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
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 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.
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 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.
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 Existence results for operator equations involving duality mappings 17 3. 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,
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 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, 1975. [2] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical points theory and applications. J. Funct. Anal. 14 (1973, 349 381.
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