On the spectrum of a nontypical eigenvalue problem

Transcription

1 Electronic Journal of Qualitative Theory of Differential Equations 18, No. 87, 1 1; On the spectrum of a nontypical eigenvalue problem Mihai Mihăilescu 1,3 and Denisa Stancu-Dumitru,3 1 Departament of Mathematics, University of Craiova, 13 A. I. Cuza Street, Craiova, 585, Romania Department of Mathematics and Computer Sciences, University Politehnica of Bucharest, 313 Splaiul Independentei Street, Bucharest, 64, Romania 3 Simion Stoilow Institute of Mathematics of the Romanian Academy, 1 Calea Grivitei Street, Bucharest, 17, Romania Received 17 August 18, appeared 9 October 18 Communicated by Patrizia Pucci Abstract. We study a nontypical eigenvalue problem in a bounded domain from the Euclidian space R subject to the homogeneous Dirichlet boundary condition. We show that the spectrum of the problem contains two distinct intervals separated by an interval where there are no other eigenvalues. Keywords: nontypical eigenvalue problem, spectrum, Trudinger s inequality, Orlicz spaces. 1 Mathematics Subject Classification: 35J, 35D3, 35P99, 49R5, 46E3. 1 Introduction 1.1 The statement of the problem Let R be an open and bounded domain with smooth boundary denoted by. We consider the following problem ux = λhux, x, 1.1 ux =, x, where λ is a real parameter and h : R R is the function given by ht = e t + t p 1, t, e t 1, t <, 1. with p, 1 a fixed real number. Note that this equation is not a typical eigenvalue problem since it has an inhomogeneous character in the sense that if u is a nontrivial solution of the equation then tu fails to be its solution for all t R. However, since in this paper we are interested in finding parameters λ R for which problem 1.1 has nontrivial solutions we will call it a nontypical eigenvalue problem. In this context, we will call such a parameter Corresponding author. mmihailes@yahoo.com

2 M. Mihăilescu and D. Stancu-Dumitru an eigenvalue of problem 1.1 and a corresponding nontrivial solution of the equation an eigenfunction. Moreover, we will refer to the set of all eigenvalues of problem 1.1 as being the spectrum of the problem. To be more precise, we will use the following definition in our subsequent analysis. Definition 1.1. We say that λ R is an eigenvalue of problem 1.1 if there exists u H 1 \ } such that u φ dx = λ huφ dx, φ H Function u from the above relation is called an eigenfunction associated to eigenvalue λ. 1. Background, motivation and main result First, we recall that in the case when ht = t, for all t R, problem 1.1 reduces to the celebrated eigenvalue problem of the Laplace operator, i.e. ux = λux, x, ux =, x. 1.4 It is well-known that the spectrum of problem 1.4 consists in an increasing and unbounded sequence of positive real numbers see, e.g. [11, Theorem 8..1.]. Moreover, each eigenvalue has a variational characterisation given by Poincaré s principle see, e.g. [11, Proposition 8..]. In particular, we just recall that the first eigenvalue of problem 1.4 is obtained by minimizing the Rayleigh quotient associated to the problem λ 1 = inf u H 1 \} u dx. 1.5 u dx Furthermore, each eigenfunction corresponding to λ 1 has constant sign in. Next, we consider the case when ht = t q t, for all t R, where q 1, \ } is a given real number. Then problem 1.1 becomes ux = λ ux q ux, x, ux =, x. 1.6 For problem 1.6 the spectrum is continuous and consists exactly in the interval, see, e.g., [1, Théorème 7.3, p. 119] or [5, Theorem 1]. On the other hand, in the case when the function h involved in problem 1.1 is of the form f t, t, ht = t, t <, with f satisfying the properties I there exists a positive constant C, 1 such that f t Ct for any t ; II there exists t > such that t f s ds > ; III lim t f t t = ;

3 On the spectrum of a nontypical eigenvalue problem 3 it was proved in [6, Theorem 1] that the spectrum of problem 1.1 contains, on the one hand, the isolated eigenvalue λ 1 given by relation 1.5 and, on the other hand, a continuous part, consisting in an interval µ 1, with µ 1 > λ 1. Finally, we consider the case when ht = e t, for all t R. Then problem 1.1 reads as follows ux = λe ux, x, 1.7 ux =, x. Problems of type 1.7 have been extensively studied in the literature see, e.g. [] or [3] and the reference therein. For instance in [, Theorem 1.3 & Theorem 5.8] it was proved that there exist two positive constants µ 1 and µ with µ 1 < µ such that each λ, µ 1 is an eigenvalue of problem 1.7 while any λ µ, can not be an eigenvalue of problem 1.7. Motivated by the above results, in this paper we study equation 1.1 when function h involved in its formulation is given by relation 1.. We reveal a new situation which can occur in the description of the spectrum of this problem, namely the fact that it contains two separate intervals. More precisely, we prove the following result. Theorem 1.. Assume function h from problem 1.1 is given by relation 1. and λ 1 is given by relation 1.5. Then there exist two positive real numbers λ and λ with λ < λ such that each λ, λ λ, is an eigenvalue of problem 1.1. Moreover, any λ λ 1, λ 1 is not an eigenvalue of problem 1.1. Proof of the main result In order to prove Theorem 1. we start by recalling a series of known results that will be essential in the analysis of problem Auxiliary results Given an N-function Φ : R R + i.e., Φ is even, convex; Φt = iff t = ; lim t t 1 Φt = and lim t t 1 Φt =, see [1, Chapter 8] for more details we can define the Orlicz space L Φ := u : R : u is measurable and } Φ ux dx <. We point out a few examples of N-functions: Φt = t q, with q 1,, or Φt = e t 1, or Φt = e t 1. Moreover, in the case when Φt = t q, with q 1,, the Orlicz space L Φ is, actually, the classical Lebesgue space L q. A well-known result see, e.g. [8, pp. 1 ] asserts that the Sobolev space H 1 is continuously embedded in the Orlicz space L Φ, where Φ t := e t 1, for all t R. This result is a consequence of Trudinger s inequality see [9] or [4, Theorem 7.15] which ensures that there exist two positive constants c 1 and c independent of such that ux c e 1 u L 1 dx c, u H 1.

4 4 M. Mihăilescu and D. Stancu-Dumitru Actually, the above inequality can be improved see, e.g. [7], since there exists a constant C >, which is independent of, such that ux ux dx u e L 1 dx C C ux dx π, u H1 \ }..1 Finally, note that for any N-function Ψ that satisfies the property: lim t Ψkt Φ t =, k > we know that H 1 is compactly embedded in LΨ see, [8, pp. 1 ]. Setting Ψ t := e t 1, for all t R, we observe that Ψ kt lim t Φ t = lim e kt 1 t e t 1 =, and consequently H 1 is compactly embedded in LΨ. Similarly, it can be checked that H 1 is compactly embedded in each Lebesgue space Lq, with q 1,.. Proof of Theorem 1. First, we note that the embedding of H 1 into the Orlicz spaces LΦ, L Ψ and L p+1 guarantees the fact that the integrals involved in Definition 1.1 are well-defined and, thus, problem 1.1 is well-posed. Next, in order to go further, it is convenient to observe that problem 1.1 can be reformulated as follows ux = λ[e u+x + u p ux =, +x 1 + e u x 1], x, x,. where u ± x := max±ux, } for all x. Recalling that for each u H 1 we have that ux = u + x u x and ux = u + x + u x, for all x, and furthermore, u ± H 1 and u + x =, if [ux ], ux, if [ux > ] and u x =, if [ux ], ux, if [ux < ] for all x see, e.g. [4, Lemma 7.6], we can also rewrite relation 1.3 in the following manner u + φdx u φdx = λ e u + + u p + 1 φdx + λ e u 1 φdx, φ H 1..3 In other words, λ R is an eigenvalue of problem 1.1 if and only if there exists u H 1 \ } such that relation.3 holds true. The proof of Theorem 1. will be a simple consequence of the conclusions of Propositions.1,.3 and.5 below. Proposition.1. Any λ λ 1, λ 1 is not an eigenvalue for problem 1.1, where λ1 is the first eigenvalue for the Laplace operator with homogeneous Dirichlet boundary condition given by relation 1.5.

5 On the spectrum of a nontypical eigenvalue problem 5 Proof. Let λ > be an eigenvalue for problem 1.1 with its corresponding eigenfunction u H 1 \ }. Note that since u = then at least one of the functions u + and u is nontrivial in. Taking φ = u in.3 we obtain u dx = λ e u 1 dx, which, in view of the fact that 1 e y y for all y, yields λ 1 Thus, if u = then u dx u dx = λ 1 e u u dx λ u dx. u dx > and the above facts imply λ λ 1..4 Otherwise, if u and λ > is an eigenvalue for problem 1.1 then u + = and relation.3 reads as follows u + φ dx = λ e u + + u p + 1 φ dx, φ H. 1.5 Let e 1 H 1 \ }, e 1x >, x, be an eigenfunction associated to the eigenvalue λ 1 defined in relation 1.5, i.e. e 1 φ dx = λ 1 Testing with φ = u + in.6 and φ = e 1 in.5 we find λ 1 u + e 1 dx = e 1 φ dx, φ H 1..6 e 1 u + dx = λ e u + + u p + 1 e 1 dx λ u + e 1 dx, since e y + y p 1 e y 1 y, y. Taking into account that u +e 1 dx > we deduce that λ 1 λ..7 Collecting the above pieces of information we find out that if λ > is an eigenvalue for problem 1.1 then either relation.4 holds true or relation.7 holds true. In conclusion, any λ λ 1, λ 1 cannot be an eigenvalue for problem 1.1. The proof of Proposition.1 is complete. Next, we consider the problem ux = λe u x 1, x, ux =, x..8 Definition.. We say that λ is an eigenvalue for problem.8 if there exists u H 1 \ } such that u φ dx = λ e u 1 φ dx, φ H. 1

6 6 M. Mihăilescu and D. Stancu-Dumitru Testing in the above relation with φ = u + we find u + dx = λ e u 1 u + =, which implies u + H 1 =, or u +. Consequently, problem.8 possesses only nonpositive eigenfunctions. Thus, it is enough to analyse the problem u x = λ 1 e u x, x,.9 ux =, x. Taking into account the definition of an eigenvalue for problem 1.1 see relation.3 we observe that if λ > is an eigenvalue of problem.8 then it is an eigenvalue of problem 1.1. Now, let us define h 1 : [, R by h 1 t = 1 e t for all t. It is easy to check that h 1 satisfies the following properties h 1 t t, t ; lim t t h 1sds = lim t t 1 e s ds = lim t t + e t 1 = +. It follows that there exists t > such that t h 1sds > ; lim t h 1 t t = lim t 1 e t t =. In other words, conditions H1 H3 from [6, page 3] are fulfilled with hx, t = h 1 t. Similar arguments as those used in the proofs of [6, Lemmas 4 & 5] can be considered in order to show that following result. Proposition.3. There exists λ > such that every λ λ, is an eigenvalue for problem.8 having the corresponding eigenfunction nonpositive. Consequently, such a λ is an eigenvalue of problem 1.1. Finally, we consider the problem ux = λe u + x + u + x p 1, x, ux =, x..1 Definition.4. We say that λ is an eigenvalue for problem.1 if there exists u H 1 \ } such that u φ dx = λ e u + + u p + 1 φ dx, φ H 1. Testing in the above relation with φ = u we obtain u dx = λ e u + + u p + 1 u =. We infer that u H 1 = which implies that u. Thus, problem.1 possesses only nonnegative eigenfunctions. Taking into account the definition of an eigenvalue for problem 1.1 see relation.3 we deduce that an eigenvalue of problem.1 is in fact an eigenvalue for problem 1.1. In order to go further we introduce the Euler Lagrange functional associated to problem.1, i.e. J λ : H 1 R defined by J λ u := 1 u dx λ 1 e u + 1 dx + 1 u p+1 + p + 1 dx u + dx.

7 On the spectrum of a nontypical eigenvalue problem 7 Standard arguments assure that J λ C 1 H 1 ; R and its derivative is given by J λ u, φ = u φ dx λ e u + + u p + 1 φ dx, u, φ H 1. We note that the weak solutions of problem.1 are exactly the critical points of the functional J λ. In view of Definition.4, λ is an eigenvalue of problem.1 if and only if the functional J λ has a nontrivial and nonnegative critical point. Proposition.5. There exists λ > such that every λ, λ is an eigenvalue for problem.1 with the corresponding eigenfunction nonnegative. Consequently, such a λ is an eigenvalue of problem 1.1. Remark.6. Since p, 1, the Hilbert space H 1 is compactly embedded in the Lebesgue space L p+1 with p + 1 1, which implies that there exists a positive constant C such that u L p+1 C u H 1, u H1..11 In order to prove Proposition.5 it is useful to first establish two auxiliary results. Lemma.7. Define λ := 1 [ ] >,.1 8 C π + C p+1 e p p + 1 where C is given by relation.11. Then, for every λ, λ we have J λ u > 1 16, u H1 with u H 1 = 1. Proof. By relation.1 we deduce that for each u H 1 with u H 1 = 1 we have ux e 4 ux u 1 dx = e L 1 dx C π. Since e y e y + e for all y, we deduce that for all u H 1 with u H 1 = 1 the following estimates hold true e u + dx e u + + e dx e u + e dx C π + e Taking into account inequalities.13 and.11, it follows that for all u H 1 with

8 8 M. Mihăilescu and D. Stancu-Dumitru u H 1 = 1 and all λ, λ where λ is given by relation.1 we get J λ u = 1 u H 1 λ e u + u + 1 dx λ p u H 1 λ e u + dx λ u p+1 + p + 1 dx 1 u H 1 λ C π + e u H 1 λ C π + e + 1 λ 1 8 λ C π + e + 1 C + p+1 p+1 p λ C π + e + 1 C + p+1 p+1 p + 1 = The proof of Lemma.7 is complete. u p+1 + dx λ u p+1 dx p + 1 p + 1 C p+1 u p+1 H 1 Lemma.8. Fix λ, λ where λ is given by relation.1. There exists t 1 > sufficiently small such that J λ t 1 e 1 <, where e 1 is a positive eigenfunction associated to λ 1 following relation 1.5. Proof. Taking into account that e y y 1, for all y, we deduce that for any t, 1 we have J λ te 1 = 1 te 1 dx λ e te 1 te 1 1 dx λ te p + 1 p+1 dx 1 t e 1 dx λtp+1 e p+1 p dx. Therefore Jte 1 <, for all t, δ 1/1 p with δ a given real number satisfying λ e p+1 1 dx < δ < p + 1 e 1. H 1 The proof of Lemma.8 is complete. Proof of Proposition.5. Let λ be defined as in.1 and fix λ, λ. Define by B 1/ the ball centred at the origin and of radius 1 from H1 and denote by B 1/ its boundary. By Lemma.7 it follows that inf J λ >..14 B 1/ On the other hand, by Lemma.8 there exists t 1 > sufficiently small such that J λ t 1 e 1 <, where e 1 is a positive eigenfunction associated to λ 1 from relation 1.5. Moreover, taking into account relations.11 and.13 we deduce that J λ u 1 u H 1 λ C π + e + 1 λ p + 1 C p+1 u p+1 H 1

9 On the spectrum of a nontypical eigenvalue problem 9 for any u B 1/. It follows that < c := inf B 1/ J λ <. We consider < ɛ < inf B1/ J λ inf B1/ J λ. Applying Ekeland s variational principle to the functional J λ : B 1/ R, we find u ɛ B 1/ such that J λ u ɛ < inf J λ + ɛ, B 1/ J λ u ɛ < J λ v + ɛ v u ɛ H 1, v = u ɛ. Since J λ u ɛ inf B1/ J λ + ɛ lim B1/ J λ + ɛ < inf B1/ J λ, we deduce that u ɛ B 1/. Now, we introduce I λ : B 1/ R defined by I λ v = J λ v + ɛ v u ɛ H 1. It is clear to see that u ɛ is a minimum point of I λ and thus I λ u ɛ + tu I λ u ɛ t for small positive t and any u B 1. The above relation yields J λ u ɛ + tu J λ u ɛ t + ɛ u H 1. Letting t we infer that J λ u ɛ, u + ɛ u H 1 and this implies that J λ u ɛ ɛ. Thus, there exists a sequence u n } B 1/ such that J λ u n c and J λ u n, as n..15 It is clear that sequence u n } is bounded in H 1 which implies that there exists u H1 such that, up to a subsequence, still denoted by u n }, u n } converges weakly to u in H 1. It follows that u H 1 lim inf u n n H 1 which implies that u B 1/. By the compact embedding of H 1 in LΨ and L p+1, we deduce that u n } converges strongly to u in L Ψ and L p+1. It follows that or lim n J u, u n u =, [ lim u u n u dx λ e u + + u p n + 1 ] u n u dx =..16 On the other hand, by relation.15 we conclude that or lim n J λ u n, u n u =, [ ] lim u n u n u dx λ e u n + + u n p n + 1 u n u dx =..17 Subtracting.16 from.17 and using the above pieces of information we deduce that lim u n u dx =. n

10 1 M. Mihăilescu and D. Stancu-Dumitru Therefore, we obtain that u n } converges strongly to u in H 1, and using.15 we deduce that J λ u = c < and J λ u =. We conclude that u is a nontrivial critical point of functional J λ. Since J λ v J λ v for any v H 1, it follows that u is a nonnegative and nontrivial critical point of J λ. Thus, any λ, λ is an eigenvalue of problem.1. The proof of Proposition.5 is complete. Acknowledgements The authors were partially supported by CNCS-UEFISCDI Grant No. PN-III-P4-ID-PCE References [1] R. Adams, Sobolev spaces, Academic Press, New York, MR45957 [] J. A. Aguilar Crespo, I. Peral Alonso, On an elliptic equation with exponential growth, Rend. Sem. Mat. Univ. Padova , MR143896; Zbl [3] J. Garcia Azorero, I. Peral Alonso, On an Emden Fowler type equation, Nonlinear Anal , MR [4] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Springer, Berlin, MR [5] M. Fărcășeanu, Eigenvalues for Finsler p-laplacian with zero Dirichlet boundary condition, An. Științ. Univ. Ovidius Constanța Ser. Mat. 416, /auom-16-13; MR [6] M. Mihăilescu, V. Rădulescu, Sublinear eigenvalue problems associated to the Laplace operator revisited, Israel J. Math , s y; MR77345 [7] T. Ogawa, A proof of Trudinger s inequality and its applications to nonlinear Schödinger equation, J. Math. App. Anal , X9914-O; MR [8] B. Ruf, Superlinear elliptic equations and systems, in: M. Chipot Ed., Handbook of differential equations: stationary partial differential equations, Vol. V, Elsevier/North- Holland, Amsterdam, 8, pp ; MR49798 [9] N. S. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech , MR1686 [1] M. Willem, Analyse fonctionnelle élémentaire, Cassini, 3. [11] M. Willem, Functional analysis. Fundamentals and applications, Birkhäuser, New York, MR311778