EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM

Transcription

1 EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM JENICĂ CRÎNGANU We derive existence results for operator equations having the form J ϕu = N f u, by using the Mountain Pass Theorem. Here J ϕ is a duality mapping on a real reflexive and smooth Banach space, compactly imbedded in a L q -space, and N f is the Nemytskii operator generated by a Carathéodory function f which satisfies some appropriate growth conditions. AMS 2 Subject Classification: 35J2, 35J6. Key words: Duality mapping, Mountain Pass Theorem, Nemytskii operator, p- Laplacian. 1. INTRODUCTION In [9] the existence of the weak solution in W 1,p (, 1 < p < +, for the Dirichlet problem (1.1 (1.2 p u = f(x, u in R N, u = on was obtained using (among other methods the Mountain Pass Theorem. It is well known that the operator p : W 1,p ( W 1,p ( is in fact the duality mapping on W 1,p ( corresponding to the gauge function ϕ(t = t p 1. In this paper we generalize the results from [9] by considering operator equations of the form (1.3 J ϕ u = N f u, where, instead of the special duality mapping ( p, we consider the case of an arbitrary duality mapping. More precisely, we consider equation (1.1 in the functional framework below. MATH. REPORTS 11(61, 1 (29, 11 2

2 12 Jenică Crînganu 2 (H 1 X is a real reflexive and smooth Banach space, compactly imbedded in L q (, where 1 < q < + and R N, N 2, is a bounded domain with smooth boundary. (H 2 J ϕ : X X is the duality mapping corresponding to the gauge function ϕ : R + R + ; we assume that J ϕ is continuous and satisfies the (S + condition: if x n x (weakly in X and lim sup J ϕ x n, x n x, n then x n x (strongly in X. (H 3 N f : L q ( L q (, 1 q + 1 q = 1, defined by (N f u(x = f(x, u(x, for u L q (, x, is the Nemytskii operator generated by the Carathéodory function f : R R, which satisfies the growth condition (1.4 f(x, s c s q 1 + b(x, x, s R, where c > is constant and b L q (. By solution of equation (1.3 we mean an element u X which satisfies (1.5 J ϕ u = (i N f iu, where i is the compact imbedding of X into L q ( and i : L q ( X is its adjoint. Notice that by (1.4 the operator N f is well-defined, continuous and bounded from L q ( into L q ( (see, e.g., [8], so that i N f i is also welldefined and compact from X into X. Moreover, the functional ψ : L q ( R, defined by ψ(u = F (x, udx, where F (x, s = s f(x, t dt is C1 on L q (, hence on X and ψ (u = N f u for all u X. Notice also that F is a Carathéodory function and there exists a constant c 1 > and a function c L q (, c, such that (1.6 F (x, s c 1 s q + c(x, x, s R. Let us remark that (1.5 is equivalent to (1.7 J ϕ u, v = N f (iu, iv L q (,L q ( = f(x, uvdx, v X. By Asplund s theorem, J ϕ u = Φ(u for each u X, where Φ(u = u ϕ(tdt and Φ : X P(X is the subdifferential of Φ in the sense of convex analysis, i.e., Φ(u = {x X : Φ(v Φ(u x, v u for all v X} (see, e.g., [4], [5] or [12]. Since Φ is convex and X is smooth, Φ is Gâteaux differentiable on X and Φ(u = {Φ (u} for all u X. So, J ϕ u = Φ (u

3 3 Existence results for operator equations involving duality mappings 13 for all u X and, by continuity of J ϕ (see (H 2, we have Φ C 1 (X, R. Consequently, the functional F : X R defined by u F(u = Φ(u Ψ(u = ϕ(tdt F (x, udx, u X, is C 1 on X and F (u = Φ (u Ψ (u = J ϕ u N f u for all u X. Therefore, u X is a solution of equation (1.3 if and only if u is a critical point for F, i.e., F (u =. To prove that F has at least a critical point in X we use the Mountain Pass Theorem, which we recall below. Theorem 1.1. Let X be a real Banach space and assume I C 1 (X, R satisfies the Palais-Smale (PS condition. Suppose I( = and that (i there are constants ρ, α > such that I u =ρ α; (ii there is an element e X, e > ρ such that I(e. Then I possesses a critical value c α. Moreover, c = inf max I(u g Γ u g([,1] where Γ = {g C([, 1], X : g( =, g(1 = e}. (It is obvious that each critical point u at level c (I (u =, I(u = c is a nontrivial one. For the proof see, e.g., [2], [11] or [13]. Let us recall that the functional I C 1 (X, R is said to satisfy the (PS condition if any sequence (u n X for which (I(u n is bounded and I (u n as n, possesses a convergent subsequence. 2. THE MAIN EXISTENCE RESULT First, we establish some preliminary results. Proposition 2.1. If (u n X is bounded and F (u n as n, then (u n has a convergent subsequence. Proof. This result is proved in [9], Lemma 2. Theorem 2.1. Assume that there are θ > 1 and s > such that (2.1 θf (x, s sf(x, s, x, s s, (2.2 lim inf t Then F satisfies the (PS condition. t ϕ(sds > 1 tϕ(t θ.

4 14 Jenică Crînganu 4 Proof. According to Proposition 2.1 it is sufficient to show that any sequence (u n X for which (F(u n is bounded and F (u n, is bounded. As in the proof of Theorem 15 in [9], there exists a constant k > such that (2.3 Φ(u n 1 f(x, u n u n dx k. θ By F (u n, as in the proof of the same theorem, there exists n N such that J ϕ u n, u n f(x, u n u n un, ( n n, which implies (2.4 1 θ ϕ( u n u n + 1 f(x, u n u n dx 1 θ θ u n, ( n 1. From (2.3 and (2.4 we obtain so that un or, equivalently, (2.5 ϕ( u n u n Φ(u n 1 θ ϕ( u n u n 1 θ u n k, ( n n, ϕ(sds 1 θ ϕ( u n u n 1 θ u n k, ( n n [ un ] ϕ(sds ϕ( u n u n 1 θ 1 k. θϕ( u n By condition (2.2 there are constants γ > 1 θ and t > such that t ϕ(sds γ, t t. tϕ(t If u n there exists n 1 N such that u n t for n n 1 and then un ϕ(sds u n ϕ( u n γ, n n 1. From (2.5 we obtain ϕ( u n u n ( γ 1 θ 1 k, n n 1. θϕ( u n a contradiction (because γ > 1 θ and ϕ( u n. Consequently, (u n is bounded in X.

5 5 Existence results for operator equations involving duality mappings 15 Theorem 2.2. Assume that there exists θ > 1 such that t (2.6 lim ϕ(sds t t θ = and either (i there exists s 1 > such that (2.7 < θf (x, s sf(x, s, x, s s 1, or (ii there exists s 1 < such that (2.8 < θf (x, s sf(x, s, x, s s 1. Then F is unbounded from below. Proof. We are going to prove the sufficiency of condition (i (similar arguments may be used if (ii holds. Let u X, u > (in fact i(u > be such that meas (M 1 (u >, where M 1 (u = {x : u(x s 1 } (in fact i(u(x s 1. We shall prove that F(λu as λ. For λ 1, from the proof of Theorem 16 in [9] we have F (x, λudx λ θ k 1 (u k 2, where k 1 (u >, k 2 > are constants. Therefore, F(λu = Φ(λu F (x, λudx Φ(λu λ θ k 1 (u + k 2 [ λ u ] = λ θ ϕ(tdt λ θ k 1 (u + k 2. By condition (2.6 there are constants α (, k 1 (u and t > such that t ϕ(sds t θ u θ α, t t, and then [ λ u ] F(λu λ θ ϕ(tdt (λ u θ u θ k 1 (u + k 2 λ θ (α k 1 (u + k 2 for λ u t. Since α k 1 (u <, we have F(λu as λ. Remark 2.1. By Theorem 2.2, since F is unbounded from below, for each ρ > there exists e X with e > ρ such that F(e. Now we are in a position to prove the main result of this section.

6 16 Jenică Crînganu 6 Theorem 2.3. Assume that hypotheses (H 1, (H 2 and (H 3 hold. Moreover, assume that (i there are θ > 1 and s > such that (2.9 < θf (x, s sf(x, s, x, s s ; t (ii lim inf ϕ(sds > 1 t tϕ(t θ ; t (iii lim ϕ(sds t t θ = ; (iv there are r 1, r 2 (1,, r 1 < r 2, and c > such that F(u c 2 u r 1 X c 3 u r 2 X ( u X with u X < c, where c 2, c 3 > are constants. Then equation (1.3 has at least a nontrivial solution in X. Proof. Let us notice that condition (iv is suggested by [1]. It is sufficient to show that F has at least a nontrivial critical point u X. To do it, we shall use Theorem 1.1. Clearly, F( =. By (i, (ii and Theorem 2.1, F satisfies the (PS condition. By (iv we have F(u u r 1 X ( c2 c 3 u r 2 r 1 X for all u X, with u X < c, hence there are constants ρ, α >, ρ < c such that F(u α > provided that u X = ρ is small enough. Finally, from (i, (iii and Theorem 2.2 (see also Remark 2.1 there is an element e X, e > ρ, such that F(e, and the proof is complete. Remark 2.2. A sufficient condition in order to have (iv is the existence of real numbers r 1, r 2, 1 < r 1 < r 2, such that s (iv 1 lim ϕ(tdt s s r >, 1 f(x, s (iv 2 lim s s r 2 1 <. Indeed, by (iv 1, Φ(u c 2 u r 1 provided that u > is small enough while, by (iv 2, F (x, udx c 3 u r 2, provided that u > is small enough (here, c 2, c 3 > are constants. Consequently, F(u c 2 u r 1 c 3 u r 2 for u > sufficiently small.

7 7 Existence results for operator equations involving duality mappings EXAMPLES Example 3.1 Consider the Dirichlet problem (1.1, (1.2, where 1 < p <, R N, N 2, is a bounded domain with smooth boundary, p u = div( u p 2 u = ( N i=1 x i u p 2 u x i is the so-called p-laplacian and f : R R is a Carathéodory function which satisfies the growth condition (3.1 f(x, s c ( s q 1 + 1, x, s R, { Np with c constant, 1 < q < p = N p if N > p, + if N p. Let us remark that the p-laplacian operator p : W 1,p ( W 1,p ( defined by N ( p 2 u p u = u, u W 1,p (, x i x i i=1 or, equivalently, p u, v = is the duality mapping u p 2 u v, u, v W 1,p (, J ϕ : W 1,p ( W 1,p ( corresponding to the gauge function ϕ(t = t p 1 (see, e.g., [9] or [12]. On the other hand, the Nemytskii operatorn f is continuous and bounded from L q ( into L q (. By solution of the Dirichlet problem (1.1, (1.2 we mean an element u W 1,p ( which satisfies (3.2 p u = ( i N f i u in W 1,p (, or, equivalently, u p 2 u v = f(x, uv, v W 1,p (, where i is the compact imbedding of W 1,p ( into L q ( and i : L q ( W 1,p ( is its adjoint. Consequently, (3.2 may be equivalently written as (3.3 J ϕ u = N f u (here, by N f we mean i N f i. We shall formulate sufficient conditions for equation (3.3 to admit a nontrivial solution, via Theorem 2.3. Take X = W 1,p ( with R N, N 2, a bounded domain with smooth boundary, 1 < p <, q (1, p, ϕ(t = t p 1,

8 18 Jenică Crînganu 8 t, f : R R a Carathéodory function which satisfies the conditions below. The growth condition (3.1. There are θ > p and s > such that (3.4 < θf (x, s sf(x, s, x, s s. (3.5 lim sup s where λ 1 = inf { v p 1,p v p,p f(x, s s p 2 s < λ 1 uniformly in x, : v W 1,p (, v } is the first eigenvalue of p in W 1,p (. Since W 1,p ( is a reflexive and smooth Banach space, compactly imbedded in L q (, hypothesis (H 1 of Theorem 2.3 is satisfied. Since J ϕ = p : W 1,p ( W 1,p ( is continuous and satisfies condition (S + (see, e.g., [9] hypothesis (H 2 of Theorem 2.3 is satisfied, too. By the growth condition (3.1, hypothesis (H 3 of Theorem 2.3 is also satisfied. Since θ > p, conditions (ii and (iii are satisfied. Finally, condition (3.5 implies condition (iv in Theorem 2.3 (see, e.g., [9]. Consequently, Theorem 2.3 applies and gives the already known result (see, e.g., [9] on the existence of a nontrivial solution for problem (1.1, (1.2. (3.6 (3.7 Example 3.2. Consider the Neumann problem p u + u p 2 u = f(x, u in, p 2 u u n = on, where 1 < p <, R N, N 2, is a bounded domain with smooth boundary and f : R R is a Carathéodory function which satisfies the growth condition (3.1. By solution of the Neumann problem (3.6, (3.7 we mean an element u W 1,p ( which satisfies (3.8 u p 2 u v + u p 2 uv = f(x, uv, v W 1,p (. Assume X = W 1,p ( is endowed with the norm u p 1,p = u p,p + u p,p, u W 1,p (, which is equivalent to the standard norm on the space W 1,p ( (see [6].

9 9 Existence results for operator equations involving duality mappings 19 In this case, the duality mapping J ϕ on ( W 1,p (, 1,p corresponding to the gauge function ϕ(t = t p 1 is defined (see [6] by J ϕ : ( W 1,p (, 1,p ( W 1,p (, 1,p, J ϕ u = p u + u p 2 u, u W 1,p (. It is easy to see that u W 1,p ( is a solution of problem (3.6, (3.7 in the sense of (3.8 if and only if (3.9 J ϕ u = N f u (here, by N f we also mean i N f i, where i is the compact imbedding of (W 1,p (, 1,p into L q ( and i : L q ( ( W 1,p (, 1,p is its adjoint. We shall formulate sufficient conditions for equation (3.9 to admit a nontrivial solution, via Theorem 2.3. Take X = W 1,p ( with R N, N 2, a bounded domain with smooth boundary, 1 < p <, q (1, p, ϕ(t = t p 1, t and f : R R a Carathéodory function which satisfies the conditions below. The growth condition (3.1. There are θ > p and s > such that (3.1 < θf (x, s sf(x, s, x, s s. f(x, s (3.11 lim sup s s p 2 s < λ 1 uniformly in x, { v p } 1,p where λ 1 = inf : v W 1,p (, v. v p,p Since ( W 1,p (, 1,p is a reflexive and smooth Banach space, compactly imbedded in L q ( (see [6], hypothesis (H 1 of Theorem 2.3 is satisfied. Since J ϕ : ( W 1,p (, 1,p ( W 1,p (, 1,p is continuous and satisfies condition (S + (see, e.g., [6], hypothesis (H 2 of Theorem 2.3 is satisfied, too. By the growth condition (3.1, hypothesis (H 3 of Theorem 2.3 is also satisfied. Since θ > p, conditions (ii and (iii are satisfied. Finally, condition (3.11 implies condition (iv in Theorem 2.3 (see, e.g., [6]. Consequently, Theorem 2.3 applies and gives the already known result (see, e.g., [6] on the existence of a nontrivial solution for problem (3.6, (3.7. REFERENCES [1] R.A. Adams, Sobolev Spaces. Academic Press, New York San Francisco London, [2] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical points theory and applications. J. Funct. Anal. 14 (1973,

10 2 Jenică Crînganu 1 [3] H. Brezis, Analyse fonctionelle. Masson, Paris, [4] F.E. Browder, Problèmes non-lineaires. Les Presses de l Université de Montreal, [5] Ioana Cioranescu, Duality Mapping in Nonlinear Functional Analysis. Publishing House of the Romanian Academy, Bucharest, (Romanian [6] J. Crînganu, Variational and topological methods for Neumann problems with p-laplacian. Comm. Appl. Nonlinear Anal. 11 (24, [7] J. Crînganu and G. Dinca, Multiple solutions for a class of nonlinear equations involving a duality mapping. Differential Integral Equations 21 (28, [8] D.G. de Figueiredo, Lectures on Ekeland variational principle with applications and detours. Tata Institut of Fundamental Research, Springer-Verlag, [9] G. Dinca, P. Jebelean and J. Mahwin, Variational and Topological methods for Dirichlet problems with p-laplacian. Portugaliae Math. 58 (21, [1] G. Dinca and P. Matei, Multiple solutions for operator equations involving duality mappings on Orlicz-Sobolev spaces via the Mountain Pass Theorem. Rev. Roumaine Math. Pures Appl. 53 (28, [11] O. Kavian, Introduction a la théorie des points critiques. Springer Verlag, [12] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non-lineaires. Dunod-Gauthier, Villars, Paris, [13] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems. Springer, [14] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Ser. Math. 65. Amer. Math. Soc., Providence, RI, Received 14 November 28 University of Galaţi Department of Mathematics 88 Galaţi, Romania jcringanu@ugal.ro