Variational approach to semilinear elliptic equation with nonlinear boundary

Accepted Manuscript Variational approach to semilinear elliptic equation with nonlinear boundary

A. Nowakowski, A. Orpel

PII: DOI: Reference:

S0022-247X(15)01099-9 http://dx.doi.org/10.1016/j.jmaa.2015.11.049 YJMAA 19995

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

20 November 2014

Please cite this article in press as: A. Nowakowski, A. Orpel, Variational approach to semilinear elliptic equation with nonlinear boundary, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2015.11.049

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Variational approach to semilinear elliptic equation with nonlinear boundary A. Nowakowski & A. Orpel Faculty of Math & Comp. Sciences, University of Lodz, Banacha 22, 90-238 Lodz Abstract We consider the existence of positive solutions for a certain class of semilinear elliptic equation with nonlinear boundary. Our approach is based on the variational methods.

Key words: semilinear elliptic equation, nonlinear boundary conditions, variational methods, positive solution 2000 Mathematics Subject Classification. 35J20, 35J66.

1

Introduction

Throughout the paper we will consider Ω to be an open bounded domain in Rn , n ≥ 2, with boundary Γ being sufficiently smooth. The goal of the paper is to study the second order elliptic equation with nonlinearity containing sources terms in the interior of the domain and nonlinear sources terms on its boundary: Δx(y) = Fx (y, x(y)) − Gx (y, x(y)), in Ω, −∂ν x(y) = Hx (y, x(y)) − Qx (y, x(y)) on Γ.

(1.1)

That type of problem describes many facts in nonlinear elasticity [5], nonNewtonian fluid mechanics [6], glaciology [16], mathematical biology [2], but also some problems in differential geometry [9]. There exist a large amount of papers devoted to problem like (1.1) (see e.g. [8], [21] and the references therein) mostly investigating positive or non-negative solutions under a variety of the nonlinear term Fx with Dirichlet boundary conditions (see e.g. [1], [13]-[15], [17]). In last ten years considerable more attention were made to problem like (1.1) with nonlinearity on the boundary and right hand side of 1

interior equation being of concave convex type. The existence of positive solutions of the problem in bounded domains is strongly dependent on a priori estimates of the solutions [10], so fewer results are known for n ≥ 2. We can find many papers dealing with the existence of positive solutions of the problems in unbounded domains (see e.g. [4], [18]- [20] and the references cited therein). Very recently, Tsung-fang Wu (in [20]), investigated the existence and nonexistence of positive solutions of problems of types: −Δx(y) = (1 + λa(y)) |x(y)|p−2 x(y) − x(y), in Rn+ , ∂ν x(y) = −f (y) |x(y)|q−2 x(y) in ∂Rn+ ,

(1.2)

if n ≥ 3, 2∗ = ∞ if n = 2), 2 < p < 2∗ where 1 < q < min{2∗ , p} (2∗ = 2(n−1) n−2 2n (2∗ = n−2 if n ≥ 3). Essential point of most of those papers with bounded domain as well unbounded domains is to have nonlinearities in the interior equations consisting of two terms convex and concave. The aim of this paper is to extend that type of nonlinearity to more general case simply difference of the derivative of two convex functions Fx − Gx and to consider the same type of nonlinearities on the boundary equation Hx − Qx . To this end we apply the approach based on the idea similar to that described in [11] and [12] One of the most popular method to study (1.2) in bounded domain as well unbounded domains are the variational methods (see e.g. [20] and references therein). Usually the energy functional Jλ   1 1 1 2 q Jλ (x) = xH 1 + f |x| dσ − (1 + λa) |x|p dy, (1.3) 2 q ∂Rn+ p Rn+  is associated with (1.2), where x2H 1 = Rn (|∇x|2 + |x|2 )dy. That shows: + both equations from (1.2) are included in Jλ . The functional (1.3) is well defined on C 1 if p and q verify 2 < p < 2∗ and 1 < q < 2∗ . These significantly confine classical variational approach to (1.2) and so also to (1.1). Our variational approach to (1.1) is different. We treat both equation in (1.1) separately. First we investigate the second equation on the boundary −∂ν x(y) = Hx (y, x(y)) − Qx (y, x(y)) on Γ,

(1.4)

with the help of the functional  JΓ (w) = (Φ(y, w(y)) + Φ∗ (y, −∂ν w(y)) − w(y), −∂ν w(y))dy, Γ

−ω 0 ν,y

x(y), where w(y) = e  ω |ν|2   2 0 −2ω 0 ν,y ω 0 ν,y ω 0 ν,y |w| + e ) − Q(y, we ) − c H(y, we 0 for w ≥ 0, 2 Φ(y, w) = +∞ for w < 0, 2

Φ∗ denotes its Fenchel conjugate, obtaining the solution x¯Γ (y) on Γ. Next, we solve Δx(y) = Fx (y, x(y)) − Gx (y, x(y)) in Ω, (1.5) x(y) = x¯Γ (y) on Γ, treating it as the critical points of the functional    1 2 |∇x(y)| + F (y, x(y)) − G(y, x(y)) dy. J (x) = 2 Ω Let us note that J is not bounded on its natural domain. Moreover, we have nonlinear boundary conditions. Taking into account both fact it is not possible, according to our best knowledge, to apply variational tools in a standard way. Thus we are going to describe a new duality which allows us to describe links between critical points of J considered on some subspace of the space H 1 (Ω) and solutions of (1.1). Roughly speaking our approach can be divided into two steps. First we show the existence of solutions x¯Γ of the second equation. Precisely, in order to obtain the existence result of (1.4), we minimize an auxiliary functional JΓ defined on some subspace of H 1 (Γ). Then having a solution of (1.4) we shall study the first equation of (1.1) on some subspace of the space H 1 (Ω) consisting of the functions satisfying boundary condition x(y) = x¯Γ (y) on Γ, i.e. equation (1.5).

2

Main results

We will focus on the case when n ≥ 3 ( n ≤ 2 being the least interesting, since the concept of criticality of Sobolev’s embedding is much less pronounced). We start with results concerning problem (1.4), its relation to the functional JΓ (x) and problem (1.1). To this effect define

˜ 1 (Γ) = x ∈ H 2 (Ω) : x |Γ ∈ H 1 (Γ) . H Then, by the normal derivative ∂ν x(y) in (1.4) we mean normal derivative ˜ 1 (Γ). Now, we put of x(·) ∈ H 2 (Ω) ∩ H ˜ 1 (Γ)} ⊂ H ˜ 1 (Γ), U¯ = {w : w(y) = e−ω0 ν,y z(y), y ∈ Γ, z ∈ H where ν = (ν 1 , ..., ν n ) is the unit outward normal to Γ and ν, y is a scalar product of ν and y. As the problem (1.4), in general, is not of variational type we will study it indirectly i.e. first we use the functional  JΓ (w) = (Φ(y, w(y)) + Φ∗ (y, −∂ν w(y)) − w(y), −∂ν w(y))dy, (2.1) Γ

3

where



  2 −2ω 0 ν,y ω 0 ν,y ω 0 ν,y |w| + e ) − Q(y, we ) − c H(y, we 0 for w ≥ 0, Φ(y, w) = +∞, for w < 0, (2.2) ∗ Φ denotes its Fenchel conjugate. The definition of Φ means that we are looking for minimizers in U¯ which are positive. We assume the following conditions concerning problem (1.4): (HΓ) H(·, x) and Q(·, x) are measurable in Γ for each x ∈ R, H(y, ·) and Q(y, ·) are differentiable and convex in R, Q(y, ·) is twice differentiable, for y ∈ Γ and for some b3 > 0 ω 0 |ν|2 2

|Qxx (y, x)| ≤ b3 for each y ∈ Γ, x ∈ R. H(·, ·) − Q(·, ·) is bounded below i.e. c0 := inf Ω×R (H(y, x) − Q(y, x)) > −∞. Moreover, we assume that ω 0 in (2.2) is sufficiently large to make Φ(y, ·) a convex function for each y ∈ Ω. Proposition 2.1. Under assumption (HΓ) there is a wΓ (·) ∈ U¯ , wΓ ≥ 0, which affords a minimum to the functional JΓ defined on U¯ and JΓ (wΓ ) = 0. Proof. Let us notice that JΓ (w(·)) is bounded below in U¯ , in fact, JΓ (w(·)) ≥ 0 and weakly lower semicontinuous in U¯ . Moreover, let us observe, by (HΓ), JΓ (w(·)) → ∞ when w(·)H 1 (Γ) → ∞. Really, it is enough to notice that 

 H(y, weων,y ) − Q(y, weων,y ) − c0 ≥ 0

and multiplying (2.2) by |w|−1 one has |w|−1 Φ(y, w) ≥

ω |ν|2 |w| . 2

Hence and from (HΓ) we infer that wn (·)L2 (Γ) is bounded for minimizing sequence {wn (·)} of JΓ and next that ∇wn (·)L2 (Γ) is bounded. Thus, there is a subsequence of {wn (·)} (which we again denote by {wn (·)}) such that it is weakly in H 1 (Γ) convergent to some wΓ and pointwise convergent to it. Therefore lim inf n→∞ JΓ (wn (·)) ≥ JΓ (wΓ (·)). Thus, we obtain that there exists a certain wΓ (·) ∈ U¯ minimizing JΓ .in U¯ and further JΓ (wΓ ) ≥ 0. To prove equality JΓ (wΓ ) = 0 we need to define the dual functional to JΓ by  (2.3) JΓD (w) = (Φ(y, −w(y)) + Φ∗ (y, ∂ν w(y)) Γ

− w(y), ∂ν w(y))dy. 4

It is clear that JΓ (w(·)) = JΓD (−w(·)) for all w(·) ∈ U¯ and so inf JΓ (w(·)) = inf JΓD (w(·)).

¯ w∈U

¯ w∈U

By the duality theory for convex functionals (see [7] and [11]) we have that inf w∈U¯ JΓ (w(·)) = − inf w∈U¯ JΓD (w(·)). Hence we infer that JΓ (wΓ (·)) = 0. Minimizers of JΓ (w) are solutions of the equation −∂ν w(y) = Φw (y, w(y)) on Γ.

(2.4)

˜ 1 (Γ). We shall consider the problem (1.4) in H Remark 2.1. One sees that w ∈ U¯ is a positive solution to (2.4) if and only ˜ 1 (Γ). if x(t, y) = w(t, y)eων,y is a positive solution to (1.4) and x ∈ H The main difficulty of the problem under study is the fact that the Neumann problem does not satisfy Lopatinski condition and therefore, the map from the boundary data in L2 (Γ) into finite energy space is not bounded (unless dimension of Ω is equal to one). We convert the problem (1.4) into variational one. However, the price we pay for that is the type of nonlinearity for the boundary source term Hx − Qx (see (HΓ)). Being inspired by Auchmuty [3] (see also [11]) we formulate the variational principle for problem (1.4). ˜ 1 (Γ) is a positive Proposition 2.2. x¯Γ (y) = wΓ (y)eω0 ν,y , y ∈ Ω, x¯Γ (·) ∈ H solution to (1.4) if and only if wΓ (·) affords a minimum to the functional JΓ defined on U¯ and JΓ (wΓ ) = 0. Proof. If wΓ (·) is a minimizer of JΓ defined on U¯ then JΓ (wΓ ) = 0. This means that we have the Fenchel equality Φ(y, wΓ (y)) + Φ∗ (y, −∂ν wΓ (y)) − wΓ (y), −∂ν wΓ (y) = 0 a.e. in Γ.

(2.5)

which gives −∂ν wΓ (y) ∈ ∂Φ(y, wΓ (y)), where ∂Φ denotes the subdifferential (in the sense of the convex analysis) of the function Φ(y, ·). Since Φ(y, ·) is convex and differentiable we state that −∂ν wΓ (y) = Φx (y, wΓ (y))   = ω 0 |ν|2 wΓ (y) + e−ω0 ν,y Hx (y, wΓ eω0 ν,y ) − Qx (y, wΓ eω0 ν,y ) . On the other hand, for wΓ (y) = x¯Γ (y)e−ω0 ν,y , we have −∂ν wΓ (y) = − < ∇¯ xΓ (y)e−ω0 ν,y − νω 0 x¯Γ (y)e−ω0 ν,y , ν > . 5

Summarizing one obtains < −∇¯ xΓ (y), ν > +ω 0 x¯Γ (y)|ν|2   = ω 0 |ν|2 wΓ (y)eω0 ν,y + Hx (y, wΓ eω0 ν,y ) − Qx (y, wΓ eω0 ν,y ) = ω 0 |ν|2 x¯Γ (y) + (Hx (y, x¯Γ (y)) − Qx (y, x¯Γ (y))) . Finally we get −∂ν x¯Γ (y) = Hx (y, x¯Γ (y)) − Qx (y, x¯Γ (y)). ˜ 1 (Γ) is a solution to (1.4), we can show Conversely, if we have that x¯Γ ∈ H −ω 0 ν,y that for wΓ (y) := x¯Γ (y)e , (2.5) holds, which means that JΓ (wΓ ) = 0, and consequently wΓ is a minimizer of JΓ in U¯ . As a consequence of propositions 2.2 and 2.1 we obtain Corollary 2.1. The problem (1.4) has a positive solution x¯Γ (y) = wΓ (y)eω0 ν,y , ˜ 1 (Γ). y ∈ Ω, x¯Γ (·) ∈ H We introduce the definition of a weak solution to (1.1): ˜ 1 (Γ). By a weak Definition 2.1. (weak solution). Let us fix any x¯Γ (·) ∈ H 2 1 ˜ solution to (1.5), we mean a function x ∈ H (Ω) ∩ H (Γ), x(y) = x¯Γ (y), y ∈ ˜ 1 (Γ) Γ such that for all ϕ ∈ H 2 (Ω) ∩ H    − ∇x∇ϕdy = (Fx − Gx )ϕdy + (Hx − Qx )ϕdy. Ω

Ω

Γ

The main contribution of this paper is a relaxation of assumptions on nolinearities of interior and boundary sources. Assumptions concerning equation (1.1). (As) Let F and G of the variable (y, x) be given. F and G are measurable in y and continuously differentiable and convex with respect to the second variable in R, both satisfy growth conditions F (y, x) ≥ a1 (y)x + b1 (y), G(y, x)) ≥ a2 (y)x + b2 (y), for some a1 , a2 , b1 , b2 ∈ L2 (Ω), for all x ∈ R and y ∈ Ω. Assume that y → F (y, 0) − G(y, 0) is integrable in Ω and F (·, ·) − G(·, ·) is bounded below on Ω × R. Remark 2.2. We would like to stress that F and G are convex. We can treat Gx as nonlinear perturbation of Fx . Usually in literature that perturbation is linear (or earlier sublinear) occurring as a kind of absorption for Fx (see e.g. [8]). 6

Theorem 2.1. (Main theorem). Under (As) there exists x ∈ X := {x + x¯Γ ; x ∈ H01 (Ω) and x ≥ 0 on Ω} such that J (x) = inf J (x) x∈X

and x is a weak nontrivial solution to (1.5). Our variational approach use duality theory which we will build for functional J. Just new duality will allow to weakend the smoothness of J and thus to omit the problem of Sobolev exponent (see Introduction - boundedness on p and q). To this effect let F ∗ be the Fenchel conjugate of F with respect to the second variable. Define a dual to J functional JD : X D → R,as:   JD(q, z) = − Ω F ∗ (y,div q (y) + z(y))dy − 12 Ω |q (y)|2 dy + Ω G∗ (y, z(y))dy + Γ x¯Γ (y) q (y) , ν(y) dy.

(2.6)

where x¯Γ is a positive solution to (1.4), ν is the outer normal to Γ and X D : = {(q, z) ∈ (L2 (Ω))n × L2 (Ω); there exist x, xˆ ∈ X such that x (y) and z(y) = Gx (y, x(y))} . div q (y) − Fx (y, xˆ(y)) = −z(y) with q (y) = ∇ˆ In the next section we show that X D = ∅. Now we can formulate theorem which gives us additional informations on solutions to (1.1) important in classical mechanics. This theorem is absolutely new for problem (1.1). Theorem 2.2. (Variational principle and duality result). Assume (As). Let x ∈ X be such that J (x) = inf J (x) . x∈X

Then there exists (¯ q , z¯) ∈ X D such that for a.e. y ∈ Ω, q¯(y) = ∇¯ x(y),

(2.7)

div q (·) = Fx (·, x (·)) − Gx (·, x¯ (·))

(2.8)

and J(¯ x) = JD (¯ q , z¯) , where z¯(y) = Gx (y, x¯ (y)) . The proofs of the above theorems are given in section below. 7

(2.9)

3

Proof of existence for problem (1.5)

In order to prove Theorem 2.1 we need to investigate first the convex functional  1 (3.1) Jv (x) = ( |∇x(y)|2 + F (y, x(y)) − x(y)Gx (y, v(y)))dy Ω 2 considered on X, for any fixed v ∈ X and the corresponding equation Δx(y) = Fx (y, x(y)) − Gx (y, v(y)) on Ω, x(y) = x¯Γ (y) on Γ.

(3.2)

Following the standard method of convex analysis (see e.g. [7]), we have the following statement concerning problem (3.2). Proposition 3.1. The function xv (y) for y ∈ Ω such that xv (·) ∈ X is a solution to (3.2) if and only if the functional Jv defined on X obtains its minimum at xv (·). Remark 3.1. Notice that since the equation (3.2) is the Euler equation of the functional Jv the function xv (·) is an element of X. Proposition 3.2. Under assumption (As) there exists an element xv (·) where the functional Jv defined on X attains its minimum. Proof. The assumptions (As) ensure that the functional Jv (·) is convex and coercive in H 1 (Ω). Indeed, by convexity of G we have that F (y, x(y)) − x(y)Gx (y, v(y)) ≥ F (y, x(y)) − G(y, x(y)) + G(y, v(y)) −v(y)Gx (y, v(y)).  Thus the functional x → Ω (F (y,  x(y))−x(y)Gx (y, v(y)))dy is bounded below in H 1 (Ω). The functional x → Ω 21 |∇x(y)|2 dy is quadratic and nonnegative, therefore x → Jv (x) is coercive in H 1 (Ω). Moreover, x → Jv (x) is weakly lower semicontinuous in H 1 (Ω). Hence, there exists an element xv (·) ∈ X where the functional Jv obtains its minimum. Remark 3.2. It is clear that X D = ∅. Indeed, by Proposition (3.2), Jv for v = x attains its minimum xˆ ∈ X, which satisfies (3.2). When we put q (y) = ∇ˆ x (y) and z(y) = Gx (y, x(y)) we obtain div q (y) = Fx (y, xˆ(y)) − z(y). 8

Both propositions and remarks are essential to the proof of Theorem 2.1. Proof. (of Theorem 2.1). By assumptions (As) the functional  x → (F (y, x(y)) − G(y, x(y)))dy Ω

is bounded below in X ⊂ H 1 (Ω). Hence the mapping x → J(x) is bounded below in H 1 (Ω). Moreover it is weakly lower semicontinuous in H 1 (Ω). Let us observe, similarly as in the proof of Proposition 3.2, that J(x) → ∞ when x(·)H 1 (Ω) → ∞. Hence, we infer that ∇xn (·)L2 (Ω) is bounded for minimizing sequence {xn (·)} of J and next that xn (·)L2 (Ω) is bounded. Thus, there is a subsequence of {xn (·)} (which we again denote by {xn (·)}) such that it is weakly, in H 1 (Ω), convergent to some x¯ and pointwise convergent to it. Therefore, we get that lim inf n→∞ J(xn ) ≥ J(¯ x). It clear that that x¯ ∈ X. Now let us investigate the dual functional JD (q, z) to J on the dual set consisting of dual pairs. By Remark 3.2 for x ¯ there exits xˆ ∈ X such that the following assertion holds div qˆ (y) − Fx (y, xˆ (y)) = −¯ z (y) ,

(3.3)

on Ω, for (ˆ q , z) ∈ X D given by qˆ(y) = ∇ˆ x(y),

(3.4)

z¯(y) = Gx (y, x¯ (y)) .

(3.5)

By the definitions of J, JD , relations (3.3), (3.5) and the Fenchel-Young inequality we have that   1 |∇¯ x(y)|2 − G (y, x¯ (y)) + F (y, x¯ (y)) dy J(¯ x Ω 2   ) = x(y)|2 dy + Ω G∗ (y, z (y)) dy − ¯ x (y) , z (y) dy + F (y, x¯ (y)) dy = Ω 21 |∇¯ Ω Ω  x (y) , qˆ (y) dy− 12 Ω |ˆ q (t, y)|2 dy  ≥  Ω ∇¯ + Ω G∗ (y, z (y)) dy − Ω ¯ x (y)  dy  , z (y) dy + Ω F (y, x¯ (y)) = − Ω ¯ x (y) , div qˆ (y) dy + Γ x¯ (y) ˆ q (y) ,ν(y) dy − 12 Ω |ˆ q (t, y)|2 dy + Ω G∗ (y, z (y)) dy − Ω ¯ x (y) , z (y)dy + Ω F (y, x¯ (y)) dy q (y)|2 dy ≥ − Ω F ∗ (y, div qˆ (y)+ z¯(y)) dy − 12 Ω |ˆ q (y) , ν(y) dy = JD (ˆ q , z¯) . + Ω G∗ (y, z¯ (y))dy + Γ x¯Γ (y) ˆ Hence we obtain that q , z¯) . J(ˆ x) ≥ J(¯ x) ≥ JD (ˆ 9

(3.6)

Next observe that    x (y) , qˆ (y) dy + ˆ x (y) , z¯ (y) dy − ˆ x(y), div qˆ (y) + z¯ (y) dy − ∇ˆ Ω Ω Ω   = − xˆ (y) ˆ q (y) , ν(y) dy + xˆ (y) div qˆ (y) dy + ˆ x (y) , z¯ (y) dy Ω Ω Γ − ˆ x(y), div qˆ (y) + z¯ (y) dy Ω   = − xˆ (y) ˆ q (y) , ν(y) dy = − x¯Γ (y) ˆ q (y) , ν(y) dy Γ

Γ

and further x) ≤ J(ˆ x) = inf J (x) = J(¯

x∈X

  = Ω



 1 2 x (y) , qˆ (y) dy |∇ˆ x(y)| − ∇ˆ 2

+ Ω + Ω

(ˆ x (y) , z¯ (y) − G (y, xˆ (y))) dy (F (y, xˆ (y)) − ˆ x(y), div qˆ (y) + z¯ (y)) dy

x¯Γ (y) ˆ q (y) , ν(y) dy  1 2 ≤ − |ˆ q (y)| dy + G∗ (y, z¯ (y))dy 2 Ω Ω   ∗ − F (y, qˆ (y) + z¯ (y)) dy + x¯Γ (y) ˆ q (y) , ν(y) dy = JD (ˆ q , z¯). +

Γ

Γ

Γ

Hence q , z¯) . J(¯ x) ≤ J(ˆ x) ≤ JD (ˆ

(3.7)

Thus, we get equality J(ˆ x) = JD (ˆ q , z¯) , namely     1 1  (y)) + G∗ (y, z(y))) dy | q (y)|2 + |∇ x(y)|2 dy − (G (y, x 2 2 Ω Ω   ∗ + F (y, x  (y)) + F (y, div q (y) + z(y)) dy Ω  x¯Γ (y) ˆ q (y) , ν(y) dy. = Γ

This equality together with (3.3) and (3.4) imply that 10



 ∇ x (y) , qˆ (y) dy +

Ω = Γ

Ω

  x(y), div qˆ (y) + z¯ (y) dy −

Ω

(G (y, x  (y)) + G∗ (y, z(y))) dy

x¯Γ (y) ˆ q (y) , ν(y) dy

which gives, by the fact that x |Γ = x¯|Γ , 

 x¯Γ (y)  q (y) , ν(y) dy −

Γ

−  = Γ

Ω

Ω

 xˆ(y) div qˆ (y) dy +

Ω

ˆ x(y), div qˆ (y) + z¯ (y) dy

G (y, x  (y)) + G∗ (y, z(y))dy

x¯Γ (y)  q (y) , ν(y) dy

and finally  Ω

G (y, x  (y)) + G∗ (y, z(y)) − ˆ x(y), z¯ (y) dy = 0.

Therefore, by standard convexity arguments we get the equality z¯(y) = Gx (y, xˆ (y)) . From (3.6) and (3.7) we infer that J(¯ x) = JD (ˆ q , z¯) . This relation implies 

2  

1 1

q (y) + |∇x(y)|2 dy + (F (y, x (y)) + F ∗ (y, div q (y) + z(y))) dy

2 2 Ω Ω   ∗ q (y) , ν(y) dy − (G (y, x (y)) + G (y, z(y))) dy = x¯Γ (y)  Ω

Γ

and further, by (3.5), we have 

2  

1 1

q (y) + |∇x(y)|2 dy + (F (y, x (y)) + F ∗ (y, div q (y) + z(y))) dy

2 2 Ω Ω   < x (y) , z(y) > dy − x¯Γ (y)  q (y) , ν(y) dy = 0. − Ω

Γ

11

Finally the following chain of equalities takes place 

2  

1 1 2 ∗

0 =

2 q (y) + 2 |∇x(y)| dy + (F (y, x (y)) + F (y, div q (y) + z(y))) dy Ω Ω    < x (y) , div qˆ (y) + z(y) > dy + x(y) div qˆ (y) dy − x¯Γ (y)  q (y) , ν(y) dy − Ω Ω Γ 

2  

1 1 2 ∗

=

2 q (y) + 2 |∇x(y)| dy + (F (y, x (y)) + F (y, div q (y) + z(y))) dy Ω Ω   < x (y) , div qˆ (y) + z(y) > dy + x¯Γ (y)  q (y) , ν(y) dy − Ω Γ   − ∇x(y)ˆ q (y) dy − x¯Γ (y)  q (y) , ν(y) dy Ω Γ 

2 

1 1

q (y) + |∇x(y)|2 − ∇x(y)ˆ = q (y) dy

2 2 Ω  + (F (y, x (y)) + F ∗ (y, div q (y) + z(y)) − x(y)(div qˆ (y) + z(y))) dy Ω

Since each of these terms is nonnegative, we get

2 

1

q (y) + 1 |∇x(y)|2 − ∇x(y)ˆ q (y) dy = 0,

2 Ω 2  F (y, x (y)) + F ∗ (y, div q (y) + z(y)) − x(y)(div qˆ (y) + z(y))dy = 0 Ω

and consequently qˆ (y) = ∇¯ x (y) and div q (y) + z(y) = Fx (y, x¯ (y)). Thus, by (3.3), x¯ is a solution to (1.5) and so the assertions of the theorem are satisfied.

References [1] A. Ambrosetti, J. Garcia-Azorero, I. Peral, Existence and multiplicity results for some nonlinear elliptic equations: A survey, Rend. Mat. Appl. (7) 20 (2000) 167–198. [2] D.G. Aronson, H.F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math. 30 (1978) 33–76. 12

[3] G. Auchmuty, Variational principles for finite dimensional initial value problems, Contemporary Math. 426, 45-56, 2007. [4] M.Chipot, M. Chledik, M. Fila, I. Shafrir, Existence of positive solun tions of a semilinear elliptic equation in R+ with nonlinear boundary condition, J. Math. Anal. Appl. 223 (1998) 429-471 [5] P.G. Ciarlet, Mathematical Elasticity, vol. I. Three-Dimensional Elasticity, North-Holland, Amsterdam, 1988. [6] J.I. Diaz, Nonlinear Partial Differential Equations and Free Boundaries, vol. I. Elliptic Equations, Res. Notes Math., vol. 106, Pitman, Boston, MA, 1985. [7] I. Ekeland, R. Temam, Convex analysis and variational problems, Studies Math. Appl. I, 1976, Noth-Holland Publ. Comp. [8] J. Garcia-Azorero, I. Peral, J.D. Rossi, A convex–concave problem with a nonlinear boundary condition, J. Differential Equations 198 (2004) 91–128. [9] J.L. Kazdan, Prescribing the Curvature of a Riemannian Manifold, CBMS Reg. Conf. Ser. Math., vol. 57, Amer. Math. Soc., Providence, RI, 1985. [10] O.A. Ladyzhenskaya, N.N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968 [11] A. Nowakowski, Nonlinear parabolic equations associated with subdifferential operators, periodic problems, Bull. Polish Acad. Sc. Math. 16, (1998), 615-621. [12] A. Nowakowski, Solvability and stability of a semilinear wave equation with nonlinear boundary conditions. Nonlinear Anal. 73 (2010), no. 6, 1495–1514. [13] A. Orpel, Superlinear Dirichlet problems, Nonlinear Anal. Series A 56 (2004) (06), 937-957. [14] A. Orpel, On the existence of positive solutions and their continuous dependence on functional parameters for some class of elliptic problems, J. Differential Eqns 204 (2004), 247-264. [15] A. Orpel, Nonlinear BVPS with functional parameters, J. Differential Eqns 246 (2009), 1500-1522. 13

´ [16] M.C. P´elissier, L. Reynaud, Etude d’un modˇcle math´ematique d’´ecoulement de glacier, C. R. Acad. Sci. Paris S´er. A 279 (1974) 531– 534. [17] Tsung-fang Wu, On semilinear elliptic equations involving concaveconvex nonlinearities and sing-changing weight function, J. Math. Anal. Appl. 318 (2006) 253-270 [18] Tsung-fang Wu, Multiple positive solutions for a class of concave-convex elliptic problems in RN involving a sign-changing weight, J. Functional Analysis 258 (2010) 99-131 [19] Tsung-fang Wu, Multiple positive solutions of a nonlinear boundary value problem involving a sign-changing weight, Nonlinear Analysis 74 (2011) 4223-4233 [20] Tsung-fang Wu, Existence and multiplicity of positive solutions for a class of nonlinear boundary value problems, J. Differential Equations, 252 (2012) 3403–3435. [21] M. Warma, Regulariry and well-posedness of some quasi-linear elliptic and parabolic problems with nonlinear general Wentzell boundary conditions on nonsmooth domains, Nonlinear Analysis 75 (2012) 5561-5588

14