- Email: [email protected]
Nonexistence of solutions to Hele-Shaw equations
Nonlinear Analysis 63 (2005) 686 – 691 www.elsevier.com/locate/na
Nonexistence of solutions to Hele-Shaw equations Ming Fanga , R.P. Gilbertb,∗ a Department of Mathematics, Norfolk State University, Norfolk, VA 23504, USA b Department of Mathematical Science, University of Delaware, Newark, DE 19716, USA
Abstract We study nonexistence of the solutions to quasilinear elliptic differential equations arising from nonisothermal, non-Newtonian Hele-Shaw flows. The proof is based on the trial function method developed by Pohozaev without recourse to comparison theorems and to the maximum principle. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: primary 35J85; 35D05; secondary 35R35; 35B45; 35J65; 35K55; 76D27 Keywords: p-Laplace operator; Hele-Shaw flows; Pohozaev identity
1. Introduction In 1992–1996, Gilbert and Shi [7,8] derived the following nonisothermal, non-Newtonian Hele-Shaw equations for injection molding: − = k()|∇p|r + f
in ,
−div{k()|∇p|r−2 ∇p} = g
in ,
(1.1) (1.2)
= 0
on j,
(1.3)
p = p0
on 0 ,
(1.4)
−k()|∇p|r−2
jp =l j
on 1 .
∗ Corresponding author.
E-mail addresses: [email protected] (M. Fang), [email protected] (R.P. Gilbert). 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.02.046
(1.5)
M. Fang, R.P. Gilbert / Nonlinear Analysis 63 (2005) 686 – 691
687
Here they assume that is a bounded domain in R n with a C 1 boundary and j is decomposed into two parts j = 0 ∪ 1 , where 0 and 1 are C 1 manifolds which do not intersect: 0 ∩ 1 = ∅. The outward unit normal of j is denoted by . f , g, k, 0 , p0 , l are given functions, and r is a given positive constant related to the power law index n. ˆ The unknown p is the pressure of the flow and is the temperature. They obtained existence of the weak solutions to certain boundary value problems associated with the above equations. Similar Hele-Shaw equations are also obtained by Aronsson and Evans [2], Fang and Gilbert [4] for compression molding and by Advani and Sozer [1, pp. 275–277], Fang and Gilbert [6] for injection molding. In 1965, Pohozaev [13], established an identity 2 ju x, ds = 2n G(u) dx − (n − 2) g(u) u dx, (1.6) j j where u ∈ C 2 () ∩ C 1 (), for semi-linear elliptic equation with Dirichlet boundary − u = g(u), x ∈ u = 0, x ∈ j.
(1.7)
Here is a bounded domain in R n with a C 1 boundary j. The outward unit normal of u j is denoted by . G(u) = 0 g(t) dt, a, b = i=n i=1 ai bi , where a = (a1 , . . . , an ) and b = (b1 , . . . , bn ). Due to the above identity (1.6), Pohozaev proved that (1.7) admits no positive solution for some starshaped domains. Since then, Figueiredo, Lions, Nussbaum; Pucci and Serrin; Guedda and Veron; Otami; Brezis and Peletier; Ogawa and Suzuki; Schaaf; McGough and Mortensen [5,14,9,12,3,11,15,10] have demonstrated that Pohozaev type identities can be useful not only in nonexistence proof but also in uniqueness, asymptotics, a priori bounds of solutions in different domain geometry. This paper is devoted to establish a generalized Pohozaev Identity and show the absence of solutions for the following quasilinear Hele-Shaw equations: −div{k()|∇p|r−2 ∇p} = g p = p0 on
in ,
0
−k()|∇p|r−2
jp =l j
(1.2) (1.7)
on 1 .
(1.8)
When k() = 1, both Guedda and Veron [9] and Otami [12] have proved in the different ways that Eq. (1.2) with Dirichlet boundary satisfies Pohozaev-type identity. We use Guedda–Veron’s [9] ideas for our Pohozaev-type identity.
2. The Pohozaev-type relation p Theorem 2.1. Assume g(x, p) and Gx are continuous on ×R, where G= 0 kg(x, t) dt, 1,p 0 ∈ and p ∈ L∞ () ∩ W0 () is a solution to (2) and (5). Then the following relation
688
holds:
M. Fang, R.P. Gilbert / Nonlinear Analysis 63 (2005) 686 – 691
n kpg dx [nG(x, p) + x, Gx ] dx + 1 − r n − 1− (k|∇p|2 )(r−2)/2 p∇p, ∇k dx r 1 − x, |∇p|2 ∇k(k|∇p|2 )(r−2)/2 dx 2 1 2 (r−2)/2 + ∇p, ∇k∇p, x(k|∇p| ) dx = 1 − x, (k|∇p|2 )(r−2)/2ds. r j (2.1)
Proof. If the data are not regular enough it is traditional to approximate the above equation by the following equation (where is some positive constant). −div{( + k |∇p |2 )(r−2)/2 ∇p } = g p = p0 on
x ∈ ,
0 ,
−k()|∇p|r−2
jp =l j
(2.2) (2.3)
on 1 .
(2.4)
Tolksdorf [16] has proved that there exist ∈ (0, 1) depending on r and n and C 0 depending on p, n, and g such that for any ∈ (0, 1)
p C 1,() C.
(2.5)
From this estimate, there exists a subsequence (we again label p ) such that {p } −→ p in C 1, () for any 0 < < , as tends to 0. Consider the following C 2 vector field P = ( + k |∇p |2 )(r−2)/2 k ∇p , x∇p .
(2.6)
div P = k ∇p , xdiv[( + k |∇p |2 )(r−2)/2 ∇p ] + ( + k |∇p |2 )(r−2)/2 ∇p , ∇k ∇p , x
(2.7)
Then
and ∇p , ∇k ∇p , x = k |∇p |2 +
k x, ∇|∇p |2 + ∇p , ∇k x, ∇p . 2
(2.8)
Moreover 1 x, k ∇|∇p |2 ( + k |∇p |2 )(r−2)/2 2 1 1 = x, ∇( + k |∇p |2 )r/2 − x, |∇p |2 ∇k ( + k |∇p |2 )(r−2)/2 . r 2
(2.9)
M. Fang, R.P. Gilbert / Nonlinear Analysis 63 (2005) 686 – 691
From Gauss formula we get 1 k x, ∇|∇p |2 ( + k |∇p |2 )(r−2)/2 dx 2 n 1 2 r/2 =− ( + k |∇p | ) dx + x, ( + k |∇p |2 )r/2 ds r r j 1 − x, |∇p |2 ∇k ( + k |∇p |2 )(r−2)/2 dx. 2 Therefore j
689
(2.10)
P , ds =
k ∇p , xdiv[( + k |∇p |2 )(r−2)/2 ∇p ]
n ( + k |∇p |2 )r/2 r + ∇p , ∇k x, ∇p ( + k |∇p |2 )(r−2)/2 1 − x, |∇p |2 ∇k ( + k |∇p |2 )(r−2)/2 ] dx 2 1 x, ( + k |∇p |2 )r/2 ds. + r j + ( + k |∇p |2 )(r−2)/2 k |∇p |2 −
Moreover j
(2.11)
P , ds =
j
( + k |∇p |2 )(r−2)/2 k |∇p |2 x, ds.
Since −div{( + k |∇p |2 )(r−2)/2 ∇p } = g for x ∈ , we obtain n − k ∇p , xg dx + k p g dx + ( + k |∇p |2 )r/2 r 1 − ∇p , ∇k x, ∇p ( + k |∇p |2 )(r−2)/2 2 − ( + k |∇p |2 )(r−2)/2 p ∇p , ∇k dx jp 2 1 1− = − ( + k |∇p |2 )(r−2)/2 ds. x, k j r r j
(2.12)
(2.13)
Since ( + k |∇p |2 )r/2 = ( + k |∇p |2 )(r−2)/2 + div[( + k |∇p |2 )(r−2)/2 k p ∇p ] + k p g − p ( + k |∇p |2 )(r−2)/2 ∇k , ∇p (2.14) and k ∇p , xg = div(xG (x, p ) − nG (x, p ) − x, Gx ),
(2.15)
690
M. Fang, R.P. Gilbert / Nonlinear Analysis 63 (2005) 686 – 691
Fig. 1. Domains without positive solutions, for more geometric domains, see the recent paper by McGough and Mortensen [10].
p p where G = 0 k g (x, t) dt, jG/jxk = 0 (jk g/jxk ) dt and Gx = (Gx1 , . . . , Gxn ). As p vanishes on j and we obtain n [nG (x, p ) + x, Gx ] dx + 1 − k p g dx r n − 1− (k |∇p |2 )(r−2)/2 p ∇p , ∇k dx r 1 − x, |∇p |2 ∇k (k |∇p |2 )(r−2)/2 dx 2 + ∇p , ∇k ∇p , x(k |∇p |2 )(r−2)/2 dx 1 = 1− x, (k|∇p |2 )(r−2)/2 ds. (2.16) r j Let tends to 0, which ends the proof.
As a conclusion, we have the following nonexistence result. Theorem 2.2. Assume g is independent of x and continuous on R and is starshaped with respect to some point. If
n n nG(x, p) + x, Gx + 1 − kpg − 1 − p, ∇k r r 1 + x, ||2 ∇k − , ∇k, x (k||2 )(r−2)/2 0 (2.17) 2 holds for all (x, p, ) ∈ × R × R n with equality only if p = 0 and = 0, then there exists 1,p no nonzero solution of (2) and (6) belonging to L∞ () ∩ W0 () A recent work by McGough and Mortensen [10] pointed out that the “Pohozaev Obstruction” depends on domain geometry. For example, when k=1, r =2, g=ps , and s (n+2)/(n−2),
M. Fang, R.P. Gilbert / Nonlinear Analysis 63 (2005) 686 – 691
691
starshaped domains as shown in Fig. 1 do not have any positive solution. Our paper also proved that solvability of Hele-Shaw equations depends on domain geometry. References [1] S.G. Advani, E.M. Sozer, Process Modeling in Composites Manufacturing, Marcel Dekker, Inc., New York, 2003. [2] G. Aronsson, L.C. Evans, An asymptotic model for compression molding, Indiana Univ. Math. J. 51 (1) (2002) 1–36. [3] H. Brezis, L.A. Peletier, Asymptotics for elliptic equations involving critical exponents, in: F. Columbini, A. Marino, L. Modica, S. Spagnolo (Eds.), Partial Differential Equations and Calculus of Variations, Birkhäuser, Basel, 1989, pp. 149–192. [4] M. Fang, R.P. Gilbert, Compression molding I: a generalized non-isothermal, non-Newtonian Hele-Shaw flow, in: R.P. Agarwal, D.O. Regan (Eds.), Nonlinear Analysis and Applications. To V. Lakshmikantham on his 80th Birthday, Kluwer Academic Publishers, Dordrecht, 2004, pp. 513–531. [5] D.De. Figueiredo, P.L. Lions, R. Nussbaum,A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. 61 (1982) 41–63. [6] R.P. Gilbert, M. Fang, Nonlinear systems arising from non-isothermal, non-Newtonian Hele-Shaw flows in the presence of body forces and sources, Math. Comput. Modeling 35 (2002) 1425–1444. [7] R.P. Gilbert, P. Shi, Nonisothermal, nonNewtonian Hele-Shaw flows, Part I: mathematical formulation, in: Proceeding on Transport Phenomenon, Technomic, Lancaster, 1992, pp. 1067–1088. [8] R.P. Gilbert, P. Shi, Nonisothermal, nonNewtonian Hele-Shaw flows, Part II: asymptotics and existence of weak solution, J. Nonlinear Analysis. Theory, Methods Appl. 27 (1996) 539–559. [9] M. Guedda, L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. TMA 13 (1989) 879–902. [10] Jeff McGough, Jeff Mortensen, Pohozaev obstructions on non-starlike domains, Calc. Var. Partial Differential Equations 18 (2003) 189–205. [11] T. Ogawa, T. Suzuki, Two dimensional elliptic equation with critical nonlinear growth, Trans. Amer. Math. Soc. 350 (1998) 4897–4918. [12] M. Otami, Existence and nonexistence of nontrivial solution of some nonlinear degenerate elliptic equations, J. Funct. Anal. 76 (1988) 140–159. [13] I. Pohozaev, Eigenfunctions of the equation u + f (u) = 0, Dokl. Akad. Nauk. SSSR 165 (1965) 33–36. [14] P. Pucci, A. Serrin, General variational identity, Indiana Univ. Math. J. 35 (1986) 681–703. [15] R. Schaaf, Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry, Adv. Differential Equations 5 (10–12) (2000) 1201–1220. [16] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984) 126–150.
Nonexistence of solutions to Hele-Shaw equations Ming Fanga , R.P. Gilbertb,∗ a Department of Mathematics, Norfolk State University, Norfolk, VA 23504, USA b Department of Mathematical Science, University of Delaware, Newark, DE 19716, USA
Abstract We study nonexistence of the solutions to quasilinear elliptic differential equations arising from nonisothermal, non-Newtonian Hele-Shaw flows. The proof is based on the trial function method developed by Pohozaev without recourse to comparison theorems and to the maximum principle. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: primary 35J85; 35D05; secondary 35R35; 35B45; 35J65; 35K55; 76D27 Keywords: p-Laplace operator; Hele-Shaw flows; Pohozaev identity
1. Introduction In 1992–1996, Gilbert and Shi [7,8] derived the following nonisothermal, non-Newtonian Hele-Shaw equations for injection molding: − = k()|∇p|r + f
in ,
−div{k()|∇p|r−2 ∇p} = g
in ,
(1.1) (1.2)
= 0
on j,
(1.3)
p = p0
on 0 ,
(1.4)
−k()|∇p|r−2
jp =l j
on 1 .
∗ Corresponding author.
E-mail addresses: [email protected] (M. Fang), [email protected] (R.P. Gilbert). 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.02.046
(1.5)
M. Fang, R.P. Gilbert / Nonlinear Analysis 63 (2005) 686 – 691
687
Here they assume that is a bounded domain in R n with a C 1 boundary and j is decomposed into two parts j = 0 ∪ 1 , where 0 and 1 are C 1 manifolds which do not intersect: 0 ∩ 1 = ∅. The outward unit normal of j is denoted by . f , g, k, 0 , p0 , l are given functions, and r is a given positive constant related to the power law index n. ˆ The unknown p is the pressure of the flow and is the temperature. They obtained existence of the weak solutions to certain boundary value problems associated with the above equations. Similar Hele-Shaw equations are also obtained by Aronsson and Evans [2], Fang and Gilbert [4] for compression molding and by Advani and Sozer [1, pp. 275–277], Fang and Gilbert [6] for injection molding. In 1965, Pohozaev [13], established an identity 2 ju x, ds = 2n G(u) dx − (n − 2) g(u) u dx, (1.6) j j where u ∈ C 2 () ∩ C 1 (), for semi-linear elliptic equation with Dirichlet boundary − u = g(u), x ∈ u = 0, x ∈ j.
(1.7)
Here is a bounded domain in R n with a C 1 boundary j. The outward unit normal of u j is denoted by . G(u) = 0 g(t) dt, a, b = i=n i=1 ai bi , where a = (a1 , . . . , an ) and b = (b1 , . . . , bn ). Due to the above identity (1.6), Pohozaev proved that (1.7) admits no positive solution for some starshaped domains. Since then, Figueiredo, Lions, Nussbaum; Pucci and Serrin; Guedda and Veron; Otami; Brezis and Peletier; Ogawa and Suzuki; Schaaf; McGough and Mortensen [5,14,9,12,3,11,15,10] have demonstrated that Pohozaev type identities can be useful not only in nonexistence proof but also in uniqueness, asymptotics, a priori bounds of solutions in different domain geometry. This paper is devoted to establish a generalized Pohozaev Identity and show the absence of solutions for the following quasilinear Hele-Shaw equations: −div{k()|∇p|r−2 ∇p} = g p = p0 on
in ,
0
−k()|∇p|r−2
jp =l j
(1.2) (1.7)
on 1 .
(1.8)
When k() = 1, both Guedda and Veron [9] and Otami [12] have proved in the different ways that Eq. (1.2) with Dirichlet boundary satisfies Pohozaev-type identity. We use Guedda–Veron’s [9] ideas for our Pohozaev-type identity.
2. The Pohozaev-type relation p Theorem 2.1. Assume g(x, p) and Gx are continuous on ×R, where G= 0 kg(x, t) dt, 1,p 0 ∈ and p ∈ L∞ () ∩ W0 () is a solution to (2) and (5). Then the following relation
688
holds:
M. Fang, R.P. Gilbert / Nonlinear Analysis 63 (2005) 686 – 691
n kpg dx [nG(x, p) + x, Gx ] dx + 1 − r n − 1− (k|∇p|2 )(r−2)/2 p∇p, ∇k dx r 1 − x, |∇p|2 ∇k(k|∇p|2 )(r−2)/2 dx 2 1 2 (r−2)/2 + ∇p, ∇k∇p, x(k|∇p| ) dx = 1 − x, (k|∇p|2 )(r−2)/2ds. r j (2.1)
Proof. If the data are not regular enough it is traditional to approximate the above equation by the following equation (where is some positive constant). −div{( + k |∇p |2 )(r−2)/2 ∇p } = g p = p0 on
x ∈ ,
0 ,
−k()|∇p|r−2
jp =l j
(2.2) (2.3)
on 1 .
(2.4)
Tolksdorf [16] has proved that there exist ∈ (0, 1) depending on r and n and C 0 depending on p, n, and g such that for any ∈ (0, 1)
p C 1,() C.
(2.5)
From this estimate, there exists a subsequence (we again label p ) such that {p } −→ p in C 1, () for any 0 < < , as tends to 0. Consider the following C 2 vector field P = ( + k |∇p |2 )(r−2)/2 k ∇p , x∇p .
(2.6)
div P = k ∇p , xdiv[( + k |∇p |2 )(r−2)/2 ∇p ] + ( + k |∇p |2 )(r−2)/2 ∇p , ∇k ∇p , x
(2.7)
Then
and ∇p , ∇k ∇p , x = k |∇p |2 +
k x, ∇|∇p |2 + ∇p , ∇k x, ∇p . 2
(2.8)
Moreover 1 x, k ∇|∇p |2 ( + k |∇p |2 )(r−2)/2 2 1 1 = x, ∇( + k |∇p |2 )r/2 − x, |∇p |2 ∇k ( + k |∇p |2 )(r−2)/2 . r 2
(2.9)
M. Fang, R.P. Gilbert / Nonlinear Analysis 63 (2005) 686 – 691
From Gauss formula we get 1 k x, ∇|∇p |2 ( + k |∇p |2 )(r−2)/2 dx 2 n 1 2 r/2 =− ( + k |∇p | ) dx + x, ( + k |∇p |2 )r/2 ds r r j 1 − x, |∇p |2 ∇k ( + k |∇p |2 )(r−2)/2 dx. 2 Therefore j
689
(2.10)
P , ds =
k ∇p , xdiv[( + k |∇p |2 )(r−2)/2 ∇p ]
n ( + k |∇p |2 )r/2 r + ∇p , ∇k x, ∇p ( + k |∇p |2 )(r−2)/2 1 − x, |∇p |2 ∇k ( + k |∇p |2 )(r−2)/2 ] dx 2 1 x, ( + k |∇p |2 )r/2 ds. + r j + ( + k |∇p |2 )(r−2)/2 k |∇p |2 −
Moreover j
(2.11)
P , ds =
j
( + k |∇p |2 )(r−2)/2 k |∇p |2 x, ds.
Since −div{( + k |∇p |2 )(r−2)/2 ∇p } = g for x ∈ , we obtain n − k ∇p , xg dx + k p g dx + ( + k |∇p |2 )r/2 r 1 − ∇p , ∇k x, ∇p ( + k |∇p |2 )(r−2)/2 2 − ( + k |∇p |2 )(r−2)/2 p ∇p , ∇k dx jp 2 1 1− = − ( + k |∇p |2 )(r−2)/2 ds. x, k j r r j
(2.12)
(2.13)
Since ( + k |∇p |2 )r/2 = ( + k |∇p |2 )(r−2)/2 + div[( + k |∇p |2 )(r−2)/2 k p ∇p ] + k p g − p ( + k |∇p |2 )(r−2)/2 ∇k , ∇p (2.14) and k ∇p , xg = div(xG (x, p ) − nG (x, p ) − x, Gx ),
(2.15)
690
M. Fang, R.P. Gilbert / Nonlinear Analysis 63 (2005) 686 – 691
Fig. 1. Domains without positive solutions, for more geometric domains, see the recent paper by McGough and Mortensen [10].
p p where G = 0 k g (x, t) dt, jG/jxk = 0 (jk g/jxk ) dt and Gx = (Gx1 , . . . , Gxn ). As p vanishes on j and we obtain n [nG (x, p ) + x, Gx ] dx + 1 − k p g dx r n − 1− (k |∇p |2 )(r−2)/2 p ∇p , ∇k dx r 1 − x, |∇p |2 ∇k (k |∇p |2 )(r−2)/2 dx 2 + ∇p , ∇k ∇p , x(k |∇p |2 )(r−2)/2 dx 1 = 1− x, (k|∇p |2 )(r−2)/2 ds. (2.16) r j Let tends to 0, which ends the proof.
As a conclusion, we have the following nonexistence result. Theorem 2.2. Assume g is independent of x and continuous on R and is starshaped with respect to some point. If
n n nG(x, p) + x, Gx + 1 − kpg − 1 − p, ∇k r r 1 + x, ||2 ∇k − , ∇k, x (k||2 )(r−2)/2 0 (2.17) 2 holds for all (x, p, ) ∈ × R × R n with equality only if p = 0 and = 0, then there exists 1,p no nonzero solution of (2) and (6) belonging to L∞ () ∩ W0 () A recent work by McGough and Mortensen [10] pointed out that the “Pohozaev Obstruction” depends on domain geometry. For example, when k=1, r =2, g=ps , and s (n+2)/(n−2),
M. Fang, R.P. Gilbert / Nonlinear Analysis 63 (2005) 686 – 691
691
starshaped domains as shown in Fig. 1 do not have any positive solution. Our paper also proved that solvability of Hele-Shaw equations depends on domain geometry. References [1] S.G. Advani, E.M. Sozer, Process Modeling in Composites Manufacturing, Marcel Dekker, Inc., New York, 2003. [2] G. Aronsson, L.C. Evans, An asymptotic model for compression molding, Indiana Univ. Math. J. 51 (1) (2002) 1–36. [3] H. Brezis, L.A. Peletier, Asymptotics for elliptic equations involving critical exponents, in: F. Columbini, A. Marino, L. Modica, S. Spagnolo (Eds.), Partial Differential Equations and Calculus of Variations, Birkhäuser, Basel, 1989, pp. 149–192. [4] M. Fang, R.P. Gilbert, Compression molding I: a generalized non-isothermal, non-Newtonian Hele-Shaw flow, in: R.P. Agarwal, D.O. Regan (Eds.), Nonlinear Analysis and Applications. To V. Lakshmikantham on his 80th Birthday, Kluwer Academic Publishers, Dordrecht, 2004, pp. 513–531. [5] D.De. Figueiredo, P.L. Lions, R. Nussbaum,A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. 61 (1982) 41–63. [6] R.P. Gilbert, M. Fang, Nonlinear systems arising from non-isothermal, non-Newtonian Hele-Shaw flows in the presence of body forces and sources, Math. Comput. Modeling 35 (2002) 1425–1444. [7] R.P. Gilbert, P. Shi, Nonisothermal, nonNewtonian Hele-Shaw flows, Part I: mathematical formulation, in: Proceeding on Transport Phenomenon, Technomic, Lancaster, 1992, pp. 1067–1088. [8] R.P. Gilbert, P. Shi, Nonisothermal, nonNewtonian Hele-Shaw flows, Part II: asymptotics and existence of weak solution, J. Nonlinear Analysis. Theory, Methods Appl. 27 (1996) 539–559. [9] M. Guedda, L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. TMA 13 (1989) 879–902. [10] Jeff McGough, Jeff Mortensen, Pohozaev obstructions on non-starlike domains, Calc. Var. Partial Differential Equations 18 (2003) 189–205. [11] T. Ogawa, T. Suzuki, Two dimensional elliptic equation with critical nonlinear growth, Trans. Amer. Math. Soc. 350 (1998) 4897–4918. [12] M. Otami, Existence and nonexistence of nontrivial solution of some nonlinear degenerate elliptic equations, J. Funct. Anal. 76 (1988) 140–159. [13] I. Pohozaev, Eigenfunctions of the equation u + f (u) = 0, Dokl. Akad. Nauk. SSSR 165 (1965) 33–36. [14] P. Pucci, A. Serrin, General variational identity, Indiana Univ. Math. J. 35 (1986) 681–703. [15] R. Schaaf, Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry, Adv. Differential Equations 5 (10–12) (2000) 1201–1220. [16] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984) 126–150.