Stationary Solutions and Connecting Orbits for p-Laplace Equation

We deal with one dimensional p-Laplace equation of the form ut=(|ux|p-2ux)x+f(x,u),x∈(0,l),t>0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_t = (|u_x|^{p-2} u_x )_x + f(x,u), \ x\in (0,l), \ t>0, \end{aligned}$$\end{document}under Dirichlet boundary condition, where p>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>2$$\end{document} and f:[0,l]×R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:[0,l]\times {\mathbb {R}}\rightarrow {\mathbb {R}}$$\end{document} is a continuous function with f(x,0)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x,0)=0$$\end{document}. We will prove that if there is at least one eigenvalue of the p-Laplace operator between limu→0f(x,u)/|u|p-2u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{u\rightarrow 0} f(x,u)/|u|^{p-2}u$$\end{document} and lim|u|→+∞f(x,u)/|u|p-2u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{|u|\rightarrow +\infty } f(x,u)/|u|^{p-2}u$$\end{document}, then there exists a nontrivial stationary solution. Moreover we show the existence of a connecting orbit between stationary solutions. The results are based on Conley index and detect stationary states even when those based on fixed point theory do not apply. In order to compute the Conley index for nonlinear semiflows deformation along p is used.


Introduction
We shall study a nonlinear p-Laplace equation with p > 2, l > 0 a continuous f : [0, l] × R → R, which is locally Lipschitz with respect to the second variable, i.e.
for any R > 0 there is The stationary version of (1), i.e. the elliptic problem −(|u (x)| p−2 u (x)) = f (x, u(x)), x ∈ (0, l), is subject of extensive studies by many authors-see earlier papers [8][9][10]14] as well as more recent examples [4,5,12,15,17] or [13]. Usually topological degree/index techniques or variational approach are applied. Here we use a dynamical system approach based on the Conley type index from [20] and [18] and Rybakowski's techniques from [19]. In the case of semiflows determined by nonlinear equations with p-Laplace operator, the main difficulty is to compute the Conley index. This task reduces to studying the nonlinear semiflow generated by u t = (|u x | p−2 u x ) x + λ|u| p−2 u with real λ. In order to compute its Conley index, inspired by [8], we use deformation along p. As a result we shall prove the following existence criterion. f (x, u) |u| p−2 u = f 0 (x) (4) and lim for some f 0 , f ∞ ∈ C([0, l]) uniformly with respect to x ∈ [0, l]. Suppose there are k 0 , k ∞ ∈ N such that λ k ∞ +1 , for all x ∈ (0, l), with the strict inequalities on set of positive measure. If k 0 = k ∞ , then there exists a nontrivial solutionū ∈ C 1 ([0, l]) of (3).
Moreover, there exists a connecting orbit betweenū and 0, i.e. a full solution u of (1) such that either u(t n , ·) →ū for some t n → +∞ and u(t, ·) → 0 as t → −∞ or u(t n , ·) →ū for some t n → −∞ and u(t, ·) → 0 as t → +∞ (with respect to the max norm of the space C(0, l)).
Here recall that λ ∈ R, for which the problem −(|u (x)| p−2 u (x)) = λ|u(x)| p−2 u(x), x ∈ (0, l), u(0) = u(l) = 0, has nonzero solutions, make a sequence of positive numbers λ ( p) n , n ≥ 1, such that λ ( p) n → +∞ and, for any n ≥ 1, λ (q) n → λ ( p) n whenever q → p (see [16]). We also put λ ( p) 0 := −∞. In this paper we consider a local semiflow (a sort of dynamical system) ( p, f ) on the space X = C 0 (0, l) := {u ∈ C(0, l) | u(0) = u(l) = 0} associated with the equation (1). To find a stationary solution and related connecting trajectory we use the theory of irreducible sets due to Rybakowski [20], where we need to find Conley indices of the zero K 0 := {0} and the set K ∞ made by all full bounded trajectories of (1). The main difficulty lies in the fact that both the differential operator and continuous term are nonlinear. In order to consider and compute Conley index (due to Rybakowski) for ( p, f ) we need to study the existence, compactness and continuity properties of solutions. We shall also exploit the Lyapunov function for the problem and related regularity to find stationary solutions at the ends of full trajectories. What we gain by use of Conley index and what we could not obtain with topological degree techniques is that we show the existence under the condition k 0 = k ∞ while topological degree works in the case where k 0 and k ∞ are of different parities. In addition, we have a full trajectory between two stationary solutions.
The paper is organized as follows. In the rest of the section we give some notation and basic preliminaries on Conley index. Section 2 is devoted mainly to continuity and compactness issues for abstract evolution equations governed by perturbations of m-accretive operators and subdifferentials of convex functionals. In Sect. 3 we study the existence and regularity of solutions together with Lyapunov function theory. The continuity and compactness properties with respect to p and f , which are crucial for computing Conley index via its continuation property, are explored in Sect. 4. Finally, we compute the Conley indices of K 0 and K ∞ and prove Theorem 1.1 in Sect. 5.

Preliminaries
Two pointed topological spaces (X, x 0 ) and (Y, y 0 ) are said to be homotopically equivalent or have the same homotopy type if and only if there are maps f : (X, x 0 ) → (Y, y 0 ) and g : (Y, y 0 ) → (X, x 0 ) such that f • g is homotopic via a mapping keeping y 0 fixed to the identity of (Y, y 0 ) and g • f is homotopic via a mapping keeping x 0 fixed to the identity of (X, x 0 ). The homotopy class represented by a space is a pair of topological spaces with a nonempty and closed A ⊂ X , then X/A denotes the quotient space, obtained by collapsing the subset A to a point [A]. l] |u|. By L p (0, l) and W k, p (0, l) and W 1, p 0 (0, l) we denote the standard Lebesgue and Sobolev spaces on the interval (0, l) and we put H 1 0 (0, l) := W 1,2 0 (0, l). In the same way, by L p (0, T ; X ) and W 1, p (0, T ; X ) we denote the spaces with values in a Banach space X .

Conley Index Due to Rybakowski
Here we briefly present homotopy index theory from [20] (see also [18]). Let : D → X , where D is an open subset of [0, ∞) × X , be a local semiflow on a metric space X . Let Tū = sup{t > 0 | (t,ū) ∈ D}. A continuous function u : J → X , where J ⊂ R is an interval, is called a solution of if and only if u(t + s) = t (u(s)) for any t ≥ 0 and s ∈ J such that t + s ∈ J . If u : [a, +∞) → X , a ∈ R, is a solution of , then by the ω-limit of u we mean the set ω(u) := ū ∈ X | ∃ (t n ) in [a, +∞) such that t n → +∞ andū = lim n→∞ u(t n ) .
The α-limit of a solution u : (−∞, a] → X of , a ∈ R, is defined by We shall say that K ⊂ X is a -invariant set or invariant with respect to provided Inv (K ) = K . A -invariant set K ⊂ X is called an isolated -invariant set if and only if there exists N ⊂ X such that K = Inv (N ) ⊂ int N . Such N is called an isolating neighborhood of K . The following concept of admissibility is crucial in Rybakowski's version of Conley theory on general metric spaces and enables us to construct a Conley type index without local compactness of X . with t n → +∞ and (v n ) in X such that [0,t n ] (v n ) ⊂ N , the sequence t n (v n ) contains a convergent subsequence.
We also need the notion of admissibility for the families of local semiflows.
contains a convergent subsequence. If additionally k does not explode in N for any k ∈ K , then N is strongly -admissible.
Let I(X ) be the family of pairs ( , K ) where is a local semiflow on a metric space X and K is an isolated invariant set having a strongly -admissible isolating neighborhood of The Conley index has the following properties In the linear case we shall use the following formula for computation of Conley index. Suppose that X is a normed space such that X = X − ⊕ X + with k := dim X + < +∞ and a C 0 -semigroup We shall use the theory of irreducible sets due to Rybakowski [20]. Recall that an isolated invariant set K (relative to a local semiflow ) is called reducible if there exist isolated invariant sets K 1 , K 2 such that K = K 1 ∪ K 2 , K 1 ∩ K 2 = ∅, ( , K 1 ), ( , K 2 ) ∈ I(X ) and both h( , K 1 ) = 0 and h( , K 2 ) = 0. We say that K is irreducible if it is not reducible. It is known that the set K is irreducible if one of the following conditions is satisfied: K is connected, h( , K ) = 0 or h( , K ) = k (see [20, Ch. I, Th. 11.6]). The concept of irreducible set turns out to be useful due to the following.

Properties of Abstract Evolution Equations with m-Accretive Operators
Let A : D(A) X defined in a Banach space X be an m-accretive operator, f : [0, T ] → X, T > 0, andū ∈ D(A). We shall consider the equation

Remark 2.2 (i) It appears that for anyū ∈ D(A)
and f ∈ L 1 (0, T ; X ) the problem (6) admits a unique integral solution (see e.g. [1]). It may be also shown that this integral solution is also a mild solution (see e.g. [1]), and as such, is a limit of discrete approximations. (ii) Now consider Banach spaces X,X such that X is continuously embedded intoX .
Suppose that a m-accretive operator A in X has an extensionÃ inX such that it is m-accretive inX . Then, for anyū ∈ X , f ∈ L 1 (0, T ; X ), the integral solution of (6) is also an integral solution of (6) in the spaceX . It follows immediately by use of discrete approximations (see (i)). (iii) Let A (ū, f ) denote the integral solution of (6). Then, for anyū 1 ,ū 2 ∈ D(A) and f 1 , f 2 ∈ L 1 (0, T ; X ) one has (see [1]) (iv) In particular, if we take f = 0, then one may define a family of operators We shall use the following continuity and compactness result.

Proposition 2.3 Let A n : D(A n )
X, n ≥ 0, be m-accretive operators.
(i) (Trotter-Kato theorem) Ifū n →ū 0 in X ,ū n ∈ D(A n ) for all n ≥ 0, A n We shall also consider nonlinear problems of the form

(ii) Assume that D(A n ) = D(A 1 ) for any n ≥ 1 and the set n≥1 S A n (t)(B) is relatively compact for any bounded B
with a locally Lipschitz F : X → X andū ∈ D(A). We shall say that a continuous u : Let us state a general existence and uniqueness theorem.
(ii) Fix T ∈ (0, Tū 0 ). Let R := max t∈[0,T ] u 0 (t) and let L R be the common Lipschitz constant for F n , n ≥ 0, on the ball B(0, 3R). Define From Proposition 2.3 (i) it follows that α n → 0 + and therefore α n e T L R < R for n ≥ n 0 for some n 0 ∈ N. Fix n ≥ n 0 . In order to prove the inequality from the conclusion it suffices to show that T < Tū n . Suppose, contrary to our claim, that Tū n ≤ T . Therefore u n ([0, t]) ⊂ B(0, 3R) and u n (t) > 2R for some t < T , which follows from (i).
Using Remark 2.2 (iii) one has, for τ ∈ [0, t], In consequence, by the Gronwall inequality we obtain the estimate This means that u n (t) ≤ u 0 (t) + R ≤ 2R. This contradicts our assumption. By the estimate u n (t) − u 0 (t) ≤ α n e T L R for t ≤ T , the convergence from the second part of the conclusion holds true.
Remark 2.5 Let D be the set of (t,ū) ∈ [0, +∞)× X such that the problem (7) has a solution on [0, t] and let : (7). It clearly follows from Proposition 2.4 that is a local semiflow on X . Now suppose that H is a Hilbert space with the scalar product ·, · and the norm · and consider a lower semicontinuous convex functional ϕ : one can consider the following problem It appears that integral solutions in this case are more regular and are strong solutions.  (10), then u is a.e. differentiable on (0, T ) and has the following properties In particular, for allū ∈ D(∂ϕ) the function ϕ • u is continuous on (0, T ].
In order to estimate time derivative of solutions we shall need the following result.
We shall need the following compactness criterion being an extension of [1, Ch. 4, Th. 2.4] to a family of semigroups generated by subdifferentials. In order to prove Proposition 2.8 we need the following lemma.
Proof of Proposition 2.8. Fix t > 0 and let B ⊂ L 2 (0, l) be bounded. From Lemma 2.9 we obtain S ∂ϕ n (t)(B) ⊂ L λ for some λ independent of n. The conclusion follows from the relative compactness of L λ .

Existence and Regularity for p-Laplace Evolution Equation
First we put the equation (1) in the abstract framework to precise the concept of solution and get information on their regularity. To this end, define Proof The proof follows directly from Proposition 2.4 applied to (15).
So as to study regularity properties of solutions, it will be convenient to use also an L 2 -extension of A p . We shall considerĀ p : D(Ā p ) → L 2 (0, l) given by Clearly, D(Ā p ) ⊂ C 1 (0, l) and Gr(A p ) ⊂ Gr(Ā p ) (by the embedding of C 0 (0, l) into L 2 (0, l)).

Lemma 3.4 (i) The operatorĀ p is m-accretive and there exists c p > 0 such that
(ii)Ā p = ∂ϕ p , where ∂ϕ p is the subdifferential of the lower semicontinuous convex functional ϕ p : L 2 (0, l) → R ∪ {+∞} given by Proof (i) If we change the space L p (0, l) with L 2 (0, l) in the proof of [4,Prop. 4.1], we obtain that the operatorĀ p is maximal monotone, which in Hilbert spaces is equivalent to being m-accretive. In particular, we have the estimate (16).
which gives the conclusion as c p = 2 2− p ≥ 2 − p .
The following result sheds more light on the regularity of solutions. Moreover, the functional ϕ p, f : W 1, p 0 (0, l) → R given by (1), since for any solution u ∈ C([0, T ], C 0 (0, l)) and 0 < s < t < T one has

is a Lyapunov function for
Proof By Remark 2.2, the function u may be viewed as an element of C([0, T ], L 2 (0, l)) and the integral solution (in L 2 (0, l)) oḟ In order to verify that ϕ p, f is a Lyapunov function observe that, by Proposition 2.6, one has u ∈ W 1,2 ((0, T ]; L 2 (0, l)) and Now take any t ∈ [0, T ] such thatu(t) exists (in L 2 (0, l)) and any sequence (h n ) in R\{0} with h n → 0. Passing to a subsequence, if necessary, we may suppose that (u(t+h n )−u(t))/ h n → u(t) a.e. on [0, l] and that there is g ∈ L 1 (0, l) such that |(u(t + h n ) − u(t))/ h n | ≤ g a.e. on [0, l]. By means of Lebesgue's dominated convergence theorem we have Furthermore, since F is Lipschitz with respect to the second variable on bounded sets, for f is bounded on bounded sets, we can use dominated convergence theorem to get d dt This together with (18) ends the proof.
We shall summarize the obtained results in the context of dynamical systems. Define ) → X is the maximal integral solution of (1) with u(0) =ū. By Remark 2.5, ( p, f ) is a local semiflow on X. Theorem 3.8 If u ∈ C(R, X) is a bounded solution of (1), then α(u) and ω(u) are nonempty, connected and compact in the space C 0 (0, l), where E is the set of all stationary solutions of (1).
(ii) We shall prove the assertion in two steps.
Step 2. Now let us take a bounded B ⊂ C 0 (0, l) and t > 0. Take any ( p n ) in [2, p] and (ū n ) in B. Put α = t/3 and define u n := S A pn (·)ū n . We are going to show that the sequence (u n (t)) has a convergent subsequence in C 0 (0, l).
We will summarize continuity and compactness properties for the equation (1) in the following At the end of this section, we express the obtained results in terms of parameterized semiflows. To this end, let X = C 0 (0, l) and let f : [0, l] × R × [0, 1] → R be continuous with f (x, 0, μ) = 0, for all x ∈ [0, l] and μ ∈ [0, 1], and such that for any R > 0 there exists where F : X × [0, 1] → X is defined by Define ) → X is the maximal integral solution of (23) with u(0) =ū. The above results imply that this family of semiflows is continuous with respect to p and μ and that bounded sets are strongly admissible. Proposition 4.4 (i) Ifū n →ū 0 in C 0 (0, l), t n → t 0 in [0, +∞), p n → p 0 in [2, +∞), μ n → μ 0 in [0, 1] and (t 0 ,ū 0 ) ∈ D ( p 0 ,μ 0 ) , then (t n ,ū n ) ∈ D ( p n ,μ n ) for large n and Proof (i) is a straightforward consequence of Theorem 4.3 (i). The proof (ii) goes along the lines of Proposition 3.10 with use of Theorem 4.3 (ii).

Proof of Theorem 1.1
We start with a computation of Conley index at zero and at infinity.   The computation of the indices of K 0 and K ∞ will be reduced to computing the Conley index of zero in a special case.

Lemma 5.2
Let g ∈ C([0, l]) be a such that for some k ≥ 1, λ for all x ∈ [0, l] with the strict inequalities on a set of positive measure and let { ( p,g) } t≥0 be the local semiflow generated on C 0 (0, l) by the problem Then u ≡ 0 is the only full bounded solution of ( p,g) and, in particular, K := {0} is an isolated invariant set relative to ( p,g) . Moreover h( ( p,g) , K ) = k .
In the proof we shall use the following lemma. has no nontrivial weak solutions.
Proof of Lemma 5.2 Given a full bounded solution u ∈ C(R, C 0 (0, l)) of (24). By Theorem 3.8, α(u) ∪ ω(u) ⊂ E. If u was nontrivial, then we would get two different equilibria, since due to Theorem 3.5 the Lyapunov function would change its value along the nontrivial solution. But according to Lemma 5.3, in this case we have E = {0}, which is a contradiction showing that there are no nontrivial bounded solutions of (24), which shows that K = {0} is an isolated invariant set. Observe that we get h ( ( p,g) , K ) = h ( ( p,λ) , K ) for any λ ∈ (λ Indeed, it is enough to consider a family of semiflows ( p,g(·,μ)) , μ ∈ Using the spectral decomposition given by the Laplace operator A 2 (andĀ 2 ) together with Theorem 1.4, we get h( (2,λ(1)) , K ) = k .

Lemma 5.4 Under the assumptions
Proof The first convergence follows directly from (4). We shall prove the second convergence. Fix ε > 0. Put V := sup{ v n p−1 | n ∈ N} and let D > 0 be such that for all x ∈ [0, l] and |u| ≥ D.
For n sufficiently large and nf (x, R n u, μ n ), x ∈ [0, l], u ∈ R, n ≥ 1. In a similar manner as in (i), there exists a subsequence of (v n ) converging uniformly on bounded intervals to some bounded v 0 ∈ C(R, X) with v 0 (0) = 1 that is an integral solution of v t = |v x | p−2 v x x + f ∞ (x)|v| p−2 v, x ∈ (0, l), t ∈ R, v(t, 0) = v(t, l) = 0, t ∈ R, which is impossible. Finally, using the homotopy invariance of Conley index and Lemma 5.2 one has h( ( p, f ) , K ∞ ) = h(˜ (1) , which completes the proof.