# EXISTENCE AND NON-EXISTENCE OF SOLUTIONS TO ELLIPTIC EQUATIONS WITH A GENERAL CONVECTION TERM

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1 EXISTENCE AND NON-EXISTENCE OF SOLUTIONS TO ELLIPTIC EQUATIONS WITH A GENERAL CONVECTION TERM SALOMÓN ALARCÓN, JORGE GARCÍA-MELIÁN AND ALEXANDER QUAAS Abtract. In thi paper we conider the nonlinear elliptic problem u + αu = g( u ) + λh(x) in Ω u = on Ω, where Ω i a mooth bounded domain of R N, α, g i an arbitrary C increaing function and h C (Ω) i nonnegative. We completely analyze the exitence and nonexitence of (poitive) claical olution in term of the parameter λ. We how that there exit olution for every λ when α = and the integral =, or α > and the integral g() =. Converely, when the repectively integral converge g() and h i nontrivial on Ω, exitence depen on the ize of λ. Moreover, nonexitence hol for large λ. Our proof mainly rely on comparion argument, and on the contruction of uitable uperolution in annuli. Our reult include ome cae where the function g i uperquadratic and till exitence hol without auming any mallne condition on λ.. Introduction The concern of the preent paper i the exitence and non-exitence of olution to the following nonlinear elliptic problem: u + αu = g( u ) + λh(x) in Ω (.) u = on Ω, where Ω i a bounded domain of cla C 2,η of R N for ome η (, ), α and g C (R) i increaing with g() =. The function h C (Ω), will be nonnegative, while λ will be regarded a a poitive parameter. We will focu our attention on general function g, obtaining harp condition which imply: either (a) problem (.) ha a unique olution for every λ or (b) there exit a critical ize of λ that divide exitence from nonexitence for (.) when h in Ω. Thi type of problem ha been extenively tudied. Here we give a quick review on the topic, other reference can be found in the paper quoted below. The pioneering work on the ubject eem to be due to Serrin [29], Amann and Crandall [6] and Lion [24]. The cae α > i conidered in [2] and [3], where exitence hol when g ha at mot quadratic growth, ee alo [4]. The cae α = and g(t) = t 2 wa tudied for example in [7] and [8] (ee alo [2] and [2]). For related reult ee [] and []. More recently alo related fully nonlinear equation are conidered in [3] (ee alo the cae α < in [22], where multiplicity reult are obtained).

2 2 S. ALARCÓN, J. GARCÍA-MELIÁN AND A. QUAAS We finally mention that a tarting point of our work can be found in [3]; actually we olve a problem given in that paper, ee Remark on page 29 there. More precie information on our contribution with repect to the known reult are given in the remark after our main theorem. By a olution to (.) we mean a function u C 2 (Ω) C (Ω) verifying the equation in the claical ene. Remark that, on one hand, tandard boottrapping give u C 2,η (Ω), while on the other hand olution are trictly poitive in Ω by the maximum principle, ince u + αu in Ω. An important remark with regard to problem (.) i that uniquene of olution hol by the comparion principle (cf. for intance Theorem. in [9] or the reult in [27] and [6]). Thu we only need to how exitence and nonexitence of olution. For other uniquene reult ee for example [8], [9] and []. Oberve alo that non-uniquene hol with le regularity on the olution, ee for intance [2]. Let u tate now our main reult. We begin with the cae α =. It turn out that the exitence of olution depen on the condition (.2) More preciely: g() =. Theorem. Aume g C (R) i increaing with g() =, while h C (Ω) i uch that h in Ω. If α = then: (i) If (.2) hol, there exit a unique olution to (.) for every λ >. (ii) If (.2) doe not hold and h on Ω, then there exit Λ > uch that for λ (, Λ) problem (.) ha a unique olution, while there are no olution when λ > Λ. Remark. (a) The non-exitence part in (ii) i already proved in the cae where g i convex (ee Theorem 2. in [3]). In the particular cae g(t) = t p with p > 2, ee [2] for exitence when h i a meaure and λ i mall. (b) Part (ii) of Theorem exten the cae g(t) = t 2 of the above quoted paper. We now turn to the cae α >. exitence of olution i (.3) In thi cae, the condition for the =. g() Oberve that condition (.3) i implied by (.2). We have: Theorem 2. Aume g C (R) i increaing with g() =, while h C (Ω) i uch that h in Ω. If α >, then: (i) If (.3) hol, there exit a unique olution to (.) for every λ >. (ii) If (.3) doe not hold and h in Ω, then there exit Λ > uch that for λ (, Λ) problem (.) admit a unique olution, while no olution exit when λ > Λ.

4 4 S. ALARCÓN, J. GARCÍA-MELIÁN AND A. QUAAS of a kind of maximum principle for u 2 + u 2, o that everything i reduced to etimating u on Ω. Thi can be done by comparing with a uitable uperolution. It i important to notice that our approach doe not rely in obtaining a uperolution ū to (.) which vanihe on the whole Ω, omething which i required to apply Theorem III. in [24]. Rather, we contruct the uperolution by analyzing problem (.) in an annulu which after a uitable tranlation i tangent to Ω at every fixed x Ω. Thi enable u to deal with a radial problem which i in ome ene integrable, o we are able to find condition which are both neceary and ufficient for exitence. The ret of the paper i organized a follow: in Section 2 we contruct uperolution to (.) in the particular cae where Ω i an annulu. Section 3 i dedicated to how nonexitence of olution to (.) when Ω i a ball. Finally, in Section 4 we deal with the proof of Theorem and Superolution for problem in annuli It will be proved in Section 4 that the exitence of a radial uperolution to problem (.) poed in an annulu when h i contant uffice to enure the exitence of a olution to (.). Thu thi ection will be dedicated to contruct a poitive radial function u verifying (2.) (r N u ) r N ( αu + g( u ) + c), R < r < R 2 u(r ) =, u(r 2 ), for uitable value of c, depending on whether α = or α > and alo on the integrability condition on g at infinity conidered in the Introduction. In what follow, R 2 > R > will be fixed. Lemma 3. Aume g C (R) i increaing with g() = and α =. Then, if (2.2) g() =, for every c > there exit a poitive radial function u verifying (2.). If (2.2) doe not hold, the exitence of uch a function alo follow provided c i mall enough. Proof. Introducing the change of variable log r N = 2 (2.3) =, N 3, N 2 rn 2 and denoting u(r) = v(), (2.) i tranformed into v r 2(N ) ( g ( ( v ) ) + c ) r N v(a) =, v(b),

5 GENERAL CONVECTION TERM 5 where a = log R, b = log R 2 when N = 2, while a = N 2 R N 2 2 N 2 R N 2, b = if N 3. Since g i increaing and poitive, it uffice to have ( ( v R 2(N ) 2 g v(a) =, v(b). R N ) ) v + c Setting w() = v( + a), thi ugget to conider the one-dimenional autonomou initial value problem ( ( ) ) w = R 2(N ) 2 g w + c (2.4) R N w() =, w () = γ >, which ha a unique olution for every γ >, and find a poitive olution in (, b a). Oberve that olution to (2.4) verify on one hand w cr 2(N ) 2, o that an integration provide w() (γ cr 2(N ) 2 /2). On the other hand, ince w i decreaing we have w ( ) = for ome >, and it follow by the ymmetry of the problem that w i ymmetric with repect to and w(2 ) =. Letting 2 be the firt zero of w and integrating the equation in (, ) we obtain ( ) N γ R R N = R 2 2 dt g(t) + c. We conclude that i an increaing function of γ which verifie ( ) N R dt (2.5) g(t) + c R 2 2 a γ +. Therefore in the cae where (2.2) hol, ince g i increaing, then the integral in (2.5) diverge. So, we can alway chooe γ large enough o that > (b a)/2, and thi provide with a poitive olution of (2.). When the integral converge, we can alo obtain > (b a)/2 if we elect c mall enough, ince the integral (2.6) dt g(t) diverge at, due to g() = and g C (R). Thi conclude the proof. Lemma 4. Aume g C (R) i increaing with g() = and α >. Then, if (2.7) =, g() for every c > there exit a poitive radial function u verifying (2.). If (2.7) doe not hold and c i mall enough, uch a function alo exit. Proof. Setting z = c/α u, we look for a function verifying (r N z ) r N (αz + g( z )) z(r ) = c α, z(r 2) c α.

6 6 S. ALARCÓN, J. GARCÍA-MELIÁN AND A. QUAAS We will look for a poitive olution z to thi inequality. With the change of variable (2.3), and letting v() = z(r), we find a before that v i a uperolution provided for intance that v R 2(N ) 2 (αv + g( R N v )) v(a) = c, v(b) =. α Setting w() = v(b ), it i thu natural to conider the initial value problem (2.8) w = R 2(N ) 2 ( αw + g ( w() =, w () = γ >, R N )) w which ha a unique olution for every γ >, and ee if we can elect γ o that w(b a) i a large a we pleae. Notice that w a long a w, o that it i not hard to ee that olution are poitive, increaing and convex for >. For every γ >, the olution i defined in an interval [, T (γ)), and when T (γ) < we have (2.9) lim w() = + or lim T (γ) T (γ) w () = +. Let u ee that when the integral condition (2.7) i atified, we alway have both condition in (2.9). Indeed, the firt one implie the econd, and if we had w(t (γ)) < +, then ( ( w R 2(N ) 2 αw(t (γ)) + g Multiplying by w and integrating we arrive at which yiel T w w αw(t (γ)) + g( γ R N R N αw(t (γ)) + g() R N w )) w ) R2(N ) 2 w(t (γ)), R2(N ) 2 R N w(t (γ)), contradicting condition (2.7). Thu w, w a T (γ). We have two cae to conider: either T (γ ) i infinite for ome γ > or T (γ) i finite for every γ >. In the firt cae, let u ee that thi implie T (γ) = for every γ >. Oberve firt that when u, v are two olution to the equation in (2.8) with u() v(), u () = γ > γ 2 = v (), then u > v in the common interval of definition, hence T (γ ) T (γ 2 ). In particular, T (γ) = for γ < γ. If γ > γ and we temporarily denote by w γ the unique olution to (2.8), there exit δ > uch that w γ (δ) > γ, ince w γ i increaing and converge to infinity. Let w(x) = w γ (x + δ). Then w i a olution to the ame equation with initial data w() = w γ (δ) >, w () = w γ (δ) > γ. It follow by the previou obervation that w > w γ, and in particular T (γ) =. Thu all olution are global in thi cae and it i eay to conclude: ince w(x) γx by convexity, we can have w(b a).

7 GENERAL CONVECTION TERM 7 a large a we pleae, o that a uperolution can be contructed with large value of c. The econd poibility i that all olution blow-up in finite time, i. e. T (γ) < for every γ >. Let u ee that in uch cae T (γ) i a continuou function of γ. Take γ n γ. By comparion we have T (γ n ) < T (γ). Moreover, we can chooe δ n uch that w γ(δ n ) > γ n. Arguing a before, w γ (x+δ n ) > w γn (x), o that T (γ) δ n < T (γ n ) and we obtain T (γ n ) T (γ). When γ n γ the proof i imilar. Next, we claim that T (γ) a γ. Indeed, aume T (γ) T when γ. Since w, we obtain w T w, o that ( ( w R 2(N ) 2 γ R N g R N w ) + αt w ) and thi lea, after an integration and a change of variable, to ( ) R 2 N g() + αt R N 2 T. R A contradiction i reached when we let γ, ince the integral then diverge. Thu lim γ T (γ) =. Let u denote T = lim γ T (γ) (which i expected to be zero). If T b a we can ue the continuity of T to obtain γ > uch that T (γ) = b a+ε for mall poitive ε. Taking ε a mall a we pleae we obtain w γ (b a) a large a we wih, and thi provide with a uperolution for large value of c. If, on the contrary, T > b a, then all olution would be defined at leat in [, b a] and ince w(b a) γ(b a), we obtain that w γ (b a) i a large a we pleae by taking large value of γ. To conclude the proof, we now conider the cae when g() < and c i mall enough. Oberve that in thi cae all olution blow up in finite time. Indeed, let T < T (γ). Since ( ) w R 2(N ) 2 g w R N we can integrate in (, T ) and let T T (γ) to arrive at: (2.) R 2(N ) 2 T (γ) R N γ g() <, ince thi lat integral alo converge. It alo follow from (2.) that T (γ) a γ (i. e. T = in the above proof). Since T (γ) i continuou with T (γ) a γ, we can chooe γ uch that T (γ) > b a and obtain a uperolution for c αw γ (T (γ)). It i worth mentioning that in the preent cae where (2.7) doe not hold we cannot guarantee that the firt equality in (2.9) hol, o that the uperolution i not valid in principle for large value of c.,

8 8 S. ALARCÓN, J. GARCÍA-MELIÁN AND A. QUAAS 3. Nonexitence of olution in ball We tackle in thi ection the quetion of nonexitence of olution to (.). We will ee in Section 4 that it uffice to how nonexitence of radial olution when Ω i a ball of R N and h i contant. Thu, under everal hypothee, we will how that the problem u N u = αu + g( u ) + c, < r < R (3.) r u () =, u(r) = doe not admit poitive olution for large value of c. Lemma 5. Aume g C (R) i increaing with g() = and α =. Then if g() <, there exit c > uch that problem (3.) doe not admit poitive olution when c c. Proof. Aume u i a olution to (3.). We firt claim that u (r) < for r (, R) and u (r) < in [, R). Oberve that u () = c/n <, o that u (r) < for r > cloe enough to zero. If we had u (r ) = for ome r (, R) with u < in (, r ), then u (r ) o that from the equation we obtain u (r ) = c <, which i impoible. Then u (r) < if < r < R. Aume now that for ome r (, R) we have u ( r ) =. Since u ( r ) in thi cae, we obtain by differentiating the equation u N u + N r r 2 u = g ( u )u o that u ( r ) <, a contradiction. Thu u (r) < for r (, R) a well. Next if we rewrite the equation a (r N u ) = r N (g( u ) + c) and integrate in (, r) we obtain, taking into account that g( u ) i increaing: r N u (r) = r N (g( u ()) + c) (g( u (r)) + c) o that plugging thi in (3.) we have r N = rn N (g( u (r)) + c), u N (g( u ) + c) in (, R). Integrating in (, R) we obtain dt g(t) + c > u (R) dt g(t) + c N R. Thi implie that c cannot be too large in order to have a poitive olution to (3.).

9 GENERAL CONVECTION TERM 9 Lemma 6. Aume g C (R) i increaing with g() = and α >. Then if (3.2) <, g() there exit c > uch that problem (3.) doe not admit poitive olution when c c. Proof. Let u be a poitive olution to (3.). We firt claim that u < c/α. Indeed, if we had u() = c/α then u c/α by uniquene, which i not poible. If u() > c/α then u () > and u initially increae. According to the boundary condition u(r) =, there hould be a point where u achieve it maximum, but thi i in contradiction with the equation. We conclude that u() < c/α and again by the equation u initially decreae and cannot reach a minimum, o u i alway decreaing. It i een much a in the previou cae that u < in [, R) alo. Thu arguing a in that proof we obtain (3.3) u N ( αu + g( u ) + c). Aume there exit a equence c n uch that a poitive olution u n to (3.) exit with c = c n (with no lo of generality we may aume that c n i increaing). Let v n = c n α u n. Then v n + N v n = αv n + g(v r n) v n() =, v n (R) = c n α, with v n >, v n >. We claim that v n () i bounded a n. Indeed, ince from (3.3) we have we can integrate to arrive at v n N (αv n + g(v n)) N (αv n() + g(v n)), N R v n (R) αv n () + g() < αv n () + g(). Therefore if v n () we arrive at a contradiction. Since olution are increaing in c (thank to uniquene), we can guarantee that v n () v for ome v >. It alo follow that v n z, the unique olution to z + N z = αz + g(z ) r z() = v, z () =, which i defined in a maximal interval [, T ). When T <, we have lim r T z(r) = or lim r T z (r) =. By comparion we alo have v n z in [, mint, R}). Let u ee that T < R. Indeed, if T R, we would have v n (R) = c nα z(r), and then T = R, z(r) = follow. Thi i impoible, ince

10 S. ALARCÓN, J. GARCÍA-MELIÁN AND A. QUAAS z N g(z ), and multiplication by z and another integration between R 2 and R ε for ome mall poitive ε yiel N (z(r ε) z(r 2 )) z (R ε) z (R/2) g() < z (R/2) g(). Letting ε we obtain a contradiction with (3.2). Thu T < R. Now chooe a mall ε >. Since v n z uniformly in [, T ε], we have v n(t ε) z (T ε) ε if n i large enough. Therefore (R T + ε) N v (R) v n(t ε) g() < z (T ε) ε g(). Letting ε, we arrive at T R, a contradiction which how that no olution to (3.) may exit if c i large enough. 4. Proof of the theorem Thi final ection will be dedicated to the proof of Theorem and 2. The idea of the proof of exitence come from [24], and it conit in truncating the function g in order to obtain a olution, and then etimating the gradient of thi olution in Ω. The eential point i to obtain a uitable uperolution. Since Ω verifie the uniform exterior phere condition, there exit R > uch that for every x Ω there exit y R N \Ω with B R (y ) Ω = x }. Chooe R 2 > R large enough o that Ω A := B R2 (y ) \ B R (y ) for every x Ω. Conider the radial problem (r N u ) = r N ( αu + g( u ) + c), R < r < R 2 (4.) u(r ) =, u(r 2 ), where c >. The firt important exitence reult i the following: Lemma 7. Aume g C (R) i increaing with g() = and h C (Ω) i nonnegative. If there exit a poitive uperolution ū to problem (4.), then for every λ (, c/ h ], problem (.) admit a poitive olution. With regard to nonexitence reult, the reference ituation i a radial problem in a ball. Oberve that, ince Ω verifie a uniform interior ball condition and h, h on Ω, we can find x Ω, y Ω and R > uch that B R (y ) Ω, B R (y ) Ω = x } and h h > in B R (y ). Conider the problem (r N u ) = r N ( αu + g( u ) + c), < r < R (4.2) u () =, u(r) =, where c >. Then: Lemma 8. Aume g C (R) i increaing with g() = and h C (Ω) i nonnegative with h h > in B R (y ). If problem (4.2) doe not admit a poitive olution for ome c >, then problem (.) doe not have olution for λ c/h. A we have quoted in the introduction, the method we follow for proving exitence (and indeed alo nonexitence) relie in obtaining good etimate for the gradient of the olution. Thi lat part i achieved by mean of a

11 GENERAL CONVECTION TERM kind of maximum principle for the gradient of olution to (.). The proof i inpired in the claical method of Berntein (ee for intance [29] or [24]). Lemma 9. Let u C 2 (Ω) C (Ω) be a olution to (.). Aume g C (R) i increaing with g() = and h C (Ω). Then there exit a contant C which depen on up Ω u, up Ω u, up Ω h and λ uch that u C in Ω. Proof. Let u be a olution to (.), and define w = u 2 +u 2. For implicity, let u denote g( ξ ) = g( ξ 2 ). By tandard regularity, it follow that u C 3 (Ω ρ ), where Ω ρ = x Ω : u 2 > ρ} for ome < ρ < u, and hence w C 2 (Ω ρ ) C(Ω ρ ). Then, it i not hard to check that in Ω ρ one ha w = 2 N D2 u 2 2 g ( u 2 ) u (w u 2 ) 2λ h u+2u u+(2+2α) u 2. On the other hand, ( N 2 ( u) 2 = ii u) N i= N ( ii u) 2 N D 2 u 2, and ince g i nondecreaing and u, o that 2 g ( u 2 ) u (u 2 ), we have w 2 N ( u)2 2 g ( u 2 ) u w 2λ h u + 2u u + (2 + 2α) u 2 in Ω ρ. An application of Cauchy-Schwarz inequality lea to w N ( u)2 2 g ( u 2 ) u w λ 2 h 2 Nu 2 + u 2 in Ω ρ. Fix M > up u 2 + 2N u 2 + λ 2 h, Ω and aume that the open et Ω = x Ω : w > M} i nonempty. It clearly follow that Ω Ω ρ ince i= u 2 u 2 > ρ in Ω. Hence Lw in Ω, where Lw := w 2g ( u ) u w, and the trong maximum principle implie w < up Ω w = M in Ω, a contradiction. Hence w M in Ω. Now we come to the proof of Lemma 7 and 8. Proof of Lemma 7. Fix for the moment x Ω and denote v(x) = ū( x y ). Take K > and let g K C (R) be a bounded increaing function verifying g K (t) = g(t) if t K. Let u conider the truncated problem u + αu = gk ( u ) + λh(x) in Ω (4.3) u = on Ω. When K > up ū, the function v i a uperolution to (4.3) and ince v = a ubolution, it follow that there exit a olution u to (4.3) (by uing the reult in [6] or [28]), which verifie u v.

12 2 S. ALARCÓN, J. GARCÍA-MELIÁN AND A. QUAAS By the maximum principle we have u > in Ω and u ν Moreover, ince v(x ) =, < on Ω. u ν (x ) v ν (x ) = ū (R ). Let u ee that the ame inequality hol for every x Ω. Indeed, if we take uch a x and A i the correponding annulu, then ince u + αu g K ( u ) + λ h in Ω, the function ū( x y ) i a uperolution to the problem (4.3), conidered in A. By comparion we obtain u(x) ū( x y ) in Ω. Thu u ν (x ) ū (R ). Hence ū (R ) u ν on Ω. We are in a poition to apply Lemma 9 to obtain a contant M >, which doe not depend on K, uch that u < M in Ω. Taking K > M, we have g K ( u ) = g( u ) in Ω and u i a olution to our original problem. Thi conclude the proof. Proof of Lemma 8. Aume problem (.) ha a poitive olution u for ome λ c/h. Then u i a uperolution to the problem v + αv = g( v ) + λh in B R (y ) v = on B R (y ). A imilar procedure a in Lemma 7 yiel the exitence of a olution to thi problem, which i unique, hence radial. Thi i in contradiction with the hypothei. It i important to remark that thi procedure work ince the uperolution vanihe at x B R (y ) Ω, which allow u to etimate v on B R (y ) in term of u(x ). Finally, we proceed to prove our main theorem. We notice that, once we have analyzed eparately the cae α = and α > in Section 2 and 3, the ret of the proof i exactly the ame in both cae. Proof of Theorem and 2. (i) By Lemma 3 and 4, there exit a uperolution to (4.) for every c >. The exitence of a poitive olution to (.) for every λ > follow thank to Lemma 7. (ii) Again by lemma 3, 4 and 7, there exit a olution for mall value of λ. On the other hand, uing Lemma 8 in conjunction with Lemma 5 and 6 we alo have that no olution to (.) exit for large value of λ. Hence, we can define Λ = upλ > : there exit a olution to (.)} and Λ i finite and poitive. By it very definition, there are no olution to (.) for λ > Λ. Now, chooe an arbitrary λ (, Λ). Then there exit µ (λ, Λ) uch that (.) with λ ubtituted by µ admit a poitive olution v. Since thi olution i a uperolution to (.), the exitence of a poitive olution follow a in Lemma 7.

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