EXISTENCE OF SOLUTIONS TO NONLINEAR, SUBCRITICAL HIGHER-ORDER ELLIPTIC DIRICHLET PROBLEMS WOLFGANG REICHEL AND TOBIAS WETH

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1 EXISTENCE OF SOLUTIONS TO NONLINEAR, SUBCRITICAL HIGHER-ORDER ELLIPTIC DIRICHLET PROBLEMS WOLFGANG REICHEL AND TOBIAS WETH Abstract. 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 m. x i x j We assume that f is superlinear at the origin and satisfies f(x,s) lim 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.. Introduction Let Ω R N be a bounded smooth domain. On Ω we consider the uniformly elliptic operator (.) L = ( N 2 ) m a ij (x) + x i x j i,j= 0 α b α (x)d α with coefficients b α C α (Ω) and a ij C 2,α (Ω) such that there exists a constant λ > 0 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 the semilinear boundary value problem (.2) Lu = f(x, u) in Ω, u = ( ) m ν u =... = u = 0 on Ω, 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. Date: June 2, Mathematics Subject Classification. Primary: 35J40; Secondary: 35B45. Key words and phrases. Higher order equation, existence, topological degree, Liouville theorems.

2 2 WOLFGANG REICHEL AND TOBIAS WETH 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 exists q > if N and < q < N+ if N > and two positive, continuous functions N, h : Ω (0, ) such that f(x, s) f(x, s) lim = h(x), lim = (x) uniformly with respect to x Ω. s + s q s s q Theorem (Reichel, Weth []). If f : Ω R R satisfies (H) then there exists a constant C > 0 depending only on the data a ij, b α, Ω, N, q, h, such that u C for every solution u C,α (Ω) of (.2). This result 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) For all x Ω the function f(x, s) is continuously differentiable with respect to s and f(x, s), s f(x, s) are α-hölder continuous in x uniformly for x Ω and s in bounded intervals. Moreover, f(x, 0) = s f(x, 0) = 0. (H3) The operator L has a bounded inverse L which maps C α (Ω) C,α (Ω) with Dirichlet boundary conditions of order up to m. Theorem 2. Suppose Ω R N is a bounded domain with Ω C,α. Let m N and assume f : Ω R R satisfies (H), (H2), (H3). Then (.2) has a nontrivial solution u C,α (Ω). We note that in many examples condition (H3) can be verified with the help of the Lax- Milgram Theorem and elliptic regularity, see Agmon, Douglis, Nirenberg []. In particular, if L is as in (.), then L + γ satisfies (H3) if γ > 0 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-Rogers [2], if γ > 0 is sufficiently large. 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. Theorem 3. Suppose Ω R N is a bounded domain with Ω C,α. Let m N and assume f : Ω R R satisfies (H). Then there exists a value Λ = Λ(Ω, L, f) such that for λ Λ the problem (.3) Lu = f(x, u) + λ in Ω, u = ( ) m ν u =... = u = 0 on Ω. ν has no solution u C,α (Ω).

3 EXISTENCE FOR NONLINEAR HIGHER-ORDER ELLIPTIC PROBLEMS 3 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 Theorems 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 (Wei, Xu [4]). Let m N and assume that q > if N and < q < N+ N if N >. If u is a classical non-negative solution of (.4) ( ) m u = u q in R N, then u 0. Theorem 5. Let m N and assume that q > if N and < q N+ N If u is a classical non-negative solution of (.5) ( ) m u = u q in R N +, u = x u =... = m then u 0. x m u = 0 on R N + if N >. Here and in the following, we set R N + := {x R N : x > 0}. 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 [0] 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 4 WOLFGANG REICHEL AND TOBIAS WETH 2. Nonexistence for the parameter dependent problem The proof of Theorem 3 uses standard L p -W,p estimates for linear problems (2.) (2.2) Lu = g(x) in Ω, u = ν u =... = ( ) m u = 0 on Ω. ν Recall the following basic estimate of Agmon, Douglis, Nirenberg []. Theorem 6 (Agmon, Douglis, Nirenberg). Let Ω R N be a bounded domain with Ω C, m N. Let a ij C 2 (Ω), b α L (Ω), g L p (Ω) for some p (, ). Suppose u W,p (Ω) W m,p 0 (Ω) satisfies (2.). Then there exists a constant C > 0 depending only on a ij C 2, b α, λ, Ω, N, p, m and the modulus of continuity of a ij such that u W,p (Ω) C( g L p (Ω) + u L p (Ω)). We will also be using the following al analogue of this result. 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 > 0}. Let m 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 = on {x R N : x < R, x = 0}. x u =... = m x m u = 0 Then there exists a constant C > 0 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 (0, ) u W,p (Ω B σr ) C ( σ) ( g L p (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 = 0 α Here a α (x) = 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 M α (i,..., i ) {,..., N} satisfying #{j : i j = l} = α l for l =,..., N. Note that a α, c α are uniformly α-hölder continuous in 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, Weth [] and similar arguments have been used by Wei and Xu in [4].

5 EXISTENCE FOR NONLINEAR HIGHER-ORDER ELLIPTIC PROBLEMS 5 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 0 α. If g : R [0, ) is convex and non-negative with g(s) > 0 for s < 0 then either v > 0 or v 0. (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 0 = v. Then we have v 0 = v, v = v 2,... v m g(v 0 ) in R N. First we show that v l 0 in R N for l =,..., m. Assume that there exists l {,..., m } and x 0 R N with v l (x 0 ) < 0 but v j 0 in R N for j = l +,..., m. We may assume w.l.o.g. that x 0 = 0. If we define for a function w W 2, (RN ) C (R N ) spherical averages w(x) = r N ω N B r(0) w(y) dσ y, r = x then the radial functions v 0, v,..., v m satisfy v 0 = v, v = v 2,... v m g( v 0 ) in R N, where we have used Jensen s inequality and the convexity of g. Since v l (0) < 0 we also have v l (0) < 0. Moreover v l(r) = (2.3) ( v l )(rξ) ξ dσ ξ ω N B (0) = = v l+ (rξ)r dξ if l < m, ω N B ( v l )(rξ)r dξ (0) ω N B (0) g(v(rξ))r dξ if l = m. ω N Since the right-hand side is non-positive in both cases we obtain v l (r) v l (0) < 0. Integrating the inequality B (0) v l = v l v l (0) > 0 we obtain r N v l (r) rn N v l(0), i.e, v l (r) r N v l(0). The unboundedness of v l yields a contradiction. Next we show that v = v 0 0. Assume that v 0 (x 0 ) < 0 and w.l.o.g. x 0 = 0. Since v 0 = v 0 we see that v 0(r) 0 and we define α := lim r v 0 (r) < 0. Thus g( v 0 (r)) g(α) > 0 for r r 2 0. As in (2.3) we find v m (r) ω N = r N ω N r B (0) r r/2 g(v 0 (rξ))r dξ = 0 B s(0) s N r N g(α) 2 ds r N ω N g(v 0 (η)) dσ η ds B r(0) r 0 g(v 0 (η)) dη s N r N g( v 0(s)) ds

6 6 WOLFGANG REICHEL AND TOBIAS WETH if r 2r 0. Since the last term converges to as r we obtain a contradiction to the boundedness of v m. Finally the alternative v > 0 or v 0 follows since v 0 by the first part of the 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) := w(y) dσ r N y = w(x + rξ) dσ ξ ω N ω N B r(x) B (0) for X R N + and 0 < r < X. For fixed X R N + the function w satisfies w (r) = ( w)(x + rξ) ξ dσ ξ = ( w)(x + rξ)r dξ r ω N ω N N B (0) for 0 < r < X. Hence, w(r) w(0) r2. Letting X 2N and r tend to infinity we find that w cannot stay bounded. We now have all the tools to complete the Proof of Theorem 3. Assume for contradiction that there exists a sequence of pairs (u, λ ) of solutions of (.3) with λ for. Let M := 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 0 Ω and B δ (0) Ω for some δ > 0. Set v (x) := u (λ / x). Then v satisfies where L := ( ) m B (0) L v (x) = λ f(λ / x, v ) + in B λ / δ (0) α = ā α(λ / x)d α + λ α 0 α c α(λ / x)d α. By standard interior regularity on the ball B R (0) for any R > 0 and any p there exists a constant C p,r > 0 such that v W,p (B R (0)) 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 > 0, where v C,α (R N ) W m,p (RN ) is a bounded wea (and hence classical) solution of Lv = in R N, where L = ( ) m α = a α (0)D α = ( N 2 ) m. a ij (0) x i x j By a linear change of variables we may assume that v is a bounded, classical, entire solution of ( ) m v = in R N. Lemma 8(b) shows that we have reached a contradiction. i,j=

7 EXISTENCE FOR NONLINEAR HIGHER-ORDER ELLIPTIC PROBLEMS 7 Case 2: u is unbounded. For this case we need to discuss various sub-cases depending on the growth of the numbers ρ := M q dist(x, Ω), N. Passing to a subsequence, we may assume that either ρ or ρ ρ 0 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 0. Then we set v (y) := M u (M q y + x ) so that v = and either v (0) = for all N (positive blow-up) or v (0) = for all 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 (0) as and they satisfy L v (y) = ) (f(m q M q y + x, M v (y)) +λ } {{ } for y B ρ (0), where this time and L := ( ) m α = =:f (y) ā α(y)d α + ā α(y) := a α (M q y + x ), c α(y) := M 0 α α (q )( ) c α(y)d α c α (M q y + x ). By our assumption (H) on the nonlinearity f(x, s) we have that f L (B ρ (0)) 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 (0) for any R > 0 and any p there exists a constant C p,r > 0 such that v W,p (B R (0)) 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 (0)) as for every R > 0, where v C,α (R N ) is bounded with v = = ±v(0). 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 if v(y) > 0, (2.4) F (y) = ( x) v(y) q if v(y) < 0, because, e.g., if v(y) > 0 then there exists 0 such that v (y) > 0 for 0 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) < 0. Finally, note that the pointwise convergence of f on the set Z + = {y R N : v(y) > 0} and Z = {y R N : v(y) < 0} determine due to the dominated convergence theorem the wea -limit F of f on the set Z + Z. Since c α(y) 0 and ā α(y) a α ( x) as and since, for any fixed p (, ),

8 8 WOLFGANG REICHEL AND TOBIAS WETH we may assume that v v in W,p solution of (2.5) Lv = F + λ in R N, where L = ( ) m (R N ) we find that v is a bounded, wea W m,p (RN )- α = a α ( x)d α = ( N 2 ) m. a ij ( x) 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 = 0 a.e. on the set {y R N : v(y) = 0} one finds that v is a strong solution of h( x)v(y) q + λ if v(y) > 0, (2.6) Lv = 0 if v(y) = 0, ( x) v(y) q + λ if v(y) < 0 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 0, (2.7) g(s) := ( x) s q if s 0. Since the function g is convex we can apply Lemma 8(a) and obtain v > 0. 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(0) =. Clearly v and all its derivatives of order are bounded. If λ = 0 then Theorem 4 tells us that this is impossible. And if λ > 0 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. Now we set v (y) := M u (M q λ / y+x ). The functions v are again well defined on a sequence of expanding balls and satisfy (2.8) L v (y) = (f(m q λ M q λ / y + x, M v (y)) +λ } {{ } ), where this time with L := ( ) m α = ā α(y)d α + =:f (y) 0 α α (q )( ) λ α c α(y)d α ā α(y) := a α (M q λ / y + x ), c α(y) := M Arguing lie before we arrive at the situation that v v in W,p i,j= c α (M q λ / y + x ). (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 = in R N. A contradiction is reached via Lemma 8(b). Case 2.2: ρ ρ 0. Then, modulo a subsequence, x x Ω as, and after translation we may assume that x = 0. By flattening the boundary through a al

9 EXISTENCE FOR NONLINEAR HIGHER-ORDER ELLIPTIC PROBLEMS 9 change of coordinates we may assume that near x = 0 the boundary is contained in the hyperplane x = 0, and that x > 0 corresponds to points inside Ω. Since Ω is ally a C,α -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 q x,. As before we need to distinguish two further possibilities by defining λ := λ /M q. Case 2.2.a: Up to selecting a subsequence assume that λ λ 0. 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 q y + x Ω} which contains B ρ (0). Then we mae another change of coordinates, defining w (z) := v (z ρ e ), ã α(z) := ā α(z ρ e ), c α(z) := c α(z ρ e ), where e = (, 0,..., 0) 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 : M q z + (0, x,2,..., x,n ) Ω}, and that w (ρ e ) = ±. We now fix R > 0 and let B + R = B R(0) R N +. By our assumptions on the boundary Ω near x, we have B + R Ω for sufficiently large. Moreover, w satisfies L w (z) = f (z) + λ in B + R, where f (z) := M q f(m q z + (0, x,2,..., x,n ), M w (z)), together with Dirichlet-boundary conditions on {z R N : z < R, z = 0}. 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 > 0 such that w W,p (B + R ) C p,r uniformly in. By the Sobolev embedding theorem, this implies that v is bounded on B + R independently of, and since = v (0) v } {{ } (ρ, 0,..., 0) ρ } {{ } v, =± =0 we see that ρ = lim ρ > 0. 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 0, 0 is determined in the same way as in Case 2..a. This time, w is a bounded, strong W,p (R N +) C,α (R N +)- solution of Lw = F + λ in R N +, z w =... = m z m w = 0 on R N +

10 0 WOLFGANG REICHEL AND TOBIAS WETH with L as in (2.5). By a linear change of variables we may assume that w solves (2.9) ( ) m w = g(w) + λ in R N +, z w =... = m z m w = 0 on R N +, 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 nonnegative, 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 = 0 or Lemma 8(b) if λ > 0. 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 : M q λ / y + x Ω} and satisfies (2.8) on this set. By our assumptions on the boundary of Ω near x, we have dist(0, Σ ) = M q λ / x, = ρ λ/ =: τ for all. Passing to a subsequence, we may assume that either τ or τ τ 0 as. In the former 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 = in R 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 3. Finding a solution u C,α (Ω) of (.3) is equivalent to finding a solution u C α (Ω) of the equation (3.) [Id K λ ](u) = 0 where for λ R the nonlinear operator K λ : C α (Ω) C α (Ω) is defined by K λ (u) = L w with w(x) = f(x, u(x)) + λ. By assumption (H3) we may regard L : C α (Ω) C,α (Ω) as a bounded linear operator. Moreover, since the embedding C,α (Ω) C α (Ω) is compact, K λ is also compact for every λ R. Let Λ > 0 be as in Theorem 3 so that (3.) has no solution for λ Λ. By Theorem there exists K > 0 such that for all λ [0, Λ] any solution u C α (Ω) of (3.) satisfies u K. By elliptic regularity and (H3) we may assume u C α (Ω) K by adjusting K. Consequently, we find that (3.2) [Id K λ ](u) 0 if (u, λ) (B 2K (0) {λ}) ( B 2K (0) [0, Λ]),

11 EXISTENCE FOR NONLINEAR HIGHER-ORDER ELLIPTIC PROBLEMS where B 2K (0) C α (Ω) denotes the 2K-ball with respect to C α (Ω). invariance of the Leray-Schauder degree and and (3.2) imply deg(id K 0, B 2K (0), 0) = deg(id K Λ ), B 2K (0), 0) = 0. The homotopy For these and other properties of the Leray-Schauder degree, we refer the reader to [2, Chapter 2.8] or [8, Chapter 2]. Next we note that 0 is an isolated solution of (3.) for λ = 0. Indeed, assume that there exists a sequence of solutions u n of (3.) with λ = 0 and u n C α 0 as n. Let v n := u n / u n. Since by (H2) f(x, s) = O(s 2 ) uniformly in x Ω for s in bounded intervals, we conclude that Lv n = f(x, u n )/ u n 0 as n so that v n 0 as n, which is a contradiction. Moreover, since s f(x, 0) = 0 by (H2), the derivative dk 0 (0) : C α (Ω) C α (Ω) of K 0 at u = 0 vanishes. Hence, for small ɛ > 0, we have by [8, Theorem 2.8.] deg(id K 0, B ɛ (0), 0) = deg(id dk 0 (0), B ɛ (0), 0) = deg(id, B ɛ (0), 0) =. The additivity property of the topological degree now implies that deg(id K 0, B 2K (0) \ B ɛ (0), 0) 0, hence there exists u B 2K (0) \ B ɛ (0) such that u K 0 (u) = 0. Therefore u is a nontrivial solution of (3.). 4. Proof of Theorem 5 In section we show how Theorem 5 can be deduced from [, Theorem 4] with the help of the doubling lemma of Poláči, Quittner and Souplet [0]. We recall the following simple special case of this useful lemma. Lemma 9. (cf. [0]) Let (X, d) be a complete metric space and M : X (0, ) be bounded on compact subsets of X. Then for any y X and any > 0 there exists x X such that M(x) M(y) and M(z) 2M(x) for all z B /M(x) (x). This follows by taing D = Σ = X in [0, 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 n. By Lemma 9, applied within the underlying complete metric space X := R N +, there exist 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 +. We then define ρ := x, M(x ), the affine halfspace H := {ζ R N function ũ : H R, ũ (ζ) = u(x + ζ ) M(x ) u(x ) : ζ > ρ } and the

12 2 WOLFGANG REICHEL AND TOBIAS WETH for N. Then ũ is a nonnegative solution of ( ) m ũ = ũ q in H, (4.) ũ = ( ) ũ = = m ũ = 0 on H ζ ζ such that ũ (0) = and ũ (ζ) 2 q for all ζ H B (0). 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(0) =. By Theorem 4 we obtain a contradiction. Case 2: ρ ρ 0 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 = (, 0,..., 0) R N is the first coordinate vector. Then v is a nonnegative solution of ( ) m v = v q in R N +, v = z v = = ( z ) m v = 0 on R N +, 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 > 0. 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. References [] S. Agmon, A. Douglis and L. Nirenberg: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 2 (959), [2] K. Deimling: Nonlinear functional analysis. Springer-Verlag, Berlin, 985. [3] D.E. Edmunds, D. Fortunato, E. Jannelli: Critical exponents, critical dimensions and the biharmonic operator. Arch. Rational Mech. Anal. 2 (990), [4] F. Gazzola, H.-C. Grunau and M. Squassina: Existence and nonexistence results for critical growth biharmonic equations. Calc. Var. Partial Differential Equations 8 (2003), [5] B. Gidas and J. Spruc: A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differential Equations 6 (98), [6] H.-C. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents. Calc. Var. Partial Differential Equations 3 (995),

13 EXISTENCE FOR NONLINEAR HIGHER-ORDER ELLIPTIC PROBLEMS 3 [7] H.-C. Grunau, G. Sweers: Classical solutions for some higher order semilinear elliptic equations under wea growth conditions. Nonlinear Anal. 28 (997), [8] L. Nirenberg: Topics in nonlinear functional analysis. Lecture Notes, Courant Institute of Mathematical Sciences, New Yor University, New Yor, 974. [9] P. Oswald: On a priori estimates for positive solutions of a semilinear biharmonic equation in a ball. Comment. Math. Univ. Carolin. 26 (985), [0] P. Poláči, P. Quittner and Ph. Souplet: Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems. Due Math. J. 39 (2007), [] W. Reichel and T. Weth: A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems. Math. Z. 26 (2009), [2] M. Renardy and R.C. Rogers: An introduction to partial differential equations. Springer-Verlag, New- Yor, 993. [3] R. Soranzo: A priori estimates and existence of positive solutions of a superlinear polyharmonic equation. Dynam. Systems Appl. 3 (994), [4] Juncheng Wei and Xingwang Xu: Classification of solutions of higher order conformally invariant equations. Math. Ann. 33 (999), [5] T. Weth: Nodal solutions to superlinear biharmonic equations via decomposition in dual cones. Topol. Methods Nonlinear Anal. 28 (2006), W. Reichel Institut für Analysis, Universität Karlsruhe, D-7628 Karlsruhe, Germany address: wolfgang.reichel@math.uni-arlsruhe.de T. Weth Institut für Mathemati, Johann Wolfgang Goethe-Universität Franfurt D Franfurt, Germany address: weth@math.uni-franfurt.de