Anisotropic singularities to semilienar elliptic equations in a measure framework
aa r X i v : . [ m a t h . A P ] A p r ANISOTROPIC SINGULARITIES TO SEMI-LINEAR ELLIPTICEQUATIONS IN A MEASURE FRAMEWORK
Huyuan Chen Department of Mathematics, Jiangxi Normal University,Nanchang, Jiangxi 330022, PR China
Abstract.
The purpose of this article is to study very weak solutions ofelliptic equation ( − ∆ u + g ( u ) = 2 k ∂δ ∂x N + jδ in B (0) ,u = 0 on ∂B (0) , (1)where k > j ≥ B (0) denotes the unit ball centered at the origin in R N with N ≥ g : R → R is an odd, nondecreasing and C function, δ is theDirac mass concentrated at the origin and ∂δ ∂x N is defined in the distributionsense that h ∂δ ∂x N , ζ i = ∂ζ (0) ∂x N , ∀ ζ ∈ C ( B (0)) . We obtain that problem (1) admits a unique very weak solution u k,j underthe integral subcritical assumption Z ∞ g ( s ) s − − N +1 N − ds < + ∞ . Furthermore, we prove that u k,j has anisotropic singularity at the origin andwe consider the odd property u k, and limit of { u k, } k as k → ∞ .We pose the constraint on nonlienarity g ( u ) that we only require integra-bility in the principle value sense, due to the singularities only at the origin.This makes us able to search the very weak solutions in a larger scope of thenonlinearity. 1. Introduction
As early as in 1977, Lieb-Simon in [10] studied the very weak solutions to equation − ∆ u + ( u − λ ) + = n X i =1 m i δ a i in R (1.1)in the description of the Thomas-Fermi theory of electric field potential determined by thenuclear charge and distribution of electrons in an atom, where λ ≥ t + = max { t, } , m i > a i ∈ R and δ a i is the Dirac mass at a i for i = 1 , , · · · , n . In fact, the solution of(1.1) turns out to be a classical singular solution of − ∆ u + ( u − λ ) + = 0 in R \ { a , · · · , a n } . [email protected] Subject Classifications: 35R06, 35B40, 35Q60.Key words: Anisotropic singularity; Very weak solution; Uniqueness. H. Chen
As a fundamental PDE’s model, the isolated singular problem − ∆ u + | u | p − u = 0 in Ω \ { } (1.2)has been studied extensively, where Ω is a domain in R N with N ≥
3. Brezis-V´eron in [3]showed that problem (1.2) admits no isolated singular solution when p ≥ NN − . A completeclassification of the isolated singularities at the origin for (1.2) was given by V´eron in [18]when N +1 N − ≤ p < NN − as follows:(i) either | x | p − u ( x ) converges to a constant which can take only two values ± [ p − ( pp − − N )] / ( p − as x → | x | N − u ( x ) converges to a constant c N k , and the couple ( u, k ) is related to the weaksolution of − ∆ u + u p = kδ in (C ∞ c (Ω)) ′ , where c N is the normalized constant of the fundamental solution Γ N of − ∆ u = δ in R N ,that is, Γ N ( x ) = ( c N | x | − N if N ≥ ,c N log | x | if N = 2 . (1.3)For 1 < p < N +1 N − , the above classification holds under the restriction of nonnegative so-lutions of (1.2) and all above singular solutions are isotropic. A conjucture states thatthere is a rich structure of the singularities for (1.2) without the restriction of nonnegativityfor 1 < p < N +1 N − . V´eron in [18] partially answered this conjucture and showed that theanisotropic singular solutions could be constructed by considering the following nonlineareigenvalue problem on S N − − ∆ S N − ω + | ω | p − ω = λω, where S N − is the sphere of unit ball in R N and ∆ S N − is the Laplace-Beltrami operator.Later on, Chen-Matano-V´eron in [5] provideds the anisotropic singular solutions of (1.2) byanalyzing the corresponding Laplace-Beltrami equations in the sphere. More singularitiesanalysis see the references [12, 16, 17, 20].In contrast with the absorption nonlinearity, the isolated singular solutions of ellipticproblem with source nonlinearity ( − ∆ u = u p in Ω \ { } ,u > \ { } , u = 0 on ∂ Ω (1.4)was classified by Lions in [11], by using Schwartz’s Theorem to build that − ∆ u − u p = ∞ X | a | =0 k a D a δ in ( C ∞ c (Ω)) ′ (1.5)and then by choosing suitable test functions in C ∞ c (Ω) to kill all D a δ with a multiple indexand | a | ≥
1, finally building the connections with the weak solutions of ( − ∆ u = u p + kδ in Ω ,u = 0 on ∂ Ω . (1.6)Lions in [11] proved that when p ∈ (1 , NN − ) with N ≥
3, any solution of (1.4) is a weaksolution of (1.6) for some k ≥
0, and when p ≥ NN − , the parameter k = 0. Essentially, D a δ with | a | ≥ nisotropic singular solutions to semi-linear elliptic equations 3 source makes the solutions anisotropic singular. For instance, the fundamental solution of − ∆ u = D a δ with a = (0 , · · · , , ∈ R N is P N ( x ) = ˜ c N x N | x | N , ∀ x ∈ R N \ { } , (1.7)where ˜ c N are the normalized constants, see [4]. Obviously, P N has anisotropic singularities.Inspired by (1.5), we observe that D a δ with | a | ≥ B (0)be the unit ball centered at the origin in R N with N ≥
2, denote by δ the Dirac massconcentrated at the origin, to be convenient, ∂δ ∂x N = D a δ with a = (0 , · · · , , ∈ R N andthen h ∂δ ∂x N , ζ i = ∂ζ (0) ∂x N , ∀ ζ ∈ C ( B (0)) . So our concern is to study the isolated singular solutions of semilinear elliptic problem ( − ∆ u + g ( u ) = 2 k ∂δ ∂x N + jδ in B (0) ,u = 0 on ∂B (0) , (1.8)where parameters k > j ≥ Definition 1.1.
A function u ∈ L ( B (0)) is called a very weak solution of (1.8), if g ( u ) be integrable in the principle value sense near the origin, g ( u ) ∈ L ( B (0) , | x | dx ) and Z B (0) [ u ( − ∆) ξ + g ( u ) ξ ] dx = 2 k ∂ξ (0) ∂x N + jξ (0) , ∀ ξ ∈ C , ( B (0)) . (1.9)We note that k ∂δ ∂x N and jδ are both visible in the distributional identity (1.9). When k = 0, the definition of very weak solution of (1.8) requires g ( u ) ∈ L ( B (0) , ρdx ), see thereferences [19]. Since both sources have the support at the origin, so in this article wepose the constraint for the very weak solution that g ( u ) is integrable in the principle valuesense near the origin means, i.e. lim ǫ → + R B (0) \ B ǫ (0) g ( u ) dx exists, that provides higherpossibility for searching the sign-changing singular solutions of (1.8).Now we are ready to state our first theorem on the existence and asymptotic behavior ofvery weak solutions to problem (1.8). Theorem 1.1.
Assume that k > , j ≥ , Γ N , P N is given in (1.3), (1.7) respectively, andthe nonlinearity g : R → R is an odd, nondecreasing and Lipschitz function satisfying Z ∞ g ( s ) s − − N +1 N − ds < + ∞ (1.10) and g ( s + t ) − g ( s ) ≤ c (cid:20) g ( s )1 + | s | t + g ( t ) (cid:21) , s ∈ R , t ≥ for some c > .Then (1.8) admits a very weak solution u k,j such that ( i ) u k,j ≥ in B +1 (0) = { x = ( x ′ , x N ) ∈ B (0) : x N > } ; ( ii ) u k,j has the following singularity at the origin lim t → + u k,j ( te ) P N ( te ) = 2 k for e = ( e , · · · , e N ) ∈ ∂B (0) , e N = 0 (1.12) H. Chen and lim t → + u k,j ( te )Γ N ( te ) = j for e ∈ ∂B (0) , e N = 0; (1.13)( iii ) u k,j is a classical solution of ( − ∆ u + g ( u ) = 0 in B (0) \ { } ,u = 0 on ∂B (0) . (1.14) In the particular case that j = 0 , denoting u k the solution u k, of (1.8) with j = 0 , u k is x N -odd, that is, u k ( x ′ , x N ) = − u k ( x ′ , − x N ) , ∀ ( x ′ , x N ) ∈ B (0) \ { } . Furthermore, the x N -odd very weak solution is unique. We note that g ( s ) = | s | p − s with p ∈ (0 , NN − ) verifies (1.10) and (1.11). It follows by(1.12) that | u k,j | p − u k,j ∈ L ( B (0)), but for p ∈ [ NN − , N +1 N − ) and k > | u k,j | p − u k,j L ( B (0)). We can’t able to obtain the uniqueness of the very weak solution to (1.8), dueto the failure of application the Kato’s inequality, which requires that the nonlinearity term g ( u ) ∈ L ( B (0)).For the existence of very weak solutions, the normal method is to approximate the Radonmeasure by C functions and consider the limit of the corresponding classical solutions.When k >
0, we use a sequence of Dirac measures δ teN − δ − teN t to approach the source ∂δ ∂x N and in this approximation, the biggest challenge is to find a uniform estimate. To overcomethis difficulty, our strategy is to to consider x N -odd property of solutions when j = 0 toderive the uniform bound in this approaching process.We next state the nonexistence of very weak solution of (1.8). Theorem 1.2.
Assume that k > , j = 0 , g ( s ) = | s | p − s with p ≥ N +1 N − , then there is no x N -odd weak solution for problem (1.8). Our strategy here is to make use of x N − odd property to deduce (1.8) into boundary dataproblem ( − ∆ u + g ( u ) = 0 in B +1 (0) ,u = kδ on ∂B +1 (0) , in the distributional sense that Z B +1 (0) u ( − ∆) ξdx + Z B +1 (0) g ( u ) ξdx = 2 k ∂ξ ( x ) ∂x N , ∀ ξ ∈ C , ( B +1 (0)) . (1.15)It is interesting but still open to derive the nonexistence when j = 0.Finally, we analyze the limit of the weak solutions { u k } k as k → ∞ . From the monotonic-ity of u k in B +1 (0) and B − (0) respectively, the limit of { u k } k as k → ∞ exists in B (0) \ { } ,denote u ∞ ( x ) = lim k →∞ u k ( x ) , ∀ x ∈ B (0) \ { } . (1.16) Theorem 1.3.
Assume that k > , j = 0 , g ( s ) = | s | p − s with p > , u k is the unique x N -odd very weak solution of (1.8) and u ∞ is given by (1.16). Then u ∞ is a classical solutionof ( − ∆ u + | u | p − u = 0 in B (0) \ { } ,u = 0 on ∂B (0) (1.17) nisotropic singular solutions to semi-linear elliptic equations 5 and satisfies that lim t → + u ∞ ( te ) t p − = ϕ ( e ) , ∀ e ∈ ∂B (0) , (1.18) where ϕ : ∂B (0) → R is a continuous x N -odd function such that for unit vector e =( e , · · · , e N ) ϕ ( e ) > e N > . The rest of this paper is organized as follows. In Section 2, we analyze the x N -oddproperty. Section 3 is devoted to study the x N -odd very weak solution in subcritical casewhen j = 0 and the nonexistence the x N -odd very weak solution in the subcritical case. InSection 4, we consider the limit of the unique x N -odd weak solutions u k of (1.8) with j = 0as k → ∞ . Finally, we prove the existence of non x N -odd very weak solution when j > Preliminary
We start this section from the x N -odd property. Notice that an x N -odd function w defined in a x N -symmetric domain B ∗ satisfies w ( x ) = 0 , ∀ x ∈ { ( x ′ , ∈ B ∗ } . In what follows, we denote by c i a generic positive constant. Lemma 2.1.
Assume that f ∈ C ( B (0)) is an x N -odd function, g ∈ C ( R ) is an odd andnondecreasing function.Then ( − ∆ u + g ( u ) = f in B (0) ,u = 0 on ∂B (0) (2.1) admits a unique classical solution w f . Moreover, ( i ) w f is x N -odd in B (0) ; ( ii ) assume more that f ≥ in B +1 (0) = { x ∈ B (0) : x N > } and f in B +1 (0) ,then w f > in B +1 (0) . Proof.
Since g is an odd and nondecreasing function, then it is standard to obtain theexistence of solution by the method of super and sub solutions. Uniqueness.
Let w f , ˜ w f be two solutions of (2.1), w = w f − ˜ w f in B (0) and A + = { x ∈ B (0) : w ( x ) > } . We claim that A + = ∅ . In fact, if A + = ∅ , we observe that w is asolution of ( − ∆ w = g ( ˜ w f ) − g ( w f ) ≤ A + ,w = 0 on ∂A + . By applying Maximum Principle, we have that w ≤ A + , which contradicts the definition of A + . Then A + = ∅ . Similarly, { x ∈ B (0) : w ( x ) < } isempty. Therefore, w f = ˜ w f in B (0) and the uniqueness holds.( i ) Let v ( x ′ , x N ) = − w f ( x ′ , − x N ), and by direct computation, we derive that − ∆ v ( x ) + g ( v ( x )) = − ∆[ − w f ( x ′ , − x N )] + g ( − w f ( x ′ , − x N ))= ∆ w f ( x ′ , − x N ) − g ( w f ( x ′ , − x N ))= − f ( x ′ , − x N ) = f ( x ) , then v is a solution of (2.1). It follows from the uniqueness of solution of (2.1) that w f ( x ′ , x N ) = − w f ( x ′ , − x N ) , ∀ x = ( x ′ , x N ) ∈ B (0) . H. Chen ( ii ) We observe that w f = 0 on ∂B +1 (0) and then w f is a classical solution of ( − ∆ u + g ( u ) = f in B +1 (0) ,u = 0 on ∂B +1 (0) . (2.2)We now claim that w f ≥ B +1 (0). Indeed, if not, we have that min B +1 (0) w f <
0. Let A − = { x ∈ B +1 (0) : w f ( x ) < min B +1 (0) w f } , then ψ := w f + min B +1 (0) w f satisfies ( − ∆ ψ ≥ A − ,ψ = 0 on ∂A − . By Maximum Principle, we have that w f ( x ) ≥
12 min B +1 (0) w f , x ∈ A − , which contradicts the definition of A − .We next prove that w f > B +1 (0). Problem (2.2) could be seen as ( − ∆ u + φu = f in B +1 (0) ,u = 0 on ∂B +1 (0) , where φ ( x ) = g ( w f ( x )) w f ( x ) if w f ( x ) = 0 and φ ( x ) = g ′ (0) if w f ( x ) = 0. It follows by g ∈ C ( R )that φ is continuous and φ ≥ B +1 (0). Since f ≥ B +1 (0), it follows by strongmaximum principle that w f > w f ≡ B +1 (0), then we exclude w f ≡ B +1 (0) bythe fact that f B +1 (0). (cid:3) Corollary 2.1.
Assume that f ∈ C ( B (0)) is an x N -odd function such that f ≥ in B +1 (0) and g ∈ C ( R ) is an odd and nondecreasing function. Let w f be the solution of (2.1) and G B (0) [ f ] be the unique solution of ( − ∆ u = f in B (0) ,u = 0 on ∂B (0) . Then G B (0) [ f ] is x N -odd and ≤ w f ≤ G B (0) [ f ] in B +1 (0) . Proof.
By applying Lemma 2.1 with g ≡
0, we have that G B (0) [ f ] is x N -odd and G B (0) [ f ] ≥ B +1 (0) . Denote v = G B (0) [ f ] − w f , then v = 0 on ∂B +1 (0) and − ∆ v = g ( w f ) ≥
0, by MaximumPrinciple, we have that v ≥ B +1 (0), which ends the proof. (cid:3) Corollary 2.2.
Assume that f ∈ C ( B (0)) is an x N -odd function such that f ≥ in B +1 (0) and g , g ∈ C ( R ) are odd and nondecreasing functions satisfying g ( s ) ≤ g ( s ) , ∀ s ≥ . Let w f,i be the solutions of (2.1) replaced by g by g i with i = 1 , respectively.Then | w f, ( x ) | ≥ | w f, ( x ) | , ∀ x ∈ B (0) . nisotropic singular solutions to semi-linear elliptic equations 7 Proof.
By applying Lemma 2.1 and Corollary 2.1, we have that w f, , w f, are x N -oddand are nonnegative in B +1 (0). We denote w = w f, − w f, , then w satisfies that ( − ∆ w = g ( w f, ) − g ( w f, ) in B +1 (0) ,w = 0 on ∂B +1 (0) . We first claim that w ≥ B +1 (0). If not, we have that min B +1 (0) w <
0. Let us define A − = ( x ∈ B +1 (0) : w ( x ) <
12 min B +1 (0) w ) , then ˜ w := w + min B +1 (0) w satisfies that ( − ∆ ˜ w ≥ A − , ˜ w = 0 on ∂A − . By Maximum Principle, we have that w ( x ) ≥
12 min B +1 (0) w, ∀ x ∈ A − , which contradicts the definition of A − . (cid:3) Proposition 2.1.
Let f and f be x N -odd functions in C loc ( B (0) \ { } ) ∩ L ( B (0) , | x | dx ) satisfying f ≥ f ≥ in B +1 (0) , then the problem ( − ∆ u = f i in B (0) ,u = 0 on ∂B (0) (2.3) admits a unique x N -odd weak solution u i with i = 1 , in the sense that u i ∈ L ( B (0)) , Z B (0) u i ( − ∆) ξdx = Z B (0) ξf i dx, ∀ ξ ∈ C , ( B (0)) , ξ (0) = 0 . (2.4) Moreover, u i is a classical solution of ( − ∆ u = f i in B (0) \ { } ,u = 0 on ∂B (0) (2.5) and ≤ u ( x ) ≤ u ( x ) ≤ Z B +1 (0) c f ( y ) | y || y − x | ( | y − x | + 2 | y | ) N − dy, ∀ x ∈ B +1 (0) . (2.6) Proof.
Uniqueness . Let w satisfy that Z B (0) w ( − ∆) ξdx = 0 , (2.7)for any ξ ∈ C , ( B (0)) such that ξ (0) = 0. Since 0 ∈ L ( B (0)), then the test functioncould be improved into C , ( B (0)) without the restriction that ξ (0) = 0. Denote by η thesolution of ( − ∆ η = sign( w ) in B (0) ,η = 0 on ∂B (0) . (2.8)Then η ∈ C , ( B (0)) and then Z B (0) | w | dx = 0 . H. Chen
This implies w = 0 in B (0) \ { } . Existence . Let f i,ǫ = f i χ B (0) \ B ǫ (0) , where i = 1 , χ B (0) \ B ǫ (0) = 1 in B (0) \ B ǫ (0)and χ B (0) \ B ǫ (0) = 0 in B ǫ (0), then f i,ǫ is an x N -odd function in L ∞ ( B (0)) such that f ,ǫ ≥ f ,ǫ ≥ B +1 (0), then ( − ∆ u = f i,ǫ in B (0) ,u = 0 on ∂B (0) (2.9)admits a unique solution u i,ǫ satisfying Z B (0) u i,ǫ ( − ∆) ξdx = Z B (0) ξf i,ǫ dx, ∀ ξ ∈ C , ( B (0)) , ξ (0) = 0 . (2.10)Moreover, from Lemma 2.1 with g ≡
0, we have that0 ≤ u i,ǫ ≤ u i,ǫ ′ in B +1 (0) for 0 < ǫ ′ ≤ ǫ, ≤ u ,ǫ ≤ u ,ǫ in B +1 (0) for any ǫ ≥ , and for any x ∈ B +1 (0), u ,ǫ ( x ) = Z B (0) G B (0) ( x, y ) f ,ǫ ( y ) dy = Z B +1 (0) [ G B (0) ( x, y ) − G B (0) ( x, ˜ y )] f ,ǫ ( y ) dy, where ˜ y = ( y ′ , − y N ). We observe that G B (0) ( x, y ) = c N | x − y | N − − ˜ G B (0) ( x, y ), where˜ G B (0) ( x, y ) is a harmonic function in B (0) with the boundary value c N | x − y | N − for y ∈ ∂B (0). Therefore, for x, y ∈ B +1 , we have that ˜ G B (0) ( x, y ) − ˜ G B (0) ( x, ˜ y ) ≥ G B (0) ( x, y ) − G B (0) ( x, ˜ y ) ≤ c (cid:20) | y − x | N − − | ˜ y − x | N − (cid:21) . Moreover, we see that | ˜ y − x | ≤ | y − x | + 2 y N ≤ | y − x | + 2 | y | and 0 ≤ | y − x | N − − | ˜ y − x | N − ≤ | y − x | N − − | y − x | + 2 | y | ) N − = ( | y − x | + 2 | y | ) N − − | y − x | N − | y − x | N − ( | y − x | + 2 | y | ) N − ≤ c | y || y − x | ( | y − x | + 2 | y | ) N − , then u ( x ) ≤ c Z B +1 (0) | y | f ( y ) | y − x | ( | y − x | + 2 | y | ) N − dy. Therefore, we obtain a uniform bound for u ,ǫ , together with monotonicity, then passingto the limit as ǫ → + in (2.10), we deduces that u i := lim ǫ → + u i,ǫ is a weak solution of(2.3) and it follows from standard stability theorem that u i is a classical solution of (2.5)for i = 1 , (cid:3) By direct extension, we have the following corollary. nisotropic singular solutions to semi-linear elliptic equations 9
Corollary 2.3.
Let f be an x N -odd function in C loc ( B (0) \ { } ) ∩ L ( B (0) , | x | i dx ) with i ∈ Z satisfying f ≥ in B +1 (0) , then the problem ( − ∆ u = f in B (0) ,u = 0 on ∂B (0) admits a unique x N -odd weak solution u f with i ∈ N , i ≥ in the sense that u f ∈ L ( B (0) , | x | i − dx ) , Z B (0) u ( − ∆) ξdx = Z B (0) ξf dx, for any ξ ∈ C , ( B (0)) s. t. | ξ ( x ) | ≤ c | x | i for any x ∈ B (0) and some c > . Moreover, u f ( x ) ≥ , ∀ x ∈ B +1 (0) . and u f is a classical solution of ( − ∆ u = f in B (0) \ { } ,u = 0 on ∂B (0) . Remark 2.1.
The arguments in Proposition 2.1 and Corollary 2.3 hold when B (0) isreplaced by R N and the boundary condition is done by lim | x |→ + ∞ u ( x ) = 0 . In order to study the convergence of weak solutions, we recall the definition and basicproperties of the Marcinkiewicz spaces.
Definition 2.1.
Let Ω ⊂ R N be a domain and µ be a positive Borel measure in Ω . For κ > , κ ′ = κ/ ( κ − and u ∈ L loc (Ω , dµ ) , we set k u k M κ (Ω ,dµ ) = inf { c ∈ [0 , ∞ ] : R E | u | dµ ≤ c (cid:0)R E dµ (cid:1) κ ′ , ∀ E ⊂ Ω Borel set } and M κ (Ω , dµ ) = { u ∈ L loc (Ω , dµ ) : k u k M κ (Ω ,dµ ) < ∞} . (2.11) M κ (Ω , dµ ) is called the Marcinkiewicz space with exponent κ or weak L κ space and k . k M κ (Ω ,dµ ) is a quasi-norm. The following property holds. Proposition 2.2. [1]
Assume that ≤ q < κ < ∞ and u ∈ L loc (Ω , dµ ) . Then there exists C ( q, κ ) > such that Z E | u | q dµ ≤ C ( q, κ ) k u k M κ (Ω ,dµ ) (cid:18)Z E dµ (cid:19) − q/κ , for any Borel set E of Ω . x N -odd very weak solution with j = 03.1. Existence of very weak solution.
In this subsection, we prove the existence anduniqueness of very weak solution to problem (1.8) when j = 0. Theorem 3.1.
Assume that k > j = 0 , the nonlinearity g : R → R is an odd, nonde-creasing and Lipchitz continuous function satisfying (1.10).Then (1.8) admits a unique x N -odd very weak solution w k such that ( i ) w k ′ ≥ w k ≥ in B +1 (0) for k ′ ≥ k > ; ( ii ) w k satisfies (1.12); ( iii ) w k is a classical solution of (1.14). H. Chen
Before proving Theorem 3.1, we need following preliminaries.
Lemma 3.1.
Assume that k > , the nonlinearity g : R → R is an odd, nondecreasingand Lipchitz continuous function and u is a very weak solution of (1.8), locally bounded in B (0) \ { } . Then u is a classical solution of (1.14). Proof.
Since ∂δ ∂x N , δ have the support in { } , so for any open sets O , O in B (0) suchthat ¯ O ⊂ O ⊂ ¯ O ⊂ B (0) \ { } , u is a very weak solution of − ∆ u + g ( u ) = 0 in O , (3.1)where u ∈ L ∞ ( O ) and g ( u ) ∈ L ∞ ( O ). By standard regularity results, we have that u satisfies (3.1) in O in the classical sense. (cid:3) For n ∈ N , we consider { g n } n of C , odd, nondecreasing functions defined in R satisfying g n ≤ g n +1 ≤ g in R + , sup s ∈ R + g n ( s ) = n and lim n →∞ k g n − g k L ∞ loc ( R ) = 0 . (3.2) Proposition 3.1.
Let g n be defined by (3.2) and µ t = δ te N − δ − te N t , t ∈ (0 , . Then for any n ∈ N , problem ( − ∆ u + g n ( u ) = kµ t in B (0) ,u = 0 on ∂B (0) (3.3) admits a unique very weak solution w k,n,t , which is a classical solution of ( − ∆ u + g n ( u ) = 0 in B (0) \ { te N , − te N } ,u = 0 on ∂B (0) . Moreover, w k,n,t is x N -odd in B (0) \ { te N , − te N } , ≤ w k,n +1 ,t ≤ w k,n,t ≤ G B (0) [ µ t ] in B +1 (0) \ { te N } and w k +1 ,n,t ≥ w k,n,t in B +1 (0) \ { te N } . (3.4) Proof.
We observe that µ t is a bounded Radon measure and g n is bounded, Lipschitzcontinuous and nondecreasing, then it follows from [19, Theorem 3.7] under the integralsubcritical assumption (1.10) replaced NN − by NN − for N ≥ w k,n,t . Moreover, w k,n,t could be approximatedby the classical solutions { w k,n,t,m } to problem ( − ∆ u + g n ( u ) = kµ t,m in B (0) ,u = 0 on ∂B (0) , (3.5)where µ t,m ( x ) = σ m ( x − te N ) − σ m ( x + te N ) t and { σ m } m is a sequence of radially symmetric, nondecreasing smooth functions convergingto δ in the distribution sense. Furthermore, Z B (0) [ w k,n,t,m ( − ∆) ξ + g n ( w k,n,t,m ) ξ ] dx = k Z B (0) µ t,m ξdx, ∀ ξ ∈ C , ( B (0)) . (3.6) nisotropic singular solutions to semi-linear elliptic equations 11 Since µ t,m is x N -odd and nonnegative in B +1 (0), so is w k,n,t,m by Lemma 2.1. We observethat ( k + 1) σ m ≥ kσ m , it follows from Lemma 2.1 that w k +1 ,n,t,m ≥ w k,n,t,m in B +1 (0) . Since g n ( s ) s ≤ g n +1 ( s ) s, ∀ s ∈ R , then it follows from Corollary 2.2 that w k,n,t,m ≤ w k,n +1 ,t,m in B +1 (0) . (3.7)From the proof of Theorem 2.9 in [19], we know that G B (0) [ µ t,m ] → G B (0) [ µ t ] in B (0) \ { te N , − te N } and in L q ( B (0)) as m → ∞ , where q ∈ [1 , NN − ). By regularity results, any compact set K and open set O in B (0) suchthat K ⊂ O , ¯ O ∩ { te N , − te N } = ∅ , there exist c , c > m such that k w k,n,t,m k C ( K ) ≤ c k k G B (0) [ µ t,m ] k L ∞ ( O ) ≤ c k k G B (0) [ µ t ] k L ∞ ( O ) . Therefore, up to some subsequence, there exists a measurable function ˜ w such that w k,n,t,m → ˜ w in B (0) \ { te N , − te N } and in L q ( B (0)) as m → ∞ , where q ∈ [1 , NN − ). Then ˜ w is x N -odd, nonnegative in B +1 (0) and g n ( w k,n,t,m ) → g n ( ˜ w ) in B (0) \ { te N , − te N } and in L ( B (0)) as m → ∞ . Passing to the limit in (3.6) as m → ∞ , we deduce that ˜ w is a weak solution of (3.3). By theuniqueness of weak solution of (3.3), we obtain that w n,t = ˜ w . Therefore, w n,t is x N -odd,nonnegative in B +1 (0) and it follows from (3.7) and (3.4) that w k,n,t ≤ w k,n +1 ,t in B +1 (0)and w k +1 ,n,t ≥ w k,n,t in B +1 (0) . This ends the proof. (cid:3)
We next passing to the limit of weak solutions as t → + . Proposition 3.2.
Let g n be defined by (3.2). Then for any n ∈ N , problem ( − ∆ u + g n ( u ) = 2 k ∂δ ∂x N in B (0) ,u = 0 on ∂B (0) (3.8) admits a unique very weak solution w k,n . Moreover, ( i ) w k,n is x N -odd for any n ∈ N in B (0) \ { } and ≤ w k,n +1 ≤ w k,n ≤ k G B (0) [ ∂δ ∂x N ] in B +1 (0) and ≤ w k,n ≤ w k +1 ,n in B +1 (0);( ii ) w k,n is a classical solution of ( − ∆ u + g n ( u ) = 0 in B (0) \ { } ,u = 0 on ∂B (0) . H. Chen
Proof.
It follows from Proposition 3.1 that problem (3.3) admits a unique very weaksolution w k,n,t , that is, Z B (0) [ w k,n,t ( − ∆) ξ + g n ( w k,n,t )] dx = k ξ ( te N ) − ξ ( − te N ) t , ∀ ξ ∈ C , ( B (0)) . (3.9)On the one hand, we have thatlim t → + ξ ( te N ) − ξ ( − te N ) t = 2 ∂ξ (0) ∂x N . On the other hand, by Proposition 3.1, we have that | w k,n,t | ≤ k | G B (0) [ µ t ] | in B (0) . By regularity results, for σ ∈ (0 ,
1) and any compact set K and open set O in B (0) suchthat K ⊂ O , ¯ O ∩ { te N : t ∈ ( − , ) } = ∅ , there exist c , c > t such that k w k,n,t k C σ ( K ) ≤ c k k G B (0) [ µ t ] k L ∞ ( O ) ≤ c k k G B (0) [ ∂δ ∂x N ] k L ∞ ( O ) . Moreover, by [4, Proposition 3.3], { G B (0) [ µ t ] } is uniformly bounded in M NN − ( B (0) , dx )if N ≥ M σ ( B (0) , dx ) for any σ ∈ (0 , ) if N = 2.Therefore, { w k,n,t } t is relatively compact in L p ( B (0)) for any p ∈ [1 , NN − ). There exists w k,n ∈ L ( B (0)) such that w k,n,t → w k,n a . e . in B (0) and in L ( B (0)) , which implies that g n ( w k,n,t ) → g n ( w k,n ) a . e . in B (0) and in L ( B (0)) as t → + . Therefore, up to some subsequence, passing to the limit as t → + in the identity (3.9), itfollows that w k,n is a very weak solution of (3.8). Moreover, w k,n is x N -odd and nonnegativein B +1 (0). Uniqueness.
Let v n be a weak solution of (3.8) and then ϕ n := w k,n − v n is a very weaksolution to ( − ∆ ϕ n + g n ( w k,n ) − g n ( v n ) = 0 in B (0) ,ϕ n = 0 on ∂B (0) . By Kato’s inequality [19, Theorem 2.4] (see also [8, 9, 17]), Z B (0) | ϕ n | ( − ∆) ξ + Z B (0) [ g n ( w k,n ) − g ( v n )]sign( w k,n − v n ) ξ dx ≤ ξ = G B (0) [1], we have that Z B (0) [ g n ( w k,n ) − g ( v n )]sign( w k,n − v n ) ξ dx ≥ Z B (0) | ϕ n | dx = 0 , then ϕ n = 0 a.e. in B (0). Then the uniqueness is obtained. (cid:3) The next estimate plays an important role in w k,n → w k in L p ( B (0)) with p ∈ [1 , N +1 N − ). Lemma 3.2.
There exists c > such that k G B (0) [ ∂δ ∂x N ] k M NN − ( B (0)) ≤ c (3.10) and k G B (0) [ ∂δ ∂x N ] k M N +1 N − ( B (0) , | x | dx ) ≤ c . (3.11) nisotropic singular solutions to semi-linear elliptic equations 13 Proof.
We observe that G B (0) [ ∂δ ∂x N ]( x ) = ∂G B (0) ( x, ∂x N and for x, y ∈ B (0), x = y , G B (0) ( x, y ) = ( c N | x − y | − N + ˜ G B (0) ( x, y ) if N ≥ , − c N log | x − y | + ˜ G B (0) ( x, y ) if N = 2 , where ˜ G B (0) is a harmonic function in B (0) × B (0). Then (cid:12)(cid:12)(cid:12)(cid:12) ∂G B (0) ( x, ∂x N (cid:12)(cid:12)(cid:12)(cid:12) ≤ c | x N || x | N + c . Therefore, we have that (cid:12)(cid:12)(cid:12)(cid:12) G B (0) [ ∂δ ∂x N ] (cid:12)(cid:12)(cid:12)(cid:12) ≤ c | x | N − , ∀ x ∈ B (0) \ { } . Proof of (3.10).
Let E be a Borel set of B (0) with | E | >
0, then there exists r ∈ (0 , | E | = | B r (0) | . We deduce that Z E | G B (0) [ ∂δ ∂x N ] | dx = Z E ∩ B r (0) c | x | N − dx + Z E \ B r (0) c | x | N − dx ≤ Z B r (0) c | x | N − dx = c r = c (cid:18)Z E | x | dx (cid:19) N . By the definition of Marcinkiewicz space, we have that k G B (0) [ ∂δ ∂x N ] k M NN − ( B (0) , dx ) ≤ c . Proof of (3.11).
Let E be a Borel set of B (0) with | E | >
0, then there exists r ∈ (0 , Z E | x | dx = Z B r (0) | x | dx. Since Z B r (0) | x | dx = c r N +12 , we deduce that Z E | G B (0) [ ∂δ ∂x N ] || x | dx ≤ Z E c | x | N − | x | dx ≤ c r = c ( Z E | x | dx ) N +1 ≤ c . This ends the proof. (cid:3)
Lemma 3.3.
Assume that g : [0 , ∞ ) → [0 , ∞ ) is continuous, nondecreasing and verifies(1.10). Then for q ≥ N +1 N − , lim s →∞ g ( s ) s − q = 0 . H. Chen
Proof.
Since Z ss g ( t ) t − − k α,β dt ≥ g ( s )(2 s ) − − N +1 N − Z ss dt = 2 − − N +1 N − g ( s ) s − N +1 N − and by (1.7), lim s →∞ Z ss g ( t ) t − − N +1 N − dt = 0 , then lim s →∞ g ( s ) s − N +1 N − = 0 . The proof is complete. (cid:3)
Now we are ready to prove Theorem 3.1.
Proof Of Theorem 3.1.
Existence.
Let { g n } be a sequence of C nondecreasing func-tions defined by (3.2). It follows that { g n } is a sequence of odd, bounded and nondecreasingLipschiz continuous functions.By Proposition 3.2, problem (3.8) admits a unique x N -odd weak solution w k,n such that0 ≤ w k,n ≤ k G B (0) [ ∂δ ∂x N ] a . e . in B +1 (0) (3.12)and Z B (0) [ w k,n ( − ∆) ξ + g n ( w k,n ) ξ ] dx = 2 k ∂ξ (0) ∂x N , ∀ ξ ∈ C , ( B (0)) . (3.13)For β ∈ (0 , K and open set O in B (0) satisfying K ⊂ O , 0 ¯ O , wehave that k w k,n k C ,β ( K ) ≤ c k G B (0) [ ∂δ ∂x N ] k C ( O ) , where c >
0. Therefore, up to some subsequence, there exists w k such thatlim n → + w k,n = w k a . e . in B (0) . Then { g n ( w k,n ) } converges to g ( w k ) a.e. in B (0). By Lemma 3.2, we have that w k,n → w k in L ( B (0)) as n → + ∞ , k g n ( w k,n ) k L ( B (0) , | x | dx ) ≤ c k G B (0) [ ∂δ ∂x N ] k L ( B (0) , | x | dx ) , by Proposition 2.2 and G B (0) [ ∂δ ∂x N ] ∈ M N +1 N − ( B (0) , | x | dx ), we have that m ( λ ) ≤ c λ − N +1 N − , ∀ λ > λ , where m ( λ ) = Z S λ | x | dx with S λ = (cid:26) x ∈ B (0) : (cid:12)(cid:12)(cid:12)(cid:12) G B (0) [ ∂δ ∂x N ] (cid:12)(cid:12)(cid:12)(cid:12) > λ (cid:27) . nisotropic singular solutions to semi-linear elliptic equations 15 For any Borel set E ⊂ B (0), we have that Z E | g n ( w k,n ) || x | dx ≤ Z E ∩ S cλ k g (cid:18) k (cid:12)(cid:12)(cid:12)(cid:12) G B (0) [ ∂δ ∂x N ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) | x | dx + Z E ∩ S λ k g (cid:18) k (cid:12)(cid:12)(cid:12)(cid:12) G B (0) [ ∂δ ∂x N ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) | x | dx ≤ g ( λ ) Z E | x | dx + Z S λ k g (cid:18) k (cid:12)(cid:12)(cid:12)(cid:12) G B (0) [ ∂δ ∂x N ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) | x | dx ≤ g ( λ ) Z E | x | dx + m (cid:18) λ k (cid:19) g ( λ ) + Z ∞ λ k m ( s ) dg (2 ks ) . On the other hand, Z ∞ λ k g (2 ks ) dm ( s ) = lim T →∞ Z T kλ k g (2 ks ) dm ( s ) . Thus, m (cid:18) λ k (cid:19) g ( λ ) + Z T kλ k m ( s ) dg (2 ks ) ≤ c g ( λ ) (cid:18) λ k (cid:19) − N +1 N − + c Z T kλ k s − N +1 N − dg (2 ks )= c T − N +1 N − g ( T ) + c Z T kλ k s − − N +1 N − g ( s ) ds, where c = c N +1 N − +1 (2 k ) N +1 N − . By assumption (1.10) and Lemma 3.3, we have that T − N +1 N − g ( T ) → T → ∞ , therefore, m (cid:18) λ k (cid:19) g ( λ ) + Z ∞ λ k m ( s ) dg (2 ks ) ≤ c Z ∞ λk s − − N +1 N − g ( s ) ds. Notice that the quantity on the right-hand side tends to 0 when λ → ∞ . The conclusionfollows: for any ǫ >
0, there exists λ > c Z ∞ λ k s − − N +1 N − g ( s ) ds ≤ ǫ . For λ fixed, there exists δ > Z E | x | dx ≤ δ = ⇒ g ( λ ) Z E | x | dx ≤ ǫ , which implies that { g n ◦ w k,n } is uniformly integrable in L ( B (0) , | x | dx ). Then g n ◦ w k,n → g ◦ w k in L ( B (0) , | x | dx ) by Vitali convergence theorem, see [7].Furthermore, for any ξ ∈ C , ( B (0)), we know that | ξ ( x ) − ξ (0) − ∇ ξ (0) · x | ≤ c | x | , (3.14)and it follows the odd prosperity of w k,n , g n and g , that Z B (0) g n ( w k,n ) ξ (0) dx = 0 and Z B (0) g ( w k ) ξ (0) dx = 0 , (3.15) H. Chen then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B (0) g n ( w k,n ) ξ dx − Z B (0) g ( w k ) ξ dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B (0) [ g n ( w k,n ) − g ( w k )] ∇ ξ (0) · x dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + c Z B (0) | g n ( w k,n ) − g ( w k ) || x | dx ≤ c k g n ( w k,n ) − g ( w k ) k L ( B (0) , | x | dx ) → n → + ∞ . Then passing to the limit as n → + ∞ in the identity (3.13), it implies that Z B (0) [ w k ( − ∆) ξ + g ( w k ) ξ ] dx = 2 k ∂ξ (0) ∂x N . (3.16)Thus, w k is a very weak solution of (1.8). The regularity results follows by Lemma 3.1. Proof of ( i ) . Since w k,n is x N -odd in B (0) \ { } and w k = lim n → + ∞ w k,n in B (0) \ { } ,then it implies that w k is x N -odd in B (0) \ { } . By the fact that w k +1 ,n ≥ w k,n ≥ B +1 (0), it follows that w k +1 ≥ w k ≥ B +1 (0) . Proof of ( ii ) . We observe that0 ≤ w k,n ≤ k G B (0) [ ∂δ ∂x N ] in B +1 (0) , then g n ( w k,n ) ≤ g (2 k G B (0) [ ∂δ ∂x N ]) for x ∈ B +1 (0) and let w g be the unique solution of (2.3)with f = g (2 k G B (0) [ ∂δ ∂x N ]), we have that2 k G B (0) [ ∂δ ∂x N ]( x ) ≥ w k,n ( x ) ≥ k G B (0) [ ∂δ ∂x N ]( x ) − w g ( x ) . (3.17)Then w k satisfies (3.17). From Proposition 2.1, it infers that for x = te with t ∈ (0 , ) e = ( e , · · · , e N ) ∈ ∂B (0) with e N > w g ( te ) ≤ Z B +1 (0) (cid:20) | y || y − te | ( | y − te | + 2 | y | ) N − (cid:21) g (cid:18) k | G B (0) [ ∂δ ∂x N ]( y ) | (cid:19) dy ≤ Z B +1 (0) (cid:20) | y || y − te | ( | y − te | + 2 | y | ) N − (cid:21) g (2 c k | y | − N ) dy := Z B +1 (0) A ( t, y ) dy. For y ∈ B t ( te ), we have that | y − te | ≥ t and t ≤ | y | ≤ t , then t N − Z B t ( te ) A ( t, y ) dy ≤ c r − N +1 N − g (2 c kr ) → r → + ∞ , where r = t − N − . nisotropic singular solutions to semi-linear elliptic equations 17 For y ∈ B t (0), we have that | y − te | ≥ t and t N − Z B + t (0) A ( t, y ) dy ≤ Z B t (0) | y | g (2 c k | y | − N ) dy = c Z t g (2 c ks − N ) s N ds = c Z ∞ ( t ) − N − g (2 c kτ ) τ − − N +1 N − dτ → t → , where we have used (1.10).For y ∈ B +1 (0) \ ( B t (0) ∪ B t ( te )), we have that | y − te | ≥ t , | y | ≥ t and t N − Z B +1 (0) \ ( B t (0) ∪ B t ( te )) A ( t, y ) dy ≤ c t Z B +1 (0) \ ( B t (0) ∪ B t ( te )) g (2 c k | y | − N ) dy = c t Z t g (2 c kr − N ) r N − dr = c t N − g (2 c kt − N ) t − = c τ − N +1 N − g ( τ ) → τ → + ∞ where τ = 2 c kt − N . Then for any e = ( e , · · · , e N ) ∈ ∂B (0) and e N = 0, we have thatlim t → + t N − w g ( te ) = 0 (3.18)and lim t → + G B (0) [ g ( G B (0) [2 k ∂δ ∂x N ])]( te ) t N − = 0 . By (3.17), lim t → + G B (0) [ ∂δ ∂x N ]( te ) t N − = e N , and x N -odd property of w k , we derive that for any e ∈ ∂B (0),lim t → + w k ( te ) t N − = 2 ke N . Finally, we prove the uniqueness. Let u k , v k be two x N − odd solutions of (1.8), fromLemma 3.4, u k , v k are two solutions of ( − ∆ u + g ( u ) = 0 in B +1 (0) ,u = kδ on ∂B +1 (0) . (3.19)Form the uniqueness of the very weak solution to (3.19), see [14, Theorem 2.1], we obtain u k = v k in B +1 (0) and combine the x N − odd property we obtain the uniqueness of the veryweak solution of (1.8) with j = 0. This ends the proof. (cid:3) Nonexistence.
This subsection is devoted to obtain the nonexistence of very weaksolutions of (1.8) in the supercritical case.
Lemma 3.4.
Assume that k > , the nonlinearity g : R → R is an odd, nondecreasing andLipchitz continuous function. Let u be an x N -odd very weak solution of (1.8).Then u is a very weak solution of (3.19). H. Chen
Proof.
From Lemma 3.1 and the x N -odd property, we have that u = 0 on ∂B +1 (0) \ { } . For any x N -odd function ξ ∈ C , ( B (0)), we deduce that ξ = 0 on ∂B +1 (0) | ξ ( x ) | ≤ c | x | , x ∈ B (0) , where c >
0. Since g ( u ) ∈ L ( B (0) , | x | dx ), then for x N -odd function ξ ∈ C , ( B (0)), wehave that g ( u ) ξ ∈ L ( B (0)) and the weak solution u satisfies that Z B (0) [ u ( − ∆) ξdx + g ( u ) ξ ] dx = 2 k ∂ξ (0) ∂x N , (3.20)for x N − odd function ξ in C , ( B (0)). By x N -odd property, we have that Z B +1 (0) [ u ( − ∆) ξdx + g ( u ) ξ ] dx = Z B − (0) [ u ( − ∆) ξdx + g ( u ) ξ ] dx, which, combined with (3.20), implies that Z B +1 (0) [ u ( − ∆) ξdx + g ( u ) ξ ] dx = k ∂ξ (0) ∂x N , ∀ ξ ∈ C , ( B +1 (0)) . So we have that u is a weak solution of (3.19). (cid:3) Proof of Theorem 1.2.
When g ( s ) = | s | p − s with p ≥ N +1 N − and k >
0, [14, Theorem 3.1]shows that the nonnegative solution of ( − ∆ u + g ( u ) = 0 in B +1 (0) ,u = 0 on ∂B +1 (0) \ { } has removable singularity at origin, so problem (3.19) has no very weak solution, whichcontradicts Lemma 3.4. Then (1.8) has no x N -odd weak solution when g ( s ) = | s | p − s with p ≥ N +1 N − . (cid:3) Strongly anisotropic singularity for p ∈ (1 , N +1 N − )In this section, we consider the limit of weak solutions w k as k → ∞ to ( − ∆ u + | u | p − u = 2 k ∂δ ∂x N in B (0) ,u = 0 on ∂B (0) , (4.1)where p ∈ (1 , N +1 N − ). By Theorem 1.1, we observe that the mapping k w k is nondecreasingin B +1 (0), then lim k → + ∞ w k ( x ) exists for any x ∈ B (0) \ { } , denoting w ∞ ( x ) = lim k → + ∞ w k ( x ) for x ∈ B (0) \ { } . (4.2)For w ∞ , we have the following result. Proposition 4.1.
Let p ∈ (1 , N +1 N − ) , then w ∞ is x N -odd, ≤ w ∞ ( x ) ≤ λ | x | − p − , ∀ x ∈ B +1 (0) for some λ > and w ∞ is a classical solution of ( − ∆ u + | u | p − u = 0 in B (0) \ { } ,u = 0 on ∂B (0) . (4.3) nisotropic singular solutions to semi-linear elliptic equations 19 Proof.
In order to obtain the asymptotic behavior of w ∞ near the origin, we constructthe function v p ( x ) = | x | − p − , ∀ x ∈ R N \ { } . For p ∈ (1 , N +1 N − ), there exists λ > λ v p is a super solution of (4.3) and − λ v p is a sub solution of (4.3).It follows by Theorem 1.1 that w k is a classical solution of (4.3) satisfying (1.12), henceby Comparison Principle, for any k , there exists r k ∈ (0 ,
1) small enough such that | w k | ≤ λ v p in B r k (0) \ { } . By Comparison Principle, we have that | w k | ≤ λ v p in B (0) \ { } . Since k is arbitrary, we deduce that | w ∞ | ≤ λ v p in B (0) \ { } . Therefore, from standard Stability Theorem, we derive that w ∞ is a classical solution of(4.3). (cid:3) We next do a precise bound for w ∞ to prove (1.18). Lemma 4.1.
Let p ∈ (1 , N +1 N − ) and w ∞ be defined by (4.2), then w ∞ satisfies (1.18). Proof.
We claim that c t p − P N ( e ) ≤ w ∞ ( te ) ≤ c t p − P N ( e ) , ∀ t > , e = ( e , · · · , e N ) ∈ ∂B (0) , e N > . (4.4)We have that2 k G B (0) [ ∂δ ∂x N ]( x ) ≥ w k ( x ) ≥ k G B (0) [ ∂δ ∂x N ]( x ) − ϕ p ( x ) , ∀ x ∈ B +1 (0) , (4.5)where ϕ p := k p G R N [ G R N [ ∂δ ∂x N ] p ] ≥ G B (0) [ w pk ], and ϕ p is a x N -odd solution of − ∆ u = ˜ c pN | x N | p − x N | x | Np in R N \ { } , lim | x |→ + ∞ u ( x ) = 0 . (4.6)Indeed, G R N [ G R N [ ∂δ ∂x N ] p ] is x N − odd and ϕ p ( x ) = c N ˜ c pN Z R N | y N | p − y N | x − y | N − | y | Np dy = c N ˜ c pN | x | (1 − N ) p +2 Z R N | y N | p − y N | e z − y | N − | y | Np dx, where e z = z | z | .We observe that ϕ p satisfies − ∆ u ( x ) = ˜ c pN | x N | p − x N | x | Np , ∀ x ∈ B + (0) \ B + (0) , and then it follows by Hopf’s Lemma (see [6]) that ϕ p ( x ) ≤ c x N , ∀ x ∈ ∂B +1 (0) . Therefore, ϕ p ( e ) ≤ c P N ( e ) , ∀ e ∈ ∂B +1 (0) . H. Chen
Proof of lower bound in (4.4).
It follows by (4.5) that w k ( x ) ≥ c k | x | − N P N ( x | x | ) − c k p | x | (1 − N ) p +2 P N ( x | x | ) , ∀ x ∈ B (0) \ { } . Set ρ k = (2 ( N − p − − c c k p − ) N − p − − , then c k p | x | (1 − N ) p +2 ≤ c k p ( ρ k (1 − N ) p +2 ≤ c k ρ − Nk ≤ c k | x | − N and k = c ρ − N − p − k ≥ c | x | N − − p − , where c > k . Thus, for t ∈ ( ρ k , ρ k ) , e ∈ ∂B (0) , e · e N > ,w k ( te ) ≥ [ c kt − N − c k p t (1 − N ) p +2 ] P N ( e ) ≥ c kt − N P N ( e ) ≥ c c t − p − P N ( e ) . Now we can choose a sequence { k n } ⊂ [1 , ∞ ) such that ρ k n +1 ≥ ρ k n and for any x ∈ B + (0) \ { } , there exists k n such that x ∈ B + ρ k (0) \ B + ρk (0) and w k n ( x ) ≥ c c | x | − p − P N ( x | x | ) . Together with w k n +1 ≥ w k n in B +1 (0), we have that w ∞ ( x ) ≥ c c | x | − p − P N ( x | x | ) , ∀ x ∈ B +1 (0) . Proof of the upper bound in (4.4).
Let ¯ w p = | x | − p − − x N , then − ∆ ¯ w p = ( 2 p − p − − N ) | x | − p − − x N , where ( p − + 1)( p − + 1 − N ) > p < N +1 N − . By Comparison Principle, there exists t > k such that u k ≤ t ¯ w p in B +1 (0) , which implies that w ∞ ≤ t ¯ w p in B +1 (0) . Proof of (1.18).
We observe there exists t ∈ (0 , t ) such that − ∆ t ¯ w p + ( t ¯ w p ) p ≤ R N + . Therefore, ( − ∆ u p + u pp = 0 in R N + ,u = 0 on ∂ R N + \ { } admits a unique solution u p and by scaling property, we have that u p ( te ) = t − p − u p ( e ) , te ∈ R N + . nisotropic singular solutions to semi-linear elliptic equations 21 By Comparison Principle, we have that u p ( te ) − max u p ( e ) ≤ u ∞ ( te ) ≤ u p ( te ) , te ∈ B +1 (0) , which implies (1.18) with ϕ ( e ) = u p ( e ). (cid:3) Proof of Theorem 1.3.
From Proposition 4.1 and Lemma 4.1, one has that u ∞ is aclassical solution of (1.17) satisfying (1.18). (cid:3) Non x N -odd solutions Existence.
Under the assumptions on g in Theorem 1.1, it shows from [19] that theproblem ( − ∆ u + g ( u ) = jδ in B (0) ,u = 0 on ∂B (0) (5.1)admits a unique weak solution, denoting by u ,j . In the approaching the weak solution ofproblem (1.8) with j >
0, a barrier will be constructed by u ,j and u k , where u k is theunique x N − odd weak solution of (1.8) with j = 0. Proof of Theorem 1.1 with j > . Step 1.
We observe that g n is bounded, Lips-chitz continuous and nondecreasing, where g n is defined by (3.2), then it follows from [19,Theorem 3.7] and the Kato’s inequality that ( − ∆ u + g n ( u ) = kµ t + jδ in B (0) ,u = 0 on ∂B (0) (5.2)admits a unique weak solution v k,j,n,t , which is a classical solution of ( − ∆ u + g n ( u ) = 0 in B (0) \ { te N , , − te N } ,u = 0 on ∂B (0) . Moreover, v k,j,n,t could be approximated by the classical solutions { v n,t,m } to ( − ∆ u + g n ( u ) = kµ t,m + jσ m in B (0) ,u = 0 on ∂B (0) , where µ t,m ( x ) = σ m ( x − te N ) − σ m ( x + te N ) t and { σ m } is a sequence of radially symmetric, nondecreasing smooth functions convergingto δ in the distribution sense. Furthermore, Z B (0) [ v n,t,m ( − ∆) ξ + g n ( v n,t,m ) ξ ] dx = Z B (0) [ kµ t,m + jσ m ] ξdx, ∀ ξ ∈ C , ( B (0)) . (5.3)Since kµ t,m + jσ m ≥ kµ t,m , it implies by Comparison Prinsiple that w k,n,t,m ≤ v n,t,m ≤ w k,n,t,m + υ j,m in B (0) , (5.4)where w k,n,t,m is the weak solution of (3.5) and υ j,n,m is the unique solution of the equation ( − ∆ u = jσ m in B (0) ,u = 0 on ∂B (0) . Therefore, v k,j,n,t satisfies w k,n,t ≤ v k,j,n,t ≤ w k,n,t + jυ j,m in B (0) \ { te N , , − te N } and v k ′ ,j ′ ,n,t ≥ v k,j,n,t in B +1 (0) \ { te N } for k ′ ≥ k, j ′ ≥ j. (5.5) H. Chen
Step 2.
From Step 1, problem (5.2) admits a unique weak solution v k,j,n,t , that is, Z B (0) [ w k,n,t ( − ∆) ξ + g n ( w k,n,t ) ξ ] dx = ξ ( te N ) − ξ ( − te N ) t + jξ (0) , ∀ ξ ∈ C ( B (0)) . (5.6)On the one hand, by Lemma 2.1, we have thatlim t → + ξ ( te N ) − ξ ( − te N ) t = 2 ∂ξ (0) ∂x N . On the other hand, by the fact that | w k,n,t | ≤ k | G B (0) [ µ t ] | in B (0) , we have that | v k,j,n,t | ≤ k | G B (0) [ µ t ] | + j | G B (0) [ δ ] | in B (0) , By interior regularity results, see [15], for σ ∈ (0 ,
1) and any compact set K and open set O in B (0) such that K ⊂ O , ¯ O ∩ { te N : t ∈ ( − , ) } = ∅ , there exist c , c > t such that k v k,j,n,t k C σ ( K ) ≤ c [ k k G B (0) [ µ t ] k L ∞ ( O ) + k j G B (0) [ δ ] k L ∞ ( O ) ] ≤ c [ k k G B (0) [ ∂δ ∂x N ] k L ∞ ( O ) + k j G B (0) [ δ ] k L ∞ ( O ) ] . Moreover, by [4, Lemma 3.6] { G B (0) [ µ t ] } is uniformly bounded in M NN − ( B (0) , dx ) if N ≥ M σ ( B (0) , dx ) for any σ ∈ (0 , ) if N = 2, { G B (0) [ δ ] } is uniformlybounded in M NN − ( B (0)) if N ≥ { G B (0) [ δ ] } , is uniformly bounded in M q ( B (0) , dx )for any q >
0. Therefore, { v k,j,n,t } t is relatively compact in L p ( B (0)) for any p ∈ [1 , NN − ).Then there exists v k,j,n ∈ L ( B (0)) such that v k,j,n,t → v k,j,n a . e . in B (0) and in L ( B (0)) , which implies that g n ( v k,j,n,t ) → g n ( v k,j,n ) a . e . in B (0) and in L ( B (0)) . Therefore, up to some subsequence, passing to the limit as t → + in the identity (5.3),it infers that v k,j,n is the unique very weak solution of ( − ∆ u + g n ( u ) = 2 k ∂δ ∂x N + jδ in B (0) ,u = 0 on ∂B (0) . Here the uniqueness follows by the Kato’s inequality. It follows by (5.4), (3.12) and x N -oddproperty of w k,n that w k,n ≤ v k,j,n ≤ w k,n + j G B (0) [ δ ] in B (0) , (5.7)where w k,n is the unique x N -odd solution of (3.8). From the proof of Theorem 3.1, weknown that for any ξ ∈ C . ( B (0)), Z B (0) g n ( w k,n ) ξ dx → Z B (0) g ( w k ) ξ dx as n → + ∞ . Step 3.
It follows by (5.7) that Z B (0) [ v k,j,n ( − ∆) ξ + g n ( v k,j,n ) ξ ] dx = 2 k ∂ξ (0) ∂x N + jξ (0) , ∀ ξ ∈ C , ( B (0)) . (5.8) nisotropic singular solutions to semi-linear elliptic equations 23 For β ∈ (0 , K and open set O in B (0) satisfying K ⊂ O , 0 ¯ O , wehave that k v k,j,n k C ,β ( K ) ≤ c [ k G B (0) [ ∂δ ∂x N ] k C ( O ) + k G B (0) [ δ ] k C ( O ) ] . Therefore, up to some subsequence, there exists v k,j such thatlim n → + ∞ v k,j,n = v k,j a . e . in B (0) . Then { g n ( v k,j,n ) } converges to g ( v k,j ) a.e. in B (0).We observe that ˜ v n := v k,j,n − w k,n is the very weak solution of ( − ∆ u + g n ( w k,n + u ) − g n ( w k,n ) = jδ in B (0) ,u = 0 on ∂B (0) . (5.9)Note that 0 ≤ ˜ v n ≤ j G B (0) [ δ ] and by (1.11), it follows that0 ≤ g n ( w k,n + ˜ v n ) − g n ( w k,n ) ≤ c (cid:20) g ( w k,n )1 + | w k,n | ˜ v n + g n (˜ v n ) (cid:21) . Thus, { g n ( w k,n + ˜ v n ) − g n ( w k,n ) } converges to g ( w k + ˜ v ) − g ( w k ) a.e. in B (0), where˜ v = v k,j − w k .Let ˜ g n ( s ) = g n ( w k,n + s ) − g n ( w k ), we see that ˜ g n (0) = 0 and function ˜ g n is nondecreasingand verifies (1.10). Then it follows by Theorem 3.7 in [19] that k ˜ g n (˜ v n ) k L ( B (0)) ≤ c k G B (0) [ δ ] k L ( B (0) dx ) and ˜ v n ≤ j G B (0) [ δ ] . Thus, it follows by (1.11) that˜ g n (˜ v n ( x )) ≤ c g ( j G B (0) [ δ ]( x )) + c g (2 k | G B (0) [ ∂δ ∂x N ] | ( x ))1 + 2 k | G B (0) [ ∂δ ∂x N ]( x ) | j G B (0) [ δ ]( x ) ≤ c g ( c | x | − N ) + c g ( c | x | − N ) | x | . Let S λ = { x ∈ B (0) : | x | > λ − } , then˜ m ( λ ) = Z S λ dx = c λ − N , ∀ λ > λ . For any Borel set E ⊂ B (0), we have that Z E | ˜ g n (˜ v n ) | dx ≤ c Z E ∩ S cλ g ( c λ N − ) dx + c Z E ∩ S λ g ( c | x | − N ) dx + c Z E ∩ S cλ g ( c λ N − ) λ − dx + c Z E ∩ S λ g ( c | x | − N ) | x | dx ≤ c g ( c λ N − ) | E | + c m ( λ ) g ( c λ N − ) + c Z ∞ λ m ( s ) dg ( c s N − )+ c g ( c λ N − ) λ − | E | + c m ( λ ) g ( c λ N − ) λ − + c Z ∞ λ m ( s ) d ( g ( c s N − ) s − ) . Since the critical index in (1.10) is N +1 N − , we have that Z ∞ λ g ( c s N − ) dm ( s ) = lim T →∞ Z Tλ g ( js ) dm ( s ) . H. Chen
Thus, m ( λ ) g ( c λ N − ) + Z Tλ m ( s ) dg ( c s N − )= c T − N g ( c T N − ) + c Z Tλ s − − N g ( c s N − ) ds = c ( T N − ) − NN − g ( c T N − ) + c Z ( c T ) / ( N − ( c λ ) / ( N − t − − NN − g ( t ) dt and m ( λ ) g ( c λ N − ) λ − + Z ∞ λ m ( s ) d ( g ( c s N − ) s − )= c T − N − g ( c T N − ) + c Z Tλ s − − N g ( c s N − ) ds = c ( T N − ) − N +1 N − g ( c T N − ) + c Z ( c T ) / ( N − ( c λ ) / ( N − t − − N +1 N − g ( t ) dt, where c ( T N − ) − NN − g ( c T N − ) → T → ∞ and c ( T N − ) − N +1 N − g ( c T N − ) → T → ∞ by the assumption (1.10) and Lemma 3.3.Therefore, c m ( λ ) g ( c λ N − ) + c Z ∞ λ m ( s ) dg ( c s N − ) ≤ c Z ( c T ) / ( N − ( c λ ) / ( N − t − − NN − g ( t ) dt and c m ( λ ) g ( c λ N − ) λ − + c Z ∞ λ m ( s ) d ( g ( c s N − ) s − ) ≤ c Z ( c T ) / ( N − ( c λ ) / ( N − t − − N +1 N − g ( t ) dt. Notice that the quantities on the right-hand side tends to 0 when λ → ∞ . The conclusionfollows: for any ǫ >
0, there exists λ > c Z ( c T ) / ( N − ( c λ ) / ( N − t − − NN − g ( t ) dt ≤ ǫ c Z ( c T ) / ( N − ( c λ ) / ( N − t − − N +1 N − g ( t ) dt ≤ ǫ . For λ fixed, there exists δ > Z E dx ≤ δ = ⇒ c g ( c λ N − ) | E | ≤ ǫ c g ( c λ N − ) λ − | E | ≤ ǫ , which implies that { ˜ g n ◦ ˜ v n } is uniformly integrable in L ( B (0)). Then˜ g n ◦ ˜ v n → g ( w k + ˜ v ) − g ( w k ) in L ( B (0))by Vitali convergence theorem. nisotropic singular solutions to semi-linear elliptic equations 25 Then passing to the limit as n → + ∞ in the identity (3.13), it implies that for any ξ ∈ C , ( B (0)), Z B (0) [ w k ( − ∆) ξ + g ( w k ) ξ ] dx = 2 k ∂ξ (0) ∂x N + jξ (0) . Thus, v k,j is a very weak solution of (1.8). The regularity results follows by Lemma 3.1. Proof of ( ii ) . It follows from (1.11) and (5.7) that v k,j,n ( x ) ≤ k G B (0) [ ∂δ ∂x N ]( x ) + j G B (0) [ δ ]( x )and v k,j,n ( x ) ≥ k G B (0) [ ∂δ ∂x N ]( x ) + j G B (0) [ δ ]( x ) − G B (0) [ g n ( G B (0) [2 k ∂δ ∂x N + jδ ])]( x ) ≥ k G B (0) [ ∂δ ∂x N ]( x ) + j G B (0) [ δ ]( x ) − c w g ( x ) − c G B (0) [ g ( G B (0) [ δ ])]( x ) , where w g is the unique solution of (2.3) andlim | x |→ + G B (0) [ δ ]( x )Γ N ( x ) = 1 . We see that for any e = ( e , · · · , e N ) ∈ ∂B (0) with e N = 0,lim t → + G B (0) [ g ( G B (0) [ δ ])]( te ) t N − = 0 , which, together with (3.18), implies thatlim t → + v k,j ( te ) t N − = 0 . Then lim t → + G B (0) [ g ( G B (0) [2 k ∂δ ∂x N ])]( te ) t N − = 0 . Thus, lim t → + G B (0) [ ∂δ ∂x N ]( te ) t N − = e N and by x N -odd property of w k , we derive thatlim t → + w k ( te ) t N − = 2 ke N . This ends the proof. (cid:3)
Acknowledgements:
H. Chen is supported by NSFC, No: 11401270, 11661045 and bythe Jiangxi Provincial Natural Science Foundation, No: 20161ACB20007 and the Project-sponsored by SRF for ROCS, SEM.
References [1] Ph. B´enilan, H. Brezis and M. Crandall, A semilinear elliptic equation in L ( R N ), Ann. Sc. Norm. Sup.Pisa Cl. Sci. 2 (1975), 523-555.[2] Ph. B´enilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation,
J. Evolution Eq.3 (2003), 673-770.[3] H. Brezis and L. V´eron, Removable singularities for some nonlinear elliptic equations,
Arch. Ration.Mech. Anal. 75 (1980), 1-6.[4] H. Chen, W. Wang and J. Wang, Anisotropic singularity of solutions to elliptic equations in a measureframework,
Elec. J. Diff. Eq., 2015, (2015), 1-12.[5] X. Chen, H. Matano and L. V´eron, Anisotropic singularities of solutions of nonlinear elliptic equationsin R N , J. Funct. Anal. 83 (1989), 50-97. H. Chen [6] L. Evans, Partial Differential Equations,
American Mathematical Society , 2000.[7] G. Folland, Real analysis,
Pure and Applied Mathematics (New York) , 1999.[8] K. Kato, Schr¨odinger operators with singular potentials,
Israel J. Math 13 , (1972) 135-148.[9] E. Lieb and M. Loss, Analysis,
Graduate Studies in Mathematics 14 , 2001.[10] E. Lieb and B. Simon, The Thomas-Fermi theory of atoms, molecules and solids,
Advances in Math. 23 (1977), 22-116.[11] P. Lions, Isolated singularities in semilinear problems,
J. Diff. Eq. 38(3) , 441-450 (1980).[12] V. Liskevich amd I. Skrypnik, Isolated singularities of solutions to quasi-linear elliptic equations withabsorption,
J. Math. Anal. Appl.338 , 536-544 (2008).[13] M. Ghergu, V, Liskevich and Z. Sobol, Singular solutions for second-order non-divergence type ellipticinequalities in punctured balls,
J. d’Anal. Math. 123 , 251-279 (2014).[14] A. Gmira and L. V´eron, Boundary singularities of solutions of some nonlinear elliptic equations,
DukeMath. J. 64 (1991), 271-324.[15] Q. Han and F. Lin, Elliptic partial differential equations,
American Mathematical Soc. vol 1,
J.Diff. Eq. 235 , 439-483 (2007).[17] M. Reed and B. Simon, Methods of modern mathematical physics: Functional analysis,
Gulf Profes-sional Publishing 1,
Nonlinear Anal. T. M. & A. 5,
Vol. I, 593-712,Handb. Differ. Eq., North-Holland, Amsterdam
Chapman and Hall CRC (1996).[21] A. Ponce, Elliptic PDEs, measures and capacities,