Journal of Differential Equations

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1 J. Differential Equations 248 (2) Contents lists available at ScienceDirect Journal of Differential Equations Existence of solutions to nonlinear, subcritical higher order elliptic Dirichlet problems Wolfgang Reichel a, Tobias Weth b, a Institut für Analysis, Universität Karlsruhe, D-7628 Karlsruhe, Germany b Institut für Mathemati, Goethe-Universität Franfurt, D-654 Franfurt, Germany article info abstract Article history: Received 2 June 29 Revised September 29 Available online 26 September 29 MSC: primary 35J4 secondary 35B45 Keywords: Higher order equation Existence Topological degree Liouville theorems We consider the -th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain Ω R N with Dirichlet boundary conditions on Ω. The operator L is a uniformly elliptic linear operator of order whose principle part is of the form ( N i, j= a ij(x) 2 x i x j ) m. We assume that f is superlinear at the origin and satisfies lim s f (x,s) s q = h(x), lim s f (x,s) s q = (x), where h, C(Ω) are positive functions and q > is subcritical. By combining degree theory with new and recently established a priori estimates, we prove the existence of a nontrivial solution. 29 Elsevier Inc. All rights reserved.. Introduction Let Ω R N be a bounded smooth domain. On Ω we consider the uniformly elliptic operator ( ) m N 2 L = a ij (x) + x i x j i, j= b α (x)d α + b (x) (.) α with coefficients b α L (Ω) and a ij C 2,α (Ω) such that there exists a constant λ> with λ ξ 2 N i, j= a ij(x)ξ i ξ j λ ξ 2 for all ξ R N, x Ω. We are interested in nontrivial solutions of * Corresponding author. addresses: wolfgang.reichel@math.uni-arlsruhe.de (W. Reichel), weth@math.uni-franfurt.de (T. Weth) /$ see front matter 29 Elsevier Inc. All rights reserved. doi:.6/j.jde

2 W. Reichel, T. Weth / J. Differential Equations 248 (2) the semilinear boundary value problem Lu = f (x, u) in Ω, u = ( ) m ν u = = u = on Ω, (.2) ν where ν is the unit exterior normal on Ω and f is a nonlinearity which is to be specified later. The main difficulties in proving existence results for this problem are the following:. (.2) has no variational structure (in general), so critical point theorems do not apply. 2. The operator L does not satisfy the maximum principle (in general) unless m =. In the second order case, the maximum principle is a basic requirement to translate (.2) into a fixed point problem for an order preserving operator, which in turn maes it possible to use topological degree (or fixed point) theory in cones or invariant order intervals given by a pair of sub- and supersolutions. 3. In the case m >, a priori bounds for (certain classes of) solutions are harder to obtain than in the second order case, which maes it difficult to find solutions to (.2) via global bifurcation theory. In a recent paper, we have proved a priori bounds for solutions of (.2) in the case of superlinear nonlinearities f (x, u) with subcritical growth satisfying an asymptotic condition. More precisely, we assumed: (H) f : Ω R R is uniformly continuous in bounded subsets of Ω R and there exist q > if N and < q < N+ if N > and two positive, continuous functions, h : Ω (, ) N such that f (x, s) f (x, s) lim = h(x), s + s q lim = (x) s s q uniformly with respect to x Ω. Theorem. (See Reichel and Weth [].) Suppose Ω R N is a bounded domain with Ω C.If f : Ω R R satisfies (H) then there exists a constant C > depending only on the data a ij, b α,ω,n, q, h, such that u C for every strong solution u of (.2) which belongs to W,p (Ω) W m,p (Ω) for every p [, ). Although in [] we did not state explicitly that we were considering strong solutions in this class, this is precisely what we used in our proof. Theorem can be seen as a first step towards existence results via degree theory. In order to state the main theorem of the present paper, we introduce additional assumptions on f. (H2) f (x, s) = o(s) as s uniformlyinx Ω. (H3) For some p, v = is the unique solution of Lv = inw,p (Ω) W m,p (Ω). Theorem 2. Suppose Ω R N is a bounded domain with Ω C.Letm N and assume f : Ω R R satisfies (H), (H2) and (H3). Then(.2) has a nontrivial solution u W,p (Ω) W m,p (Ω) for every p [, ). We note that (H3) can be verified in many examples. In particular, if L is as in (.), then L + γ satisfies (H3) if γ > is sufficiently large and if additionally b α C α m (Ω) for m < α <. This is true since the smoothness of the coefficients allows to write L in divergence form and hence the quadratic form associated with L + γ is coercive due to Garding s inequality, cf. Renardy and Rogers [2], if γ > issufficientlylarge. As an intermediate step in the proof of Theorem 2, we need to complement Theorem with the following a priori estimate for a parameter, which might be of independent interest.

3 868 W. Reichel, T. Weth / J. Differential Equations 248 (2) Theorem 3. Suppose Ω R N is a bounded domain with Ω C.If f : Ω R R satisfies (H), then there exists a value Λ = Λ(Ω, L, f ) such that for λ Λ the problem Lu = f (x, u) + λ in Ω, u = ( ) m ν u = = u = on Ω (.3) ν has no solution which belongs to W,p (Ω) W m,p (Ω) for all p [, ). Due to the lac of the maximum principle for higher order equations, we have no sign information on the solution provided by Theorem 2. By the same reason, it is important that Theorems and 3 hold with no restriction on the sign of the solutions. We also point out that we mae no assumption concerning the shape of the domain. We recall that the proof of Theorem is carried out by a rescaling method in the spirit of the seminal wor of Gidas and Spruc [5] (but without a priori information on the sign of the solutions) and by investigating the corresponding limit problems. In particular, the following Liouville type theorems are used. Theorem 4. (See Wei and Xu [4].) Let m N and assume that q > if N and < q < N+ if N >. N If u is a classical non-negative solution of then u. ( ) m u = u q in R N, (.4) Theorem 5. Let m N and assume that q > if N and < q N+ N non-negative solution of if N >. If u is a classical ( ) m u = u q in R N +, u = x u = = m x m u = on R N + (.5) then u. Here and in the following, we set R N + := {x RN : x > }. Theorem 5 is a slight generalization of Theorem 4 in our recent paper []. More precisely, it is assumed in [] that u is bounded, but an easy argument based on the doubling lemma of Poláči, Quittner and Souplet [] shows that this additional assumption can be removed. See Section 4 below for details. Theorems 4 and 5 will also be used in the proof of Theorem 3. However, the rescaling argument is somewhat more involved since both λ and the L -norm of the solutions need to be controlled. Here various cases have to be distinguished, and additional limit problems arise. Finally we comment on some previous wor related to Theorem 2. If L = ( ) m is the polyharmonic operator, then (.2) has a variational structure. In this case existence and multiplicity results for solutions of (.2) have been obtained under additional assumptions on f via critical point theory and related techniques, see e.g. [3,4,6,5] and the references therein. The approach via a priori estimates and degree theory was taen by Soranzo [3] and Oswald [9], but only in the special case where Ω is a ball. More precisely, in [9,3] the authors first prove a priori estimates for radial positive solutions before proving existence results within this class of functions. An existence result for more general operators L was obtained in [7] for a different class of nonlinearities which gives rise to coercive nonlinear operators. See also the references in [7] for earlier results in this direction. The paper is organised as follows. Section 2 is devoted to the proof of Theorem 3, while Theorem 2 is proved in Section 3. Finally, in Section 4 we show how to remove the boundedness assumption which was present in the original formulation of Theorem 5.

4 W. Reichel, T. Weth / J. Differential Equations 248 (2) Nonexistence for the parameter dependent problem The proof of Theorem 3 uses L p W,p estimates for linear problems Lu = g(x) in Ω, (2.) u = ( ) m ν u = = u = on Ω. (2.2) ν In particular, we need the following result due to Agmon, Douglis and Nirenberg []. From now on we always assume that L is an operator of type (.). Theorem 6. Let Ω R N be a bounded domain with Ω C, and let L satisfy (H3), i.e., for some p, v = is the unique solution of Lv = in W,p (Ω) W m,p (Ω). Then for any r [p, ), L has a bounded inverse L : L r (Ω) W,r (Ω) W m,r (Ω). Since this is not stated explicitly in [], we briefly note how it follows from results stated there. So let r p, and let v W,r (Ω) W m,r (Ω) solve Lv =. Then also v W,p (Ω) W m,p (Ω) and therefore v = by assumption. Hence the uniqueness holds also for r instead of p. By the remars preceding Theorem 2.7 in [], it now follows that L : W,r (Ω) W m,r (Ω) L r (Ω) is a bijection. Since L is bounded by the boundedness of the coefficients of L, the open mapping theorem implies that L is bounded as well, as claimed. We will also be using the following al a priori estimates for solutions of (2.). For a standard proof see []. Corollary 7. Let Ω be a ball {x R N : x < R} or a half-ball {x R N : x < R, x > }. Letm N, a ij C 2 (Ω),b α L (Ω), g L p (Ω) for some p (, ). Suppose u W,p (Ω) satisfies (2.) (i) either on the ball, (ii) or on the half-ball together with the boundary conditions u = u x = = m u = on {x R N : x m x < R, x = }. Then there exists a constant C > depending only on a ij C 2, b α,λ,ω,n, p,m, the modulus of continuity of a ij and R such that for any σ (, ) u W,p (Ω Bσ R) C ( ) g L p ( σ ) (Ω) + u L p (Ω). It is sometimes convenient to rewrite the operator L in the form L = ( ) m a α (x)d α + c α (x)d α. α = α Here a α (x) = I M α a i i 2 (x) a i3 i 4 (x) a i i (x), where M α is the set of all vectors I = (i,...,i ) {,...,N} satisfying #{ j: i j = l} =α l for l =,...,N. Note that the functions a α are continuous on Ω and c α L (Ω). Finally, the following lemma is used a number of times in the subsequent proof of Theorem 3. A version of part (a) of the lemma already appeared in Reichel and Weth [] and similar arguments have been used by Wei and Xu in [4].

5 87 W. Reichel, T. Weth / J. Differential Equations 248 (2) Lemma 8. (a) Let v be a strong W, (R N ) C (R N ) solution of ( ) m v g(v) in R N such that D α v is bounded for all multi-indices α with α.ifg: R [, ) is convex and non-negative with g(s)> for s < then either v > or v. (b) Let v be a strong W, (R N + ) C (R+ N ) solution of ( )m v in R+ N.Then( )m v is unbounded. Proof. Part (a): Let v l := ( ) l v for l =,...,m and set v = v. Thenwehave v = v, v = v 2,..., v m g(v ) in R N. First we show that v l inr N for l =,...,m. Assume that there exists l {,...,m } and x R N with v l (x )<butv j inr N for j = l +,...,m. We may assume w.l.o.g. that x =. If we define for a function w W 2, (RN ) C (R N ) spherical averages w(x) = r N B r () w(y) dσ y, r = x then the radial functions v, v,..., v m satisfy v = v, v = v 2,..., v m g( v ) in R N, where we have used Jensen s inequality and the convexity of g. Sincev l ()<wealsohave v l ()<. Moreover v l (r) = B () = B () ( v l )(rξ) ξ dσ ξ ( v l )(rξ)rdξ { = B () v l+(rξ)rdξ if l < m, ω N B () g(v(rξ))rdξ if l = m. (2.3) Since the right-hand side is non-positive in both cases we obtain v l (r) v l () <. Integrating the inequality v l = v l v l ()> we obtain r N v l (r) rn N v l(), i.e., v l (r) r N v l(). The unboundedness of v yields a contradiction. l Next we show that v = v. Assume that v (x )<andw.l.o.g.x =. Since v = v we see that v (r) and we define α := lim r v (r)<. Thus g( v (r)) 2 g(α)>forr r.as in (2.3) we find v m (r) B () = r r N r r/2 ( g v (rξ) ) rdξ = ( r N g v (η) ) dη B r () B s () s N r N g(α) 2 ds ( g v (η) ) r dσ η ds s N r N g( v (s) ) ds

6 W. Reichel, T. Weth / J. Differential Equations 248 (2) if r 2r. Since the last term converges to as r we obtain a contradiction to the boundedness of v m. Finally the alternative v > orv follows since v bythefirstpartofthe proof. Part (b): Let w := ( ) m v so that w is a strong W 2, (RN + ) C (R+ N ) solution of w. Let w(r; X) := r N B r (X) w(y) dσ y = B () for X R+ N and < r < X.ForfixedX R+ N the function w satisfies w (r) = B () ( w)(x + rξ) ξ dσ ξ = B () w(x + rξ)dσ ξ ( w)(x + rξ)rdξ r for < r < X.Hence, w(r) w() r2 2N. Letting X and r tend to infinity we find that w cannot stay bounded. We now have all the tools to complete the N Proof of Theorem 3. Assume for contradiction that there exists a sequence of pairs (u,λ ) of solutions of (.3) with λ for.letm := u. By considering a suitable subsequence we can assume that there exists x Ω such that either M = u (x ) for all N or M = u (x ) for all N. Case : u stays bounded. W.l.o.g. we can assume Ω and B δ () Ω for some δ>. Set v (x) := u (λ / x). Thenv satisfies where L := ( ) m L v (x) = λ f ( λ / x, v ) + in Bλ / δ () α = a α ( λ / x ) D α + λ α α ( / ) c α λ x D α. By standard interior regularity on the ball B R () for any R > and any p there exists a constant C p,r > such that v W,p (B R ()) C p,r uniformly in. For p sufficiently large and by passing to a subsequence (again denoted v ) we see that v v in C,α (R N ) and in W m,p (RN ) as for every R >, where v C,α (R N ) W m,p (RN ) is a bounded wea (and hence classical) solution of Lv = inr N, where L = ( ) ( ) m N m a α ()D α 2 = a ij (). x i x j α = By a linear change of variables we may assume that v is a bounded, classical, entire solution of ( ) m v = inr N. Lemma 8(b) shows that we have reached a contradiction. i, j=

7 872 W. Reichel, T. Weth / J. Differential Equations 248 (2) Case 2: u is unbounded. For this case we need to discuss various sub-cases depending on the growth of the numbers q ρ := M dist(x, Ω), N. Passing to a subsequence, we may assume that either ρ or ρ ρ as. Case 2.: ρ as. Again we need to distinguish two further possibilities. Let λ := λ /M q. Case 2..a: λ is bounded, i.e., up to selecting a subsequence, λ λ. Then we set v (y) := q u M (M y + x ) so that v = and either v () = forall N (positive blow-up) or v () = forall N (negative blow-up). Moreover we can assume that x x Ω. The functions v are well defined on the sequence of balls B ρ () as and they satisfy where this time and L v (y) = ( q M q f M y + x, M v (y) ) + λ for y B ρ (), }{{} =: f (y) ā α (y) := a α L := ( ) m α = ā α (y)dα + ( q ) M y + x, c α (y) := M α α (q )( ) c α (y)dα ( q ) c α M y + x. By our assumption (H) on the nonlinearity f (x, s) we have that f L (Bρ ()) is bounded in. Note that the ellipticity constant, the L -norm of the coefficients of L and the moduli of continuity of ā α are not larger then the one for the operator L. By applying Corollary 7 on the ball B R() for any R > and any p there exists a constant C p,r > such that v W,p (B R ()) C p,r uniformly in. For large enough p we may extract a subsequence (again denoted v ) such that v v in C,α (B R ()) as for every R >, where v C,α (R N ) is bounded with v = = ±v(). Taing yet another subsequence we may assume that f F in L (K ) as for every compact set K R N. Also we see that { h( x)v(y) q F (y) = ( x) v(y) q if v(y)>, if v(y)<, (2.4) because, e.g., if v(y)> then there exists such that v (y)>for and hence M v (y) as. Therefore (H) implies that f (y) h( x)v(y) q as, and a similar pointwise convergence holds at points where v(y)<. Finally, note that the pointwise convergence of f on the set Z + ={y R N : v(y)>} and Z ={y R N : v(y)<} determine due to the dominated convergence theorem the wea -limit F of f on the set Z + Z.Since c α (y) and ā α (y) a α( x) as and since, for any fixed p (, ), we may assume that v v in W,p (R N ) we find that v is

8 W. Reichel, T. Weth / J. Differential Equations 248 (2) abounded,weaw m,p (RN )-solution of Lv = F + λ in R N, where L = ( ) m α = ( a α ( x)d α = ) m N 2 a ij ( x). (2.5) y i y j Since F L (R N ) we get that v W,p (R N ) C,α (R N ) is a bounded, strong solution of (2.5). Because D v = a.e.ontheset{y R N : v(y) = } one finds that v is a strong solution of h( x)v(y) q + λ if v(y)>, Lv = if v(y) =, ( x) v(y) q + λ if v(y)< i, j= (2.6) in R N. Note that the right-hand side of (2.6) is larger or equal to g(v), where the function g is defined by { h( x)s q if s, g(s) := ( x) s q if s. (2.7) Since the function g is convex we can apply Lemma 8(a) and obtain v >. Thus v is a classical C,α (R N ) solution, and by a linear change of variables we may assume that v solves ( ) m v = h( x)v q + λ in R N, v() =. Clearly v and all its derivatives of order are bounded. If λ = then Theorem 4 tells us that this is impossible. And if λ > then Lemma 8(b) provides a contradiction. This finishes the proof in this case. Case 2..b: λ is unbounded, i.e., up to a subsequence λ.nowweset v (y) := ( q u M λ / ) y + x. M The functions v are again well defined on a sequence of expanding balls and satisfy where this time L v (y) = λ M q L := ( ) m ( q f M λ / y + x, M v (y) ) +, (2.8) } {{ } =: f (y) α = ā α (y)dα + α c α (y)dα with ā α (y) := a α ( q M λ / ) y + x, c α (y) := M α (q )( ) λ α ( q c α M λ / y + x ).

9 874 W. Reichel, T. Weth / J. Differential Equations 248 (2) Arguing lie before we arrive at the situation that v v in W,p (R N ) as, where, modulo a linear change of variables, v C,α (R N ) W,p (R N ) is a bounded strong (and hence classical) solution of ( ) m v = inr N. A contradiction is reached via Lemma 8(b). Case 2.2: ρ ρ. Then, modulo a subsequence, x x Ω as, and after translation we may assume that x =. By flattening the boundary through a al change of coordinates we may assume that near x = the boundary is contained in the hyperplane x =, and that x > corresponds to points inside Ω. Since Ω is ally a C -manifold, this change of coordinates transforms the operator L into a similar operator which satisfies the same hypotheses as L. For simplicity we call the transformed variables x, the transformed domain Ω and the transformed operator L. Note that dist(x, Ω)= x, for sufficiently large. By passing to a subsequence we may assume that this is true for every, so that ρ = M defining λ := λ /M q. q x,. As before we need to distinguish two further possibilities by Case 2.2.a: Up to selecting a subsequence assume that λ λ. In this case we define the function v, the coefficients ā α, c α and the operator L as in Case 2..a, where v is now defined on the set {y R N : M defining q y + x Ω} which contains B ρ (). Then we mae another change of coordinates, w (z) := v (z ρ e ), ã α (z) := ā α (z ρ e ), c α (z) := c α (z ρ e ), where e = (,,...,) R N is the first coordinate vector, and liewise the operator L. Note that w, ã α, c α and the operator L are defined on the set Ω := { z R N q : M z + (, x,2,...,x,n ) Ω }, and that w (ρ e ) =±. We now fix R > such that ρ e B + R for all N where B+ R := B R() R+ N. By our assumptions on the boundary Ω near x, wehaveb + R Ω for sufficiently large. Moreover, w satisfies L w (z) = f (z) + λ in B + R, where f (z) := M q ( q f M z + (, x,2,...,x,n ), M w (z) ), together with Dirichlet boundary conditions on {z R N : z < R, z = }. Hence we may apply Corollary 7 on the half-ball B + R and find that for any p there exists a constant C p,r > such that w W,p (B + R ) C p,r uniformly in. By the Sobolev embedding theorem, this implies that w is bounded on B + R Combining this with the fact that independently of. = w (ρ e ) w }{{} () ρ }{{} w L (B + R ) for all N, =± =

10 W. Reichel, T. Weth / J. Differential Equations 248 (2) we deduce that ρ = lim ρ >. As in Case 2..a we can now extract convergent subsequences w w in C,α (R N + ) and f F in L (R+ N ) as, where F, is determined in the same way as in Case 2..a. This time, w is a bounded, strong W,p Lw = F + λ in R N +, z w = = m z m (R+ N ) C,α (R N + )-solution of w = on R N + with L as in (2.5). By a linear change of variables we may assume that w solves ( ) m w = g(w) + λ in R N +, z w = = m z m w = on R N +, (2.9) where g is defined as in (2.7) of Case 2..a. The representation formula of Theorem 9 in [] applies and shows that w is non-negative, so that g(w(z)) = h( x)w(z) q.moreover, w(ρe ) = lim w (ρ e ) =, so that w is a positive, bounded and classical solution C -solution of ( ) m w = h( x)w q + λ in R+ N with Dirichlet boundary conditions on R+ N. A contradiction is reached by either Theorem 5 if λ = or Lemma 8(b) if λ >. Case 2.2.b: Up to selecting a subsequence λ. In this case we need to define v as in Case 2..b, which is now well defined on the set Σ := { y R N q : M λ / y + x Ω } and satisfies (2.8) on this set. By our assumptions on the boundary of Ω near x, wehave q dist(, Σ ) = M λ / x, = ρ λ / =: τ for all. Passing to a subsequence, we may assume that either τ or τ τ as.intheformer case we come to a contradiction as in Case 2..b, since then v is well defined and bounded on a sequence of expanding balls. In the latter case we proceed completely analogously as in Case 2.2.a with ρ replaced by τ for every. The only difference is that in this case, modulo a linear change of variables, we end up with a bounded strong classical solution of ( ) m v = inr+ N. Again a contradiction is reached via Lemma 8(b). Since in all cases we obtained a contradiction, the proof of Theorem 3 is finished. 3. Proof of the existence result In this section we complete the proof of Theorem 2. So let Ω R N be a bounded domain with Ω C, and suppose that (H3) is satisfied for L and (H) is satisfied for f. By Theorem 6, we may fix p > N such that L : W,p (Ω) W m,p (Ω) L p (Ω) has a bounded inverse L : L p (Ω) W,p (Ω) W m,p (Ω). Since W,p (Ω) W m,p (Ω) is compactly embedded into C(Ω), finding a solution u W,p (Ω) W m,p (Ω) of (.3) is equivalent to finding a solution of the equation [Id K λ ](u) =, u C(Ω), (3.)

11 876 W. Reichel, T. Weth / J. Differential Equations 248 (2) where for λ R the nonlinear operator K λ : C(Ω) C(Ω) is defined by K λ (u) = L w with w(x) = f ( x, u(x) ) + λ. Here we may regard L : C(Ω) L p (Ω) W,p (Ω) W m,p (Ω) as a bounded linear operator. Moreover, since the embedding W,p (Ω) C(Ω) is compact, K λ is also compact for every λ R. Next we note that every solution of (3.) belongs to W,r (Ω) W m,r (Ω) for all r [, ), sincel also maps L r (Ω) onto W,r (Ω) W m,r (Ω) for r p by Theorem 6. As a consequence, Theorems and 3 also apply to solutions of (3.), i.e., there exist Λ>andK > such that (3.) has no solution for λ Λ, and any solution u of (3.) with λ [,Λ] satisfies u K. Consequently, we find that [Id K λ ](u) if(u,λ) ( B 2K () {Λ} ) ( B 2K () [,Λ] ), (3.2) where B 2K () C(Ω) denotes the 2K -ball with respect to. The homotopy invariance of the Leray Schauder degree and (3.2) imply deg ( Id K, B 2K (), ) = deg ( Id K Λ, B 2K (), ) =. For these and other properties of the Leray Schauder degree, we refer the reader to [2, Chapter 2.8] or [8, Chapter 2]. Since K (u) = o( u ) as u by assumption (H2), there exists a small ε > such that for every t [, ] the operator Id tk : C(Ω) C(Ω) does not attain the value on B ε () C(Ω). By the homotopy invariance of the degree, we therefore have deg(id K, B ɛ (), ) = deg(id, B ɛ (), ) =. The additivity property of the degree now implies that deg ( Id K, B 2K () \ B ɛ (), ), hence there exists u B 2K () \ B ɛ () such that u K (u) =. Therefore u is a nontrivial solution of (.2). 4. Proof of Theorem 5 In this section we show how Theorem 5 can be deduced from [, Theorem 4] with the help of the doubling lemma of Poláči, Quittner and Souplet []. We recall the following simple special case of this useful lemma. Lemma 9. (Cf. [].) Let (X, d) be a complete metric space and M : X (, ) be bounded on compact subsets of X. Then for any y Xandany> there exists x Xsuchthat M(x) M(y) and M(z) 2M(x) for all z B /M(x) (x). This follows by taing D = Σ = X in [, Lemma 5.], so that Γ := Σ \ D = and therefore dist(y,γ)= for all y X. We now may complete the proof of Theorem 5. Suppose by contradiction that there exists an unbounded solution u of (.5), and put M := u q : R+ N R. Then there exists a sequence (y ) R+ N such that M(y ) as. By Lemma 9, applied within the underlying complete metric space X := R+ N, there exists another sequence (x ) R+ N such that M(x ) M(y ) and M(z) 2M(x ) for all z B /M(x )(x ) R N +.

12 W. Reichel, T. Weth / J. Differential Equations 248 (2) We then define ρ := x, M(x ), the affine half-space H := {ζ R N : ζ > ρ } and the function for N. Thenũ is a non-negative solution of such that ũ : H R, ũ (ζ ) = u(x + ζ M(x ) ) u(x ) ( ) m ũ = ũ q in H, ũ = ( ) m (4.) ũ = = ũ = on H ζ ζ ũ () = and ũ (ζ ) 2 q for all ζ H B (). We may now pass to a subsequence and distinguish two cases: Case : ρ as. In this case Corollary 7(i) implies that the sequence (ũ ) is ally W,p -bounded on R N, therefore we can extract a convergent subsequence ũ ũ in C,α (R N ), where ũ is a solution of (.4) satisfying u() =. By Theorem 4 we obtain a contradiction. Case 2: ρ ρ as. In the case we perform a further change of coordinates, defining v (z) := ũ (z ρ e ) for z R N +, where again e = (,,...,) R N is the first coordinate vector. Then v is a non-negative solution of ( ) m v = v q in R + N, v = ( ) m v = = v = on R+ N z z, while v (ρ e ) = and v (z) 2 q for all z B (ρ e ) R N +. Using now Corollary 7(ii), we deduce that the sequence (v ) is ally W,p -bounded in R+ N.In particular v remains bounded independently of in a neighborhood of the origin, which in view of the boundary conditions implies that ρ = lim ρ >. We can therefore extract a convergent subsequence v v in C,α (R N + ), where v is a solution of (.5) satisfying v(ρe ) = and v(z) 2 q for z R N +. This contradicts [, Theorem 4], and the proof is finished. Acnowledgment We than the referee for helpful comments and suggestions, which led to an improvement of the paper.

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