c 4 > 0. (3.37) We set f(z, u 0 (z)) g(z) = u 0 (z) if u 0 (z) = 0, fx(z, u 0 (z)) if u 0 (z) = 0. From (3.37) and hypothesis (H (iv)) we see that g L (), and we have u 0 (z) = g(z)u 0 (z) for almost all z, u 0 n + β(z)u 0 = 0 on (3.38) (see Papageorgiou and Rădulescu [17]). From Wang [22, Lemma 5.1] we have u 0 L (), and from (3.38) it follows that u 0 L (). Using the Calderon Zygmund estimates (see Wang [22, Lemma 5.2]), we have u 0 W 2,s () with s > N. Then the Sobolev embedding theorem says that W 2,s () C 1+α () with α = 1 N s > 0 u 0 C 1 ().

13 N. S. Papageorgiou and V. D. Rădulescu, Robin problems with multiple resonance 249 If have η 0 (nonresonant problem at ± ), then from Papageorgiou and Rădulescu [19, Proposition 26] we C k (φ, ) = δ k,dm Z for all k N. If η(z) = 0 for almost all z (resonant problem at ± ), then by Motreanu, Motreanu and Papageorgiou [15, Theorem 6.7.1] we have C k (φ, ) = 0 for all k [d m, d m+1 ] (3.39) (here d m+1 = dim m+1 i=1 E( λ i )). Since d l < d m (see hypothesis (H (iii))), we have C dl (φ, 0) = 0 and C dl (φ, ) = 0 (see (3.36) and (3.39)). (3.40) Then from (3.40) and Proposition 2.4 we know that we can find u K φ such that From (3.36) and (3.41) we infer that that u C 1 (). We need to show that u C dl 1(φ, u) = 0 or C dl +1(φ, u) = 0. (3.41) u is a nontrivial solution of (1.1) and the regularity theory implies = u 0. According to (3.35), it suffices to show that C dm (φ, u) = 0. (3.42) Let f(z, u(z)) if u(z) = 0, g(z) = u(z) fx(z, u(z)) if u(z) = 0. As before we have g L (). We consider the following linear eigenvalue problem: v(z) = λ g(z)v(z) in, (3.43) v (3.44) n + β(z)v = 0 on. Since u C 1 () is a solution for problem (1.1), from (3.42) it follows that λ = 1 is an eigenvalue of problem (3.44) with u C 1 () as a corresponding eigenfunction. Recall that λ l g(z) for almost all z (see hypothesis (H (iii)) and Remark 3.1). Suppose that λ l = g(z) for almost all z. Then from (3.44) it follows that u C 1 () is an eigenfunction corresponding to λ l. The UCP implies that u(z) = 0 for almost all z. Then from (3.43) and Proposition 2.5 it follows that (3.42) holds (recall l m 2). So, we may assume that the inequality λ l g(z) for almost all z is strict on a set of positive measure. Invoking Proposition 2.1, we have λ i ( g) < λ i ( λ l ) 1 Since λ = 1 is an eigenvalue of (3.44), we have By hypothesis (H (iv)) we have for all i 1,..., l}. λ l+1 ( g) 1. (3.45) σ(z) = fx(z, u(z)) g(z) for almost all z. If this last inequality is strict on a set of positive measure, then Proposition 2.1 implies that λ l+1 ( σ) < λ l+1 ( g) λ l+1 ( σ) < 1 (see (3.45)). (3.46) Recall that φ ( u)y, y = ϑ(y) fx(z, u(z))y 2 dz for all y H 1 ().

14 250 N. S. Papageorgiou and V. D. Rădulescu, Robin problems with multiple resonance Then from (3.46) it follows that m( u) > d l. From (3.41) and Proposition 2.5 we see that we must have C k (φ, u) = δ k,dl +1Z for all k N 0. (3.47) But l m 2 (see hypothesis (H (iii))). Therefore, (3.47) implies that (3.42) holds. Now suppose that σ(z) = fx(z, u(z)) = g(z) for almost all z. Hypothesis (H (iv)) implies that σ(z) λ m+1 for almost all z, with strict inequality on a set of positive measure. Hence 1 < λ m+1 ( σ) (see Proposition 2.1) m( u) + ν( u) d m. (3.48) Recall that d l + 1 < d m (see hypothesis (H (iii))). Then from (3.41), (3.48) and Proposition 2.5, we infer that (3.42) holds again. From (3.42) and (3.35) it follows that u = u 0 u C 1 () is the second nontrivial smooth solution of problem (1.1). Funding: The second author acknowledges the support through a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE References [1] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc. 915 (2008), [2] G. Barletta and N. S. Papageorgiou, Periodic problems with double resonance, NoDEA Nonlinear Differential Equations Appl. 19 (2012), [3] S. Cingolani and M. Degiovanni, Nontrivial solutions for p-laplace equations with right-hand side having p-linear growth at infinity, Comm. Partial Differential Equations 30 (2005), [4] J. N. Corvellec and A. Hantoute, Homotopical stability of isolated critical points of continuous functionals, Set-Valued Var. Anal. 10 (2002), [5] M. Degiovanni and S. Lancelotti, Linking over cones and nontrivial solutions for p-laplace equations with p-superlinear nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), [6] F. O. de Paiva, Multiple solutions for asymptotically linear resonant elliptic problems, Topol. Methods Nonlinear Anal. 21 (2003), [7] E. R. Fadell and P. Rabinowitz, Generalized cohomological index theories for Lie group actions with applications to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978), [8] L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, [9] L. Gasinski and N. S. Papageorgiou, Neumann problems resonant at zero and infinity, Ann. Mat. Pura Appl. (4) 191 (2012), [10] E. M. Landesman, S. Robinson and A. Rumbos, Multiple solutions of semilinear elliptic problems at resonance, Nonlinear Anal. 24 (1995), [11] C. Li, The existence of infinitely many solutions of a class of nonlinear elliptic equations with Neumann boundary condition for both resonance and oscillation problems, Nonlinear Anal. 54 (2003), [12] C. Li and S. Li, Multiple solutions and sign-changing solutions of a class of nonlinear elliptic equations with Neumann boundary condition, J. Math. Anal. Appl. 298 (2004), [13] S. Li and M. Willem, Multiple solutions for asymptotically linear boundary value problems in which the nonlinearity crosses at least one eigenvalue, NoDEA Nonlinear Differential Equations Appl. 5 (1998), [14] Z. Liang and J. Su, Multiple solutions for semilinear elliptic boundary value problems with double resonance, J. Math. Anal. Appl. 354 (2009),

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