EXPLICIT SOLUTIONS AND MULTIPLICITY RESULTS FOR SOME EQUATIONS WITH THE p-laplacian

QUARTERLY OF APPLIED MATHEMATICS http://dx.doi.og/10.1090/qam/1471 Aticle electoically published o Apil 19, 2017 EXPLICIT SOLUTIONS AND MULTIPLICITY RESULTS FOR SOME EQUATIONS WITH THE p-laplacian By PHILIP KORMAN Depatmet of Mathematical Scieces, Uivesity of Ciciati, Ciciati, Ohio 45221-0025 Abstact. We deive explicit goud state solutios fo seveal equatios with the p-laplacia i R, icludig (hee ϕ(z) =z z p 2, with p>1) ϕ (u ()) + 1 ϕ (u ()) + u M + u Q =0. The costat M > 0 is assumed to be below the citical powe, while Q = Mp p+1 is above the citical powe. This explicit solutio is used to give a multiplicity esult, similaly to C. S. Li ad W.-M. Ni (1998). We also give the p-laplace vesio of G. Batu s solutio, coected to combustio theoy. I aothe diectio, we peset a chage of vaiables which emoves the o-autoomous tem α i ϕ (u ()) + 1 ϕ (u ()) + α f(u) =0, while pesevig the fom of this equatio. I paticula, we study sigula equatios, whe α<0, that occu ofte i applicatios. The Coulomb case α = 1 tued out to give the citical powe. 1. Itoductio. Fo the equatio with the citical expoet (whee u = u(x), x R ) Δu + u +2 2 = 0 (1.1) thee is a well-kow explicit solutio u(x) = ( a 1+ 2 a2 2 ) 2 2, (1.2) goig back to T. Aubi [1] ad G. Taleti [15]. Hee = x, ada is a abitay positive costat. This explicit solutio is vey impotat, fo example, it played a cetal ole Received August 21, 2016 ad, i evised fom, Mach 20, 2017. 2010 Mathematics Subject Classificatio. Pimay 35J25, 35J61. Key wods ad phases. Explicit solutios, multiplicity esults. E-mail addess: komap@ucmail.uc.edu 1 c 2017 Bow Uivesity Licesed to Uiv of Ciciati. Pepaed o Mo Aug 7 12:04:45 EDT 2017 fo dowload fom IP 129.137.96.5. Licese o copyight estictios may apply to edistibutio; see http://www.ams.og/licese/jou-dist-licese.pdf

2 PHILIP KORMAN i the classical pape of H. Bézis ad L. Niebeg [4]. How does oe deive such a solutio? Radial solutios of (1.1) satisfy u + 1 u + u +2 2 =0, u (0) = 0, u () < 0. (1.3) Let us set u = au 2. (1.4) The u = au 2 + 2 a2 2 u +2 2, ad usig these expessios fo u ad u i (1.3), we get a algebaic equatio fo u, solvig of which leads to the solutio i (1.2). I ode fo such a appoach to wok, the solutio u() must satisfy the asatz (1.4), ad it does! We show that a simila appoach poduces the explicit solutio of C. S. Li ad W.-M. Ni [11] fo the equatio u + 1 u + u q + u 2q 1 =0, (1.5) with +2 2 <q< 2 < 2q 1, ad some othe equatios, ad fo the p-laplace vesios of all of these equatios. As a applicatio, we state a multiplicity esult fo the p-laplace vesio of (1.5), similaly to C. S. Li ad W.-M. Ni [11]. While studyig positive solutios of semiliea equatios o a ball i R, we oticed that fo the o-autoomous poblem (hee α>0, ad a>0aecostats) u + 1 u + α f(u) =0, u(0) = a, u (0) = 0, (1.6) oe ca pove simila esults as fo the autoomous case, whe α = 0. We wodeed if the α tem ca be emoved by a chage of vaiables. It tus out that the chage of vaiables t = 1+α/2 1+α/2 tasfoms the poblem (1.6) ito u (t)+ m du t u (t)+f(u(t)) = 0, u(0) = a, (0) = 0, (1.7) dt with m = 1+α/2 1+α/2. The poit hee is that this chage of vaiables peseves the Laplacia i the equatio. This tasfomatio allows us to get some ew multiplicity esults fo the coespodig Diichlet poblem, icludig the sigula case, whe α<0. We peset simila esults fo equatios with the p-laplacia. Such poblems, with the α tem, ofte aise i applicatios, fo example i modelig of electostatic mico-electomechaical systems (MEMS), see e.g., J. A. Pelesko [14], N. Ghoussoub ad Y. Guo [5], Z. Guo ad J. Wei [6], o P. Koma [8]. 2. Some explicit goud state solutios. Fo the poblem u + 1 u + f(, u) =0, > 0, u (0) = 0, (2.1) the cucial ole is played by Pohozhaev s fuctio P () = [ u 2 ()+2F (, u()) ] +( 2) 1 u ()u(), wheewedeotef (, u) = u f(, t) dt. Oe computes that ay solutio of (2.1) satisfies 0 P () = 1 [2F (, u()) ( 2)u()f(, u()) + 2F (, u())]. (2.2) Licesed to Uiv of Ciciati. Pepaed o Mo Aug 7 12:04:45 EDT 2017 fo dowload fom IP 129.137.96.5. Licese o copyight estictios may apply to edistibutio; see http://www.ams.og/licese/jou-dist-licese.pdf

EXPLICIT SOLUTIONS AND MULTIPLICITY RESULTS 3 I case f(, u) =u p,wehavep () =0fop = +2 2, P () < 0fop> +2 2,ad P () > 0fop< +2 2. (Itegatig (2.2), oe shows that the Diichlet poblem fo (2.1) o ay ball has o solutios if p> +2 +2 2.) The citical expoet 2 is also the cut-off fo the Sobolev embeddig. I case f(, u) = α u p,withacostatα, wehavep () =0 fo p = +2+2α 2, the ew citical expoet. Itegatig (2.2), oe sees that the Diichlet poblem fo the equatio (2.3) below, o ay ball, has o solutios if p> +2+2α 2. Let us look fo positive goud state solutios of ( >2) u + 1 u + α u +2+2α 2 =0, > 0, u (0) = 0. (2.3) Hee by goud state we mea solutios which ted to zeo as. Deotig p = +2+2α 2, we let (obsevig that u () < 0) u = a 1+α u p+1 2 = a 1+α u +α 2, (2.4) whee a>0 is a costat. The u = (1 + α)a α u p+1 2 + p +1 2 a2 2+2α u p. Usig these expessios fo u ad u i (2.3), we get a algebaic expessio, which we solve fo u: [ ] 2 [ ] 2 2+α a + aα a + aα u() = 1+ p+1 = 2 a2 2+α 1+ +α. (2.5) 2 a2 2+α I ode fo this fuctio to be a solutio of (2.3), it must satisfy the asatz (2.4), which mightlookulikely. Butisdoes,foaycostata! By choosig a, we ca satisfy the iitial coditios u(0) = A, u (0) = 0, fo ay A>0. Whe α = 0, the goud state solutio i (2.5) is the same as the well-kow oe i (1.2). Popositio 1. The fomula (2.5) povides goud state solutios of (2.3), fo ay costat a>0. We coside ext the poblem ( >2, p>1) We set u + 1 u + α ( u p + u 2) =0, > 0, u (0) = 0. (2.6) whee a>0 is a costat. The u = a 1+α u p, (2.7) u = (1 + α)a α u p + a 2 p 2+2α u 2. Usig these expessios fo u ad u i (2.6), we obtai [ ] 1 a+ aα+1 u() = 1+pa 2 2+α. (2.8) This fuctio satisfies the asatz (2.7) povided that a = p 1 α p + +2p. (2.9) Licesed to Uiv of Ciciati. Pepaed o Mo Aug 7 12:04:45 EDT 2017 fo dowload fom IP 129.137.96.5. Licese o copyight estictios may apply to edistibutio; see http://www.ams.og/licese/jou-dist-licese.pdf

4 PHILIP KORMAN I ode to have a>0, we eed p< +α 2 +2+2α,adthe2 < 2, i.e., both powes ae sub-citical. Coclusio: the fuctio u() i (2.8), with a give by (2.9) povides a goud state solutio fo (2.6). Fially, we coside the poblem ( >2, p>1) u + 1 u + α ( u p + u 2) =0, > 0, u (0) = 0. (2.10) Usig the asatz (2.7) agai, we obtai [ ] 1 a+ aα 1 u() = 1+pa 2 2+α. (2.11) This fuctio satisfies the asatz (2.7) povided that a = I ode to have a>0, we eed p> +α 2 p 1 p 2p α. (2.12) +2+2α,adthe2 > 2, the citical expoet. Coclusio: the fuctio u() i (2.11), with a give by (2.12) povides a goud state solutio fo (2.10). I case α = 0, this solutio was oigially foud by C. S. Li ad W.-M. Ni [11]. Popositio 2. The fomula (2.11), with the costat a give by (2.12), povides a goud state solutio of (2.10). A simila appoach ca be tied fo the equatios of the fom u + 1 u + Aψ(u)+Bψ(u)ψ (u) =0, > 0, u (0) = 0, (2.13) whee ψ(u) is a give fuctio, with mootoe ψ (u), so that the ivese fuctio (ψ ) 1 (u) exists. Hee A ad B ae give costats. Settig u = aψ(u), (2.14) with u = a 2 2 ψ(u)ψ (u) aψ(u), we obtai fom (2.13) ( ) u() =(ψ ) 1 a A a 2 2. (2.15) + B This fuctio gives a solutio of (2.13), povided it satisfies (2.14). If we select hee =2,A =0,adψ(u) = 2e u/2, the the last fomula gives u() =2l 2 2a a 2 2 + B. (2.16) Oe veifies that fo ay a>0, ad ay B>0 the fuctio i (2.16) solves u ()+ 1 u ()+Be u() =0, u (0) = 0. This is the famous G. Batu s [2] solutio. It immediately implies the exact cout of solutios fo the coespodig Diichlet poblem o the uit ball i R 2. Licesed to Uiv of Ciciati. Pepaed o Mo Aug 7 12:04:45 EDT 2017 fo dowload fom IP 129.137.96.5. Licese o copyight estictios may apply to edistibutio; see http://www.ams.og/licese/jou-dist-licese.pdf

EXPLICIT SOLUTIONS AND MULTIPLICITY RESULTS 5 Popositio 3. The poblem u ()+ 1 u ()+Be u() =0, u (0) = u(1) = 0 (2.17) has exactly two solutios fo 0 <B<2, exactly oe solutio fo B = 2, ad o solutios if B>2. Poof. Accodig to the fomula (2.16), the bouday coditio u(1)=0isequivalet to a 2 2 2 a + B =0. This quadatic equatio has two solutios fo 0 <B<2, oe solutio fo B =2,ad oe if B>2. It is kow that the value of u(0) uiquely idetifies the solutio pai (B,u()); see [9]. Sice solutios i (2.16) cove all possible values of u(0), o othe solutios ae possible. The equatio (2.17), kow as the Gelfad equatio, is pomiet i combustio theoy; see J. Bebees ad D. Ebely [2]. Aothe example: the equatio u + 1 u +( 2)e u + Be 2u =0, > 0, u (0) = 0, has a solutio u =l 2 2 +B, fo ay eal B. The class of ψ(u), fo which this appoach woks is ot wide. Ideed, witig (2.15) as ψ (u) = A 2 +B, diffeetiatig this equatio, ad usig (2.14), we see that ψ(u) must satisfy ψ (u)ψ(u) = 2 A ψ 2 (u). (2.18) Solutios of the last equatio ae expoetials ad powes (of c 1 u + c 2 ). If A =0,a solutio of (2.18) is ψ(u) =u k, with k = 2, which leads to the goud state solutio fo the citical powe +2 2, that we cosideed above. 3. Explicit goud states i case of the p-laplacia. Fo equatios with the adial p-laplacia i R ( p) ϕ (u ()) + 1 ϕ (u ()) + f(u) =0, (3.1) Pohozhaev s fuctio P () = [(p 1)ϕ(u ())u ()+pf (u())] + ( p) 1 ϕ(u ())u() was itoduced i P. Koma [7]. Hee ϕ(z) =z z p 2, with p>1, ad F (u) = u f(t) dt. 0 Fo the solutios of (3.1) we have P () = 1 [pf (u) ( p)uf(u)]. Compaig this P () to the oe i case p = 2, it was elatively easy fo us to make the adjustmets, except fo the p 1 facto, which we foud oly afte a lot of expeimetatio, usig Mathematica. I case f(u) =u q, oe calculates the citical powe (whe P () =0)tobeq = ()+p p. Licesed to Uiv of Ciciati. Pepaed o Mo Aug 7 12:04:45 EDT 2017 fo dowload fom IP 129.137.96.5. Licese o copyight estictios may apply to edistibutio; see http://www.ams.og/licese/jou-dist-licese.pdf

6 PHILIP KORMAN We look fo positive goud state solutios of ( >p) ϕ (u ()) + 1 ϕ (u ()) + u q =0, u (0) = 0, (3.2) whee q is the citical powe q = ()+p p.thep () = 0, so that P () =costat =0, which simplifies as ] [(p 1) u p + p uq+1 +( p)ϕ(u ())u() =0. (3.3) q +1 By maximum piciple, positive solutios of (3.2) satisfy u () 0, fo all. I (3.3) we set (a >0isacostat) ϕ (u ()) = au s (), (3.4) with the powe s to be specified. Witig (3.4) as ϕ ( u ()) = au s (), o ( u ()) = au s (), we expess u () =a 1 1 s u (). The (3.3) becomes (p 1)a p p sp p u + q +1 uq+1 = a( p)u s+1. (3.5) sp We ow choose s to get the equal powes of u o the left: = q + 1, givig (q +1)() s = = p The solvig (3.5) fo u, weget [ a( p) u() = p (p 1) p. p p +()a ] p p. (3.6) Oe veifies that this u() satisfies the asatz (3.4) fo ay a>0, ad so it gives a goud state solutio of (3.2). By choosig a, we ca satisfy the iitial coditios u(0) = A, u (0) = 0, fo ay A>0. We coside ext the equatio of Li-Ni type with the p-laplacia ϕ (u ()) + 1 ϕ (u ()) + u M + u Q =0. (3.7) Hee M>isapositive costat, ad Mp p +1 Q = >M. (3.8) p 1 Lookig fo a positive goud state, we set i (3.7) ϕ (u ()) = au M (), (3.9) with the costat a>0 to be detemied. As above, we expess u () =a 1 1 M u (), so that d d ϕ (u ()) = au M amu M 1 u = au M + Ma p p u Q. The (3.7) gives ( u() = a 1 1+a p M p ) M p+1. (3.10) Licesed to Uiv of Ciciati. Pepaed o Mo Aug 7 12:04:45 EDT 2017 fo dowload fom IP 129.137.96.5. Licese o copyight estictios may apply to edistibutio; see http://www.ams.og/licese/jou-dist-licese.pdf

EXPLICIT SOLUTIONS AND MULTIPLICITY RESULTS 7 I ode fo this fuctio to be a solutio of (3.7), it must satisfy the asatz (3.9). This happes if M p +1 a = M p + Mp. (3.11) Obseve that a > 1, povided that both the umeato ad deomiato ae positive i (3.11), o whe M> p p, (3.12) which implies that Q> ()+p p, the citical powe. Coclusio: the fuctio u() i (3.10), with a fom (3.11), gives a goud state solutio of (3.7), povided that (3.12) holds. Similaly to C. S. Li ad W.-M. Ni [11] the existece of a explicit goud state solutio implies a multiplicity esult. Theoem 3.1. Suppose that p>1, >p, M>, the coditio (3.12) holds, ad Q is defied by (3.8). The thee exists R > 0, so that fo R>R the poblem ϕ (u ()) + 1 ϕ (u ()) + u M + u Q =0, fo 0 <<R, (3.13) u (0) = u(r) =0, has at least two positive solutios. Poof. Recall that (3.12) implies: p 1 <M< ()+p p ad W.-M. Ni [11], we employ shootig, ad coside <Q. Similaly to C. S. Li ϕ (u ()) + 1 ϕ (u ()) + u M + u Q =0, fo 0 <<R, (3.14) u(0) = a, u (0) = 0. Let ρ(a) deote the fist oot of u(), ad we say ρ(a) = if u() is a goud state solutio. Whe a is small, oe sees by scalig that a multiple of the solutio of (3.14) is a abitaily small petubatio of ϕ (z ()) + 1 ϕ (z ()) + z M =0, z(0) = a, z (0) = 0. (3.15) Ideed, settig u = aw, ad = βs, with β = a M p+1 p, the poblem (3.14) is tasfomed ito ( ) d dw ds ϕ + 1 ( ) dw ϕ + w M + ɛw Q =0, w(0) = 1, w (0) = 0, ds s ds with ɛ = a Q M. Solutios of the last equatio ae deceasig (while they ae positive), ad so the ɛw Q tem is bouded by ɛw Q (0) = ɛ. Fo the poblem (3.15) it is kow (see e.g., [7] o [9]) that fo ay a>0, the solutio z() has a uique oot, this oot teds to ifiity as a 0 (by scalig), ad z() is egative ad deceasig afte the oot (because z < 0 at ay oot by uiqueess of IVP). The fist oot exists because the coespodig Diichlet poblem has a positive solutio, as follows by the moutai pass lemma. By the cotiuity i ɛ, it follows that ρ(a) < fo a small, ad ρ(a) as a 0. Now deote A = {a >0 ρ(a) < }. The set A is ope, but sice we have a explicit goud state, it follows that thee exists Licesed to Uiv of Ciciati. Pepaed o Mo Aug 7 12:04:45 EDT 2017 fo dowload fom IP 129.137.96.5. Licese o copyight estictios may apply to edistibutio; see http://www.ams.og/licese/jou-dist-licese.pdf

8 PHILIP KORMAN u(0) 1.4 1.2 1.0 0.8 0.6 2000 3000 4000 5000 λ Fig. 1. The solutio cuve fo the poblem (3.17) a maximal iteval (0, β) A, with β / A. By the cotiuous depedece o the iitial data, lim a β ρ(a) =, ad the theoem follows, with R =if{ρ(a) a (0, β)}. We ow discuss the poblem (3.13) i case p =2,wheQ =2M 1. By scalig, we ca tasfom it to a Diichlet poblem o a uit ball u + 1 u + λ ( u M + u 2M 1) =0, 0 <<1,u (0) = u(1) = 0, (3.16) with a positive paamete λ. The esult of C. S. Li ad W.-M. Ni [11] (exteded above), togethe with the bifucatio theoy developed i [10], [13] ad [9], implies the existece of a cuve of solutios i the (λ, u(0)) plae. Alog this cuve λ,wheu(0) 0, ad whe u(0) β. This cuve has a hoizotal asymptote at u(0) = β; see [13]. Based o the umeical evidece, we cojectue that the solutio cuve makes exactly oe tu to the ight i the (λ, u(0)) plae, ad it exhausts the set of positive solutios of (3.16); see Figue 1. Howeve, the pictue chages dastically eve if the lowe powe M is petubed; see Figue 2. This supisig pheomeo is simila to the oe obseved by H. Bézis ad L. Niebeg [4], i case f(u) =λu + u +2 2. Example 1. We solved umeically the poblem (3.16), with =3,M =4,2M 1 = 7 u + 2 u + λ ( u 4 + u 7) =0, u (0) = u(1) = 0. (3.17) (See [9] fo the expositio of the shoot-ad-scale algoithm that we used.) The solutio cuve is peseted i Figue 1. Obseve that the λ s i this pictue ae lage tha fo most othe f(u); see [9]. We have veified this umeical esult by a idepedet computatio. Takig a abitay poit ( λ, ū) o the solutio cuve, we solved umeically the iitial value poblem fo the equatio i (3.17), with λ = λ, usig the iitial coditios u(0) = ū, u (0) = 0. The fist oot of the solutio was always at =1. We cojectue that thee is citical λ 0 so that the Li-Ni poblem (3.16) has o positive solutios fo λ<λ 0, exactly oe positive solutio at λ = λ 0, ad exactly two positive solutios fo λ>λ 0. Licesed to Uiv of Ciciati. Pepaed o Mo Aug 7 12:04:45 EDT 2017 fo dowload fom IP 129.137.96.5. Licese o copyight estictios may apply to edistibutio; see http://www.ams.og/licese/jou-dist-licese.pdf

EXPLICIT SOLUTIONS AND MULTIPLICITY RESULTS 9 u(0) 5 4 3 2 1 200 300 400 500 λ Fig. 2. The solutio cuve fo the poblem (3.18) Example 2. We solved umeically the poblem u + 2 u + λ ( u 3 + u 7) =0, u (0) = u(1) = 0. (3.18) Compaed with Example 1, oly the lowe powe is chaged fom 4 to 3. Not oly the solutio cuve, peseted i Figue 2, has a diffeet shape, λ s ae ow much smalle, while u(0) s go highe. We cojectue that thee ae still exactly two positive solutios fo λ lage eough. We tu ext to the p-laplace vesio of Batu s equatio ϕ (u ()) + 1 ϕ (u ()) + Be u =0, (3.19) whee ϕ (z) =z z 1 (i.e., p = ), ad B>0is a costat. Set hee ϕ (u ()) = ae 1 u, whee a>0 is a costat. The u = a 1 1 1 1 1 e u. It follows that ϕ (u ()) = ae 1 u 1 1 ae u u = ae 1 u + 1 a 1 1 e u. We use these expessios i (3.19), ad solve fo u: ( ) a u() = l B + 1 a. (3.20) 1 1 Oe veifies that this fuctio is a solutio of (3.19) fo ay a>0, B>0, ad >1. This family of exact solutios immediately implies the exact cout of solutios fo the coespodig Diichlet poblem o the uit ball i R. Popositio 4. Fo the poblem ϕ (u ()) + 1 ϕ (u ()) + Be u =0, u (0) = u(1) = 0, whee ϕ (z) =z z 1 (i.e., p = ), thee is a costat B() > 0, so that thee ae exactly two solutios fo 0 <B<B(), exactly oe solutio fo B = B(), ad o solutios if B>B(). Licesed to Uiv of Ciciati. Pepaed o Mo Aug 7 12:04:45 EDT 2017 fo dowload fom IP 129.137.96.5. Licese o copyight estictios may apply to edistibutio; see http://www.ams.og/licese/jou-dist-licese.pdf

10 PHILIP KORMAN Poof. Accodig to the fomula (3.20), the bouday coditio u(1)=0isequivalet to a satisfyig 1 a 1 + B = a. O the left we have a covex supeliea fuctio of a, so that thee is a costat B = B(), such that this equatio has two solutios fo 0 <B<B(), oe solutio fo B = B(), ad oe if B>B(). We emak that exact multiplicity esults ae ae fo equatios ivolvig the p- Laplacia. 4. A chage of vaiables. Fo the o-autoomous poblem (hee α, ada>0 ae costats) u + 1 u + α f(u) =0, u(0) = a, u (0) = 0, (4.1) we peset a chage of vaiables which essetially elimiates the o-autoomous tem α (although it chages the spatial dimesio). Popositio 5. Let u() C 2 (0,b) C 1 [0,b] be a solutio of (4.1), with some b>0, ad assume that α> 1. The chage of vaiables t = 1+α/2 1+α/2 (4.1) ito with m = 1+α/2 1+α/2. u (t)+ m t u (t)+f(u(t)) = 0, u(0) = a, tasfoms the poblem du (0) = 0, (4.2) dt Poof. We have u = u t α/2, u = u tt α + α 2 u t α 2 1, ad (4.1) becomes u tt α + α 2 u t α 2 1 +( 1)u t α 2 1 + α f(u) =0. Dividig by α, we get the equatio i (4.2). To see that du dt (0) = 0, we ewite (4.1) as ( 1 u ) + α+ 1 f(u) = 0, ad the expess u () = 1 1 z α+ 1 f(u(z)) dz. 0 We have du u () (0) = lim dt 0 = lim 1 z α+ 1 f(u(z)) dz =0. α/2 0 1+α/2 0 Obseve that i case =2,wehavem = 1 = 1, which meas that the α tem is elimiated without chagig the dimesio. We also emak that fo α 1, we do ot expect the poblem (4.1) to have solutios of class C 2 (0,b) C 1 [0,b], as a explicit example below shows. Example. The poblem u (t)+ 1 t u (t)+e u =0, u(0) = a, u (0) = 0, has a solutio u(t) =a 2l ( 1+ ea 8 t2) goig back to the pape of G. Batu [3] fom 1914 (we wee dealig with this solutio i aothe fom above); see also J. Bebees Licesed to Uiv of Ciciati. Pepaed o Mo Aug 7 12:04:45 EDT 2017 fo dowload fom IP 129.137.96.5. Licese o copyight estictios may apply to edistibutio; see http://www.ams.og/licese/jou-dist-licese.pdf

EXPLICIT SOLUTIONS AND MULTIPLICITY RESULTS 11 ad D. Ebely [2]. (Lettig hee a = l8 ( 3 ± 2 2 ), oe gets two solutios of the coespodig Diichlet poblem o the uit ball, with u(1) = 0.) Settig hee t = 1+α/2 1+α/2, we see that ( ) u() =a 2l e a 1+ 8 ( α 2 +1) 2 α+2 (4.3) is the solutio of the poblem u ()+ 1 u ()+ α e u =0, u(0) = a, u (0) = 0. (4.4) This explicit solutio is of paticula impotace fo sigula equatios, whe α<0, showig us what to expect fo moe geeal o-lieaities tha e u.ithemildly sigula case, whe 1 <α<0, the fuctio i (4.3) is still a solutio of (4.4), although it is ot classical, but oly of class C 1,1+α.Ithestogly sigula case, whe α< 1, the fuctio i (4.3) has ubouded deivative as 0. The case of Coulomb potetial, whe α = 1, is vey special. The coespodig solutio fom (4.3) u() =a 2l (1+ ea2 ) still satisfies u(0) = a, but ot u (0) = 0. Istead, we have u (0) = e a = e u(0). We see that the iitial value poblem u ()+ 1 u ()+ 1 eu =0, u(0) = a, u (0) = e u(0), (4.5) is a atual substitute of the poblem (4.4) i case of the Coulomb potetial. Poblems with the Coulomb potetial occu i applicatios; see J. L. Mazuola et al. [12]. (The applicatio i [12], as well as may othes, ivolve covolutio with Coulomb potetial. Howeve, sigulaities as i (4.5) also occu i applicatios.) We ca ow exted all of the kow multiplicity esults fo autoomous equatios to the o-autoomous equatio (4.1). Fo example, we have the followig esult fo a cubic o-lieaity, which is based o a simila theoem fo α = 0 case; see [10], [13], [9]. Theoem 4.1. Assume that c>2b >0, ad α>0. The thee is a citical λ 0,such that fo λ<λ 0 the poblem u + 1 u + λ α u(u b)(c u) =0, (0, 1), u (0) = u(1) = 0, has o positive solutios, it has exactly oe positive solutio at λ = λ 0,adtheeae exactly two positive solutios fo λ>λ 0. Moeove, all solutios lie o a sigle smooth solutio cuve, which fo λ>λ 0 has two baches, deoted by u (, λ) <u + (, λ), with u + (, λ) stictly mootoe iceasig i λ, ad lim λ u + (, λ) =c fo all [0, 1). Fo the lowe bach, lim λ u (, λ) =0fo 0. (All of the solutios ae classical.) A simila tasfomatio woks fo the p-laplace case ϕ (u ()) + 1 ϕ (u ()) + α f(u()) = 0, u(0) = a, u (0) = 0, (4.6) whee ϕ(z) =z z p 2, with p>1. Licesed to Uiv of Ciciati. Pepaed o Mo Aug 7 12:04:45 EDT 2017 fo dowload fom IP 129.137.96.5. Licese o copyight estictios may apply to edistibutio; see http://www.ams.og/licese/jou-dist-licese.pdf

12 PHILIP KORMAN Popositio 6. Let u() C 2 (0,b) C 1 [0,b] be a solutio of (4.6), with some b>0, ad assume that α> 1. The chage of vaiables t = 1+α/p 1+α/p tasfoms the poblem (4.6) ito ϕ (u (t)) + m t ϕ du (u (t)) + f(u(t)) = 0, u(0) = a, (0) = 0, (4.7) dt with m = 1+α α/p 1+α/p. Poof. We have u = u t α/p, ϕ (u )= α α/p ϕ (u t ), ad d d ϕ (u )=(α α/p) α α/ ϕ (u t )+ α α/p d dt ϕ (u t) α/p, which leads us to (4.7). To see that du dt (0) = 0, we ewite (4.6) as ( 1 ϕ (u ) ) + α+ 1 f(u) = 0, ad the expess [ 1 ] 1 u () = 1 z α+ 1 f(u(z)) dz. (4.8) We have ad by (4.8) [ du u () (0) = lim = lim dt 0 α/p 0 ( u ()) lim 0 α p () = lim 0 1 1+α α/p 0 ( u ()) 0 α p () ] 1 z α+ 1 f(u(z)) dz =0, completig the poof. I case = p, wehavem = 1, which meas that the α tem is elimiated without chagig the dimesio. Refeeces [1] Thiey Aubi, Poblèmes isopéimétiques et espaces de Sobolev (Fech), J. Diffeetial Geomety 11 (1976), o. 4, 573 598. MR0448404 [2] Jeold Bebees ad David Ebely, Mathematical poblems fom combustio theoy, Applied Mathematical Scieces, vol. 83, Spige-Velag, New Yok, 1989. MR1012946 [3] G. Batu, Su les équatios itégales o liéaies (Fech), Bull. Soc. Math. Face 42 (1914), 113 142. MR1504727 [4] Haïm Bézis ad Louis Niebeg, Positive solutios of oliea elliptic equatios ivolvig citical Sobolev expoets, Comm. Pue Appl. Math. 36 (1983), o. 4, 437 477, DOI 10.1002/cpa.3160360405. MR709644 [5] Nassif Ghoussoub ad Yuji Guo, O the patial diffeetial equatios of electostatic MEMS devices: statioay case, SIAM J. Math. Aal. 38 (2006/07), o. 5, 1423 1449, DOI 10.1137/050647803. MR2286013 [6] Zogmig Guo ad Jucheg Wei, Ifiitely may tuig poits fo a elliptic poblem with a sigula o-lieaity, J.Lod.Math.Soc.(2)78 (2008), o. 1, 21 35, DOI 10.1112/jlms/jdm121. MR2427049 [7] Philip Koma, Existece ad uiqueess of solutios fo a class of p-laplace equatios o a ball, Adv. Noliea Stud. 11 (2011), o. 4, 875 888, DOI 10.1515/as-2011-0406. MR2868436 [8] Philip Koma, Global solutio cuves fo self-simila equatios, J. Diffeetial Equatios 257 (2014), o. 7, 2543 2564, DOI 10.1016/j.jde.2014.05.045. MR3228976 [9] Philip Koma, Global solutio cuves fo semiliea elliptic equatios, Wold Scietific Publishig Co. Pte. Ltd., Hackesack, NJ, 2012. MR2954053, Licesed to Uiv of Ciciati. Pepaed o Mo Aug 7 12:04:45 EDT 2017 fo dowload fom IP 129.137.96.5. Licese o copyight estictios may apply to edistibutio; see http://www.ams.og/licese/jou-dist-licese.pdf

EXPLICIT SOLUTIONS AND MULTIPLICITY RESULTS 13 [10] Philip Koma, Yi Li, ad Tiacheg Ouyag, A exact multiplicity esult fo a class of semiliea equatios, Comm. Patial Diffeetial Equatios 22 (1997), o. 3-4, 661 684, DOI 10.1080/03605309708821278. MR1443053 [11] Chag Shou Li ad Wei-Mig Ni, A couteexample to the odal domai cojectue ad a elated semiliea equatio, Poc. Ame. Math. Soc. 102 (1988), o. 2, 271 277, DOI 10.2307/2045874. MR920985 [12] J.L. Mazuola, S.G. Rayo ad G. Simpso, Existece ad stability popeties of adial boud states fo Schödige-Poisso with a exteal Coulomb potetial i thee dimesios, AXiv:1512.03665v2 (2015). [13] Tiacheg Ouyag ad Jupig Shi, Exact multiplicity of positive solutios fo a class of semiliea poblem. II, J. Diffeetial Equatios 158 (1999), o. 1, 94 151, DOI 10.1016/S0022-0396(99)80020-5. MR1721723 [14] Joh A. Pelesko, Mathematical modelig of electostatic MEMS with tailoed dielectic popeties, SIAM J. Appl. Math. 62 (2001/02), o. 3, 888 908, DOI 10.1137/S0036139900381079. MR1897727 [15] Giogio Taleti, Best costat i Sobolev iequality, A. Mat. Pua Appl. (4) 110 (1976), 353 372, DOI 10.1007/BF02418013. MR0463908 Licesed to Uiv of Ciciati. Pepaed o Mo Aug 7 12:04:45 EDT 2017 fo dowload fom IP 129.137.96.5. Licese o copyight estictios may apply to edistibutio; see http://www.ams.og/licese/jou-dist-licese.pdf