Critical points of solutions to a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions
CCritical points of solutions to a quasilinear elliptic equation withnonhomogeneous Dirichlet boundary conditions ∗ Haiyun Deng † , Hairong Liu , Long Tian School of Science, Nanjing University of Science and Technology, Nanjing, Jiangsu, 210094, China; School of Science, Nanjing Forestry University, Nanjing, Jiangsu, 210037, China;
Abstract:
In this paper, we mainly investigate the critical points associated to solutions u of a quasilinearelliptic equation with nonhomogeneous Dirichlet boundary conditions in a connected domain Ω in R . Based onthe fine analysis about the distribution of connected components of a super-level set { x ∈ Ω : u ( x ) > t } for anymin ∂ Ω u ( x ) < t < max ∂ Ω u ( x ), we obtain the geometric structure of interior critical points of u . Precisely, whenΩ is simply connected, we develop a new method to prove Σ ki =1 m i + 1 = N , where m , · · · , m k are the respectivemultiplicities of interior critical points x , · · · , x k of u and N is the number of global maximal points of u on ∂ Ω. WhenΩ is an annular domain with the interior boundary γ I and the external boundary γ E , where u | γ I = H, u | γ E = ψ ( x ) and ψ ( x ) has N local (global) maximal points on γ E . For the case ψ ( x ) ≥ H or ψ ( x ) ≤ H or min γ E ψ ( x ) < H < max γ E ψ ( x ),we show that Σ ki =1 m i ≤ N (either Σ ki =1 m i = N or Σ ki =1 m i + 1 = N ). Key Words: a quasilinear elliptic equation, critical point, multiplicity, multiply connecteddomain.
In this paper we mainly investigate the interior critical points of solutions to the following aquasilinear elliptic equation Lu = (cid:80) i,j =1 a ij ( ∇ u ) ∂ u∂x i ∂x j = 0 in Ω , (1.1)where Ω is a bounded, smooth and connected domain in R , a ij is smooth and L is uniformly ellipticin Ω.The subject of critical points is a significant research topic for solutions of elliptic equations.Until now, there are many results about the critical points. In 1992 Alessandrini and Magnanini[1] studied the geometric structure of the critical set of solutions to a semilinear elliptic equationin a planar nonconvex domain, whose boundary is composed of finite simple closed curves. Theydeduced that the critical set is made up of finitely many isolated critical points. In 1994, Sakaguchi[19] considered the critical points of solutions to an obstacle problem in a planar, bounded, smoothand simply connected domain. He showed that if the number of critical points of the obstacle is finiteand the obstacle has only N local (global) maximum points, then the inequality Σ ki =1 m i + 1 ≤ N (the equality Σ ki =1 m i + 1 = N ) holds for the critical points of one solution in the noncoincidence ∗ The work is supported by National Natural Science Foundation of China (No.11401307, No.11401310), High leveltalent research fund of Nanjing Forestry University (G2014022) and Postgraduate Research & Practice InnovationProgram of Jiangsu Province (KYCX17 0321). The second author is sponsored by Qing Lan Project of JiangsuProvince. † Corresponding author E-mail: [email protected], Tel.: +86 15877935256 a r X i v : . [ m a t h . A P ] M a y et, where m , m , · · · , m k are the multiplicities of critical points x , x , · · · , x k respectively. In 2012Arango and G´omez [3] considered critical points of the solutions to a quasilinear elliptic equationwith Dirichlet boundary condition in strictly convex and nonconvex planar domains respectively. Ifthe domain is strictly convex and u is a negative solution, they proved that such a critical pointset has exactly one nondegenarate critical point. Moreover, they obtained the similar results of asemilinear elliptic equation in a planar annular domain, whose boundary has nonzero curvature. See[2, 5, 6, 9, 10, 11, 12, 14, 15, 16, 17] for related results.Concerning the Neumann and Robin boundary value problems, there exist a few results aboutthe critical points of solutions to elliptic equations. In 1990, Sakaguchi [18] proved that a solutionof Poisson equation with Neumann or Robin boundary condition has exactly one critical point in aplanar domain. In 2017, Deng, Liu and Tian [8] showed the nondegeneracy and uniqueness of thecritical point of a solution to prescribed constant mean curvature equation with Neumann or Robinboundary condition in a smooth, bounded and strictly convex domain Ω of R n ( n ≥ −(cid:52) u = f ( u ), Cabr´e and Chanillo [4] showed that the solution u has exactlyone nondegenerate critical point in bounded, smooth and convex domains of R n ( n ≥ K has exactlyone nondegenerate critical point in a strictly convex domain of R n ( n ≥
2) and K has (respectively,has no) a rotationally symmetric critical closed surface S in a concentric (respectively, an eccentric)spherical annulus domain of R n ( n ≥ Theorem 1.1.
Let Ω be a bounded, smooth and simply connected domain in R . Suppose that ψ ( x ) ∈ C (Ω) and that ψ has N local maximal points on ∂ Ω . Let u be a non-constant solution ofthe following boundary value problem (cid:80) i,j =1 a ij ( ∇ u ) ∂ u∂x i ∂x j = 0 in Ω ,u = ψ ( x ) on ∂ Ω . (1.2) Then u has finite interior critical points, denoting by x , x , · · · , x k , and the following inequalityholds k (cid:80) i =1 m i + 1 ≤ N, (1.3) where m , m , · · · , m k are the multiplicities of critical points x , x , · · · , x k respectively. Theorem 1.2.
Let Ω be a bounded, smooth and simply connected domain in R . Suppose that ψ ( x ) ∈ C (Ω) and that ψ has only N global maximal points and N global minimal points on ∂ Ω ,i.e., all the maximal and minimal points of ψ are global. Let u be a non-constant solution of (1.2).Then u has finite interior critical points and k (cid:80) i =1 m i + 1 = N, (1.4) where m i is as in Theorem 1.1. heorem 1.3. Let Ω be a bounded smooth annular domain with the interior boundary γ I and theexternal boundary γ E in R . Suppose that ψ ( x ) ∈ C (Ω) , H is a given constant, ψ ( x ) ≥ H and that ψ has N local maximal points on γ E . Let u be a non-constant solution of the following boundaryvalue problem (cid:80) i,j =1 a ij ( ∇ u ) ∂ u∂x i ∂x j = 0 in Ω ,u | γ I = H, u | γ E = ψ ( x ) . (1.5) Then u has finite interior critical points and k (cid:80) i =1 m i ≤ N, (1.6) where m i is as in Theorem 1.1. Theorem 1.4.
Let Ω be a bounded smooth annular domain with the interior boundary γ I and theexternal boundary γ E in R . Suppose that ψ ( x ) ∈ C (Ω) , ψ ( x ) ≥ H and that ψ has only N globalmaximal points and N global minimal points on γ E , i.e., all the maximal and minimal points of ψ are global. Let u be a non-constant solution of (1.5). Then u has finite interior critical points, andeither k (cid:80) i =1 m i = N, (1.7) or k (cid:80) i =1 m i + 1 = N, (1.8) where m i is as in Theorem 1.1. Theorem 1.5.
Let Ω be a bounded smooth annular domain with the interior boundary γ I and theexternal boundary γ E in R . Suppose that ψ ( x ) ∈ C (Ω) , H is a given constant, min γ E ψ ( x ) < H < max γ E ψ ( x ) and that ψ has N local maximal points on γ E . Let u be a non-constant solution of (1.5).Then u has finite interior critical points and k (cid:80) i =1 m i ≤ N, (1.9) where m i is as in Theorem 1.1. Theorem 1.6.
Let Ω be a bounded smooth annular domain with the interior boundary γ I and theexternal boundary γ E in R . Suppose that ψ ( x ) ∈ C (Ω) , min γ E ψ ( x ) < H < max γ E ψ ( x ) and that ψ hasonly N global maximal points and N global minimal points on γ E , i.e., all the maximal and minimalpoints of ψ are global. Let u be a non-constant solution of (1.5). Then u has finite interior criticalpoints, and either k (cid:80) i =1 m i = N, (1.10) or k (cid:80) i =1 m i + 1 = N, (1.11) where m i is as in Theorem 1.1. Remark 1.7.
In particular, when a ij ( ∇ u ) = √ |∇ u | ( δ ij − u xi u xj |∇ u | ) , then the minimal surfaceequation div ( ∇ u √ |∇ u | ) = 0 in a bounded smooth domain is a particular example of a quasilinearelliptic equation in (1.1). u ( x ) = · · · = u ( x k ), andall interior critical values are not totally equal. However, when ψ ( x ) has only N global maximalpoints and N global minimal points on the boundary ∂ Ω , we prove that all interior critical valuesare equal. We obtain (1.4) and (1.7) by showing that there are the following three “ just right ”s:(i) the first “ just right ” is that the critical values for all interior critical points are equal (i.e., u ( x ) = u ( x ) = · · · = u ( x k ) = t for some t );(ii) the second “ just right ” is that all the critical points x , x , · · · , x k together with the corre-sponding level lines of { x ∈ Ω : u ( x ) = t } clustering round these points form a connected set;(iii) the third “ just right ” is that every simply connected component ω of { x ∈ Ω : u ( x ) > t } ( { x ∈ Ω : u ( x ) < t } ) has exactly one global maximal (minimal) point on the boundary ∂ Ω.The rest of this paper is organized as follows. In Section 2, we investigate the geometric structureof interior critical points of solutions in a bounded, smooth and simply connected domain in R .We show that if ψ ( x ) has only N local (global) maximal points on ∂ Ω, then Σ ki =1 m i + 1 ≤ N (Σ ki =1 m i + 1 = N ) holds for the interior critical points of a solution u . We develop a new methodto prove Σ ki =1 m i + 1 = N , we show the three “just right”s. In Section 3, we study the geometricstructure of interior critical points of solutions in a bounded smooth annular domain with the interiorboundary γ I and the external boundary γ E in R , where u | γ I = H, u | γ E = ψ ( x ) , ψ ( x ) ≥ H and ψ has N local (global) maximal points on γ E . We deduce Σ ki =1 m i ≤ N (Σ ki =1 m i = N or Σ ki =1 m i + 1 = N ).In Section 4, we investigate the case of min γ E ψ ( x ) < H < max γ E ψ ( x ), where ψ has N local (global)maximal points on γ E and show the same results as in Section 3. In order to prove Theorem 1.1, we need the following basic lemmas.
Lemma 2.1.
Let u be a non-constant solution of (1.2). For any t ∈ (min Ω u, max Ω u ) , we have thatany connected component of { x ∈ Ω : u ( x ) > t } and { x ∈ Ω : u ( x ) < t } is simply connected, whichhas to meet the boundary ∂ Ω . Proof.
Let A be a connected component of { x ∈ Ω : u ( x ) > t } and α be a non-equivalent simpleclosed curve in A. By the Jordan curve theorem there exists a bounded domain B with ∂B = α. SinceΩ is simply connected, then B is contained in Ω . The strong maximum principle implies that u > t indomain B . It shows that B is contained in A , namely A is simply connected. The strong maximumprinciple shows that u obtain its maximum points and minimal points on boundary ∂ Ω , therefore theconnected component A has to meet the boundary ∂ Ω . The proof of the case of { x ∈ Ω : u ( x ) < t } is similar. Lemma 2.2.
Suppose that x is an interior critical point of u in Ω and that m is the multiplicity of x . Then m + 1 distinct connected components of { x ∈ Ω : u ( x ) > u ( x ) } and { x ∈ Ω : u ( x ) < u ( x ) } cluster around the point x respectively.Proof. According to the results of Hartman and Wintner [13], in a neighborhood of x the levelline { x ∈ Ω : u ( x ) = u ( x ) } consists of m + 1 simple arcs intersecting at x . By the results4f Lemma 2.1, there exist m + 1 distinct connected components of { x ∈ Ω : u ( x ) > u ( x ) } and { x ∈ Ω : u ( x ) < u ( x ) } clustering around the point x respectively. This completes the proof. Lemma 2.3.
Suppose that u is a non-constant solution to (1.2). Then u has finite interior criticalpoints in Ω .Proof. We set up the usual contradiction argument. Suppose that u has infinite interior criticalpoints in Ω , denoting by x , x , · · · . The results of Lemma 2.1 and Lemma 2.2 show that there existsinfinite connected components of { x ∈ Ω : u ( x ) > u ( x i ) } and { x ∈ Ω : u ( x ) < u ( x i ) } ( i = 1 , , · · · ).The strong maximum principle implies that there exists at least a maximum point and minimal pointon ∂ Ω for any connected components of { x ∈ Ω : u ( x ) > u ( x i ) } and { x ∈ Ω : u ( x ) < u ( x i ) } ( i =1 , , · · · ) respectively. Therefore there exists infinite maximal points and minimal points on ∂ Ω , thiscontradicts with the assumption. This completes the proof. Lemma 2.4.
Let x , x , · · · , x k be the interior critical points of u in Ω . Suppose that u ( x ) = u ( x ) = · · · = u ( x k ) = t for some t ∈ R , where m , m , · · · , m k are the multiplicities of critical points x , x , · · · , x k respectively. We set M and M as the number of the connected components of thesuper-level set { x ∈ Ω : u ( x ) > t } and the sub-level set { x ∈ Ω : u ( x ) < t } respectively. Suppose thatall the critical points x , x , · · · , x k together with the corresponding level lines of { x ∈ Ω : u ( x ) = t } clustering round these points form q connected sets, where q ≥ . Then M ≥ k (cid:80) i =1 m i + 1 , M ≥ k (cid:80) i =1 m i + 1 , (2.1) and M + M = 2 k (cid:80) i =1 m i + q + 1 . (2.2) Proof.
We divide the proof into two cases.(i) Case 1: When q = 1 , by induction. Since the number of the connected components of thesuper-level set { x ∈ Ω : u ( x ) > t } equals the number of the connected components of the sub-levelset { x ∈ Ω : u ( x ) < t } . Without loss of generality, we only estimate the number of the connectedcomponents of the super-level set { x ∈ Ω : u ( x ) > t } . By induction. When k = 1 , the result holdsby Lemma 2.2. Assume that 1 ≤ k ≤ n the connected set, which consists of k critical points andthe connected components clustering round these points, contains exactly (cid:80) ki =1 m i + 1 componentsof the super-level set { x ∈ Ω : u ( x ) > t } . Let k = n + 1 . Let A be the set which consists of thepoints x , x , · · · , x n +1 together with the respective components clustering round these points. Wemay assume that the points x , x , · · · , x n together with the respective components clustering roundthese points form a connected set, denotes by B. By Lemma 2.1 we know that A cannot surrounda component of { x ∈ Ω : u ( x ) < t } . Up to renumbering, therefore there is only one component of { x ∈ Ω : u ( x ) > t } whose boundary γ contains both x n and x n +1 . Next we give the distribution forthe level lines of { x ∈ Ω : u ( x ) = t } . Figure 1.
The distribution for the level lines of { x ∈ Ω : u ( x ) = t } . A and B are connected. By using Lemma 2.2 and the inductive assumption to B, thenwe know that A contains exactly( n (cid:88) i =1 m i + 1) + ( m n +1 + 1) − n +1 (cid:88) i =1 m i + 1connected components of the super-level set { x ∈ Ω : u ( x ) > t } . This completes the proof of case 1.(ii) Case 2: When q ≥ . Since the number of connected sets of the level lines { x ∈ Ω : u ( x ) = t } together with x , x , · · · , x k increases one leading the number of connected componentsof { x ∈ Ω : u ( x ) > t } or { x ∈ Ω : u ( x ) < t } also increases one, i.e., if the number of connectedcomponents of { x ∈ Ω : u ( x ) > t } increased by 1, then the number of connected componentsof { x ∈ Ω : u ( x ) < t } unchanged, and vice versa. Figure 2 pictures the changing of connectedcomponents of { x ∈ Ω : u ( x ) > t } or { x ∈ Ω : u ( x ) < t } . Figure 2.
The distribution for the connected components of { x ∈ Ω : u ( x ) > t } and { x ∈ Ω : u ( x ) < t } . Now we put M := (cid:93) (cid:110) the connected components of the super-level set { x ∈ Ω : u ( x ) > t } (cid:111) ,M := (cid:93) (cid:110) the connected components of the sub-level set { x ∈ Ω : u ( x ) < t } (cid:111) . If all the critical points x , x , · · · , x k together with the level lines of { x ∈ Ω : u ( x ) = t } clusteringround these points form q connected sets. By the results of case 1, then we have M ≥ k (cid:80) i =1 m i + 1 , M ≥ k (cid:80) i =1 m i + 1 , and M + M = 2( k (cid:80) i =1 m i + 1) + ( q −
1) = 2 k (cid:80) i =1 m i + q + 1 . This completes the proof of case 2.We are now ready to present the proof of Theorem 1.1.
Proof of Theorem 1.1. (i) Case 1: If u ( x ) = u ( x ) = · · · = u ( x k ) = t for some t ∈ R . By the resultsof Lemma 2.4, we know that (cid:93) (cid:110) the connected components of the super-level set { x ∈ Ω : u ( x ) > t } (cid:111) ≥ k (cid:80) i =1 m i + 1 . Therefore, in this case the super-level set always has at least k (cid:80) i =1 m i + 1 connected components andat most k (cid:80) i =1 m i + q connected components. Using the strong maximum principle and Lemma 2.1, we6ave that u exists at least k (cid:80) i =1 m i + 1 local maximal points on ∂ Ω . Hence, we have k (cid:88) i =1 m i + 1 ≤ N. (ii) Case 2: The values at critical points x , · · · , x k are not totally equal. Without loss ofgenerality, we may suppose that u ( x ) = · · · = u ( x j ) < u ( x j +1 ) = · · · = u ( x j ) < · · · < · · · < u ( x j n − +1 ) = · · · = u ( x j n ) , (2.3)where x , · · · , x j , · · · , x j , · · · , x j n are different critical points in Ω, j n = k and n ≥ . Now we put E j := (cid:110) ω : open set ω is a connected component of { x ∈ Ω : u ( x ) > u ( x j ) } (cid:111) ( j = j , j , · · · , j n ) , and F j n := (cid:110) ω : open set ω is a connected component of { x ∈ Ω : u ( x ) < u ( x j ) } or ω is a connected component of { x ∈ Ω : u ( x j i ) < u ( x ) < u ( x j i +1 ) for some 1 ≤ i ≤ n − } (cid:111) . According to the definition, we know that F j n consists of disjoint components. Denotes | F j n | := (cid:93) { ω : ω is a connected component of F j n } . To illustrate F j n , let us consider an illustration for F j . Assume that u has only three criticalpoints x j , x j +1 , x j with respective multiplicity m j = 1 , m j +1 = 1 , m j = 1 in Ω and u ( x j )
The distribution of elements of F j . By the definition of F j n , we know | F j | = 6 . Now let us show that (cid:12)(cid:12) F j s (cid:12)(cid:12) ≥ (cid:80) j s i =1 m i + 1 by induction on the number s. When s = 1 , the resultholds by case 1. Assume that (cid:12)(cid:12) F j s (cid:12)(cid:12) ≥ (cid:80) j s i =1 m i + 1 for 1 ≤ s ≤ n − . Let s = n. Then, by (2.3) andthe definition of E j , we have { x j n − +1 , · · · , x j n } ⊂ (cid:91) ω ∈ E jn − ω, where ω is a connected component of { x ∈ Ω : u ( x ) > u ( x j n − ) } . Let us assume that { x j n − +1 , · · · , x j n } are contained in exactly (cid:101) q components ω , · · · , ω (cid:101) q . Then x j n − +1 , · · · , x j n together with the corresponding level lines of { x ∈ Ω : u ( x ) = u ( x j n ) } clusteringround these points at least form (cid:101) q connected sets. By Lemma 2.4, we have M := (cid:93) (cid:110) the connected components of { x ∈ Ω : u ( x j n − ) < u ( x ) < u ( x j n ) } in all ω j ( j = 1 , · · · , (cid:101) q ) (cid:111) ≥ j n (cid:80) i = j n − +1 m i + (cid:101) q.
7y using the definition of (cid:12)(cid:12) F j n (cid:12)(cid:12) and the inductive assumption to 1 ≤ s ≤ n − , then we have (cid:12)(cid:12) F j n (cid:12)(cid:12) = (cid:12)(cid:12) F j n − (cid:12)(cid:12) + M ≥ (cid:12)(cid:12) F j n − (cid:12)(cid:12) + ( j n (cid:80) i = j n − +1 m i + (cid:101) q ) − (cid:101) q ≥ j n (cid:80) i =1 m i + 1 . By the strong maximum principle and Lemma 2.1, we have that u has at least k (cid:80) i =1 m i + 1 localminimal points on ∂ Ω . Therefore, we obtain k (cid:88) i =1 m i + 1 ≤ N. This completes the proof of case 2.
In this subsection, we investigate the geometric structure of interior critical points of a solutionin a planar, bounded, smooth and simply connected domain Ω for the case of ψ having only N global maximal points and N global minimal points on ∂ Ω . We develop a new method to proveΣ ki =1 m i + 1 = N , where N ≥
2, and we show the three “just right”s.
Proof of Theorem 1.2.
We divide the proof into five steps.Step 1, we show that u has at least one interior critical point in Ω . Suppose by contradictionthat |∇ u | > z = min ∂ Ω u, Z = max ∂ Ω u . The strong maximum principle implies that u hasno interior maximum point and minimal point in Ω , then we have z < u ( x ) < Z for any x ∈ Ω . According to the assumption of Theorem 1.2, let q , · · · , q N and p , · · · , p N be the global maximalpoints and minimal points on ∂ Ω , respectively.Without loss of generality, we may assume that there only exists two different global maxi-mal points q , q and global minimal points p , p on boundary ∂ Ω . Note that u is monotonicallydecreasing on the connected components of boundary ∂ Ω from one maximal point to the near min-imal point. Therefore, by the continuity of level lines { x ∈ Ω : u ( x ) = t , z < t < Z } , weknow that { x ∈ Ω : u ( x ) = Z − (cid:15) } ( { x ∈ Ω : u ( x ) = z + (cid:15) } ) exactly exists two level lines inΩ for any (cid:15) such that 0 < (cid:15) < Z − z . This is impossible, because this would imply that either: u ( x ) = z ( u ( x ) = Z ) in interior points of Ω, or: there exists two level lines intersect in Ω, i.e., thereexists critical points in Ω , this contradicts with the assumption |∇ u | > . This completes theproof of step 1. The figure as shown in Figure 4.
Figure 4.
The distribution of some level lines { x ∈ Ω : u ( x ) = t , z < t < Z } . Step 2, the first “just right”: According to Lemma 2.3, we assume that the interior critical pointsof u are x , x , · · · , x k . We show that u ( x ) = u ( x ) = · · · = u ( x k ) = t for some t ∈ R . We set up8he usual contradiction argument. We assume that the values at critical points x , · · · , x k are nottotally equal. Without loss of generality, we suppose that u ( x ) < u ( x ) and that m , m are therespective multiplicities of x , x . Then, by Lemma 2.1, we know that any connected component B of { x ∈ Ω : u ( x ) < u ( x ) } has to meet the boundary ∂ Ω and u ( p ) < u ( x ), where p is the minimalpoint of connected component B on ∂ Ω. At the same time, we know that any connected component C of { x ∈ Ω : u ( x ) < u ( x ) } has to meet the boundary ∂ Ω and u ( x ) < u ( p ) < u ( x ), where p is the minimal point of connected component C on ∂ Ω (see Figure 5). Then u ( p ) (cid:54) = u ( p ), whichcontradicts with the assumption of Theorem 1.2. This completes the proof of step 2. Figure 5.
The distribution of the connected components.
Step 3, the second “just right”: we show that x , x , · · · , x k together with the corresponding levellines of { x ∈ Ω : u ( x ) = t } clustering round these points exactly form one connected set. Withoutloss of generality, we suppose by contradiction that x , x , · · · , x k together with the level lines of { x ∈ Ω : u ( x ) = t } clustering round these points form two connected sets. Therefore, there exists aconnected components of { x ∈ Ω : u ( x ) < t } (or { x ∈ Ω : u ( x ) > t } ), which meets two parts γ , γ of ∂ Ω, denoting by A (see Figure 6). Figure 6.
The distribution of some level lines { x ∈ Ω : u ( x ) = (cid:101) t, z < (cid:101) t < t } . Note that u is monotonically decreasing on the connected components of boundary ∂ Ω from onemaximal point to the near minimal point. Therefore, { x ∈ A : u ( x ) = t − (cid:15) } exactly exists two levellines in A for any (cid:15) such that 0 < (cid:15) < t − z . This is impossible, because this would imply thateither: u ( x ) = z in interior points of A , or: there exists two level lines intersect in A , i.e., thereexists critical points in A. This completes the proof of step 3.Step 4, the third “just right”: we show that every connected component of { x ∈ Ω : u ( x ) > t } ( { x ∈ Ω : u ( x ) < t } ) has exactly one global maximal (minimal) point on boundary ∂ Ω . In fact,we assume that some connected component B of { x ∈ Ω : u ( x ) > t } ( { x ∈ Ω : u ( x ) < t } ) existstwo global maximal (minimal) points on boundary ∂ Ω . According to ψ has only N global maximalpoints and N global minimal points on ∂ Ω, then there must exist a minimal point p between the twomaximal points on ∂ Ω such that u ( p ) = z . Since u ( x ) > t > z in B , by the continuity of solution u ,this contradicts with the definition of connected component B . This completes the proof of step 4.Step 5, By the results of step 3 and the results of case 1 in Lemma 2.4, we have (cid:93) (cid:110) the connected components of the super-level set { x ∈ Ω : u ( x ) > t } (cid:111) = k (cid:80) i =1 m i + 1 , (2.4)9nd (cid:93) (cid:110) the connected components of the sub-level set { x ∈ Ω : u ( x ) < t } (cid:111) = k (cid:80) i =1 m i + 1 . (2.5)On the other hand, using the results of step 4 and the strong maximum principle, therefore we obtain k (cid:80) i =1 m i + 1 = N. (2.6)This completes the proof of Theorem 1.2.Let N = 1, we have: Corollary 2.5.
Suppose that u is a non-constant solution of (1.2) and that ψ has exactly onemaximal point on ∂ Ω , then u has no interior critical points in Ω . ψ ( x ) ≥ H In order to prove Theorem 1.3, we need the following basic lemmas.
Lemma 3.1.
Let u be a non-constant solution of (1.5). For any t ∈ ( H, max γ E ψ ( x )) , then anyconnected component of { x ∈ Ω : u ( x ) > t } has to meet the external boundary γ E . Proof.
Let A be a connected component of { x ∈ Ω : u ( x ) > t } . According to the assumption of u | γ I = H , we know that A can not contain γ I . Then the strong maximum principle and (1.5) showthat the connected component A has to meet the external boundary γ E . Lemma 3.2.
Suppose that x is an interior critical point of u in Ω and that m is the multiplicityof x . Then m + 1 distinct connected components of { x ∈ Ω : u ( x ) > u ( x ) } cluster around the point x .Proof. According to the results of Hartman and Wintner [13], in a neighborhood of x the level line { x ∈ Ω : u ( x ) = u ( x ) } consists of m + 1 simple arcs intersecting at x . By Lemma 3.1, there exist m + 1 distinct connected components of { x ∈ Ω : u ( x ) > u ( x ) } clustering around the point x . Thiscompletes the proof. Lemma 3.3.
If there exists t ∈ ( H, max γ E ψ ( x )) such that a connected component ω of { x ∈ Ω : u ( x ) < t } is non-simply connected and the external boundary γ of ω is a simply closed curve in Ω ,i.e., the external boundary γ of ω is a simply closed curve between γ I and γ E . Then there does notexist any interior critical point in ω .Proof. Suppose by contradiction that there exists an interior critical point x in ω such that H
Suppose that u is a non-constant solution to (1.5). Then u has finite interior criticalpoints in Ω .Proof. We set up the usual contradiction argument. Suppose that u has infinite interior criticalpoints in Ω , denoting by x , x , · · · . The results of Lemma 3.1, Lemma 3.2 and Lemma 3.3 showthat there exists infinite connected components of { x ∈ Ω : u ( x ) > u ( x i ) } ( i = 1 , , · · · ). The strongmaximum principle implies that there exists at least a maximum point on γ E for any connectedcomponent of { x ∈ Ω : u ( x ) > u ( x i ) } ( i = 1 , , · · · ). Therefore there exists infinite maximal pointson γ E , this contradicts with the assumption. This completes the proof. Lemma 3.5.
Let x , x , · · · , x k be the interior critical points of u in Ω . Suppose that u ( x ) = u ( x ) = · · · = u ( x k ) ≡ t for some t ∈ ( H, max γ E ψ ( x )) and that all the critical points x , x , · · · , x k together with the corresponding level lines of { x ∈ Ω : u ( x ) = t } clustering round these points form q connected sets, where q ≥ and m , m , · · · , m k are the multiplicities of critical points x , x , · · · , x k respectively.Case 1: Suppose that there exists a non-simply connected component ω of { x ∈ Ω : u ( x ) < t } andthe external boundary γ of ω is a simply closed curve between γ I and γ E such that u has at least onecritical point on γ , then (cid:93) (cid:110) the simply connected components ω of the sub-level set { x ∈ Ω : u ( x ) < t } such that ω meet the external boundary γ E (cid:111) = k (cid:80) i =1 m i + q − . (3.1) Case 2: Suppose that there exists a non-simply connected component ω of { x ∈ Ω : u ( x ) < t } suchthat ω meets γ E . In addition, we set M and M as the number of the connected components of thesuper-level set { x ∈ Ω : u ( x ) > t } and the sub-level set { x ∈ Ω : u ( x ) < t } , respectively. Then M ≥ k (cid:80) i =1 m i + 1 , M ≥ k (cid:80) i =1 m i + 1 , and M + M = 2 k (cid:80) i =1 m i + q + 1 . (3.2) Proof. (i) Case 1: We divide the proof into two steps.Step 1: When q = 1 , by induction. When k = 1 , the result holds by Lemma 3.1 and Lemma3.2. Assume that 1 ≤ k ≤ n the connected set, which consists of k critical points and the connectedcomponents clustering round these points, contains exactly (cid:80) ki =1 m i components ω of the sub-levelset { x ∈ Ω : u ( x ) < t } such that ω meet the external boundary γ E . Let k = n + 1 . Let A be theset which consists of the points x , x , · · · , x n +1 together with the respective components clusteringround these points. We may assume that the points x , x , · · · , x n together with the respectivecomponents clustering round these points form a connected set, denotes by B. By Lemma 3.1 weknow that A cannot surround a component of { x ∈ Ω : u ( x ) > t } . Up to renumbering, therefore11here is only one component of { x ∈ Ω : u ( x ) < t } whose boundary α contains both x n and x n +1 .By Lemma 3.3, next we give the distribution for the level lines of { x ∈ Ω : u ( x ) = t } . Figure 8.
The distribution for the level lines of { x ∈ Ω : u ( x ) = t } . Since both A and B are connected. By using Lemma 3.2 and the inductive assumption to B, thenwe know that A contains exactly n (cid:88) i =1 m i + ( m n +1 + 1) − n +1 (cid:88) i =1 m i connected components ω of the sub-level set { x ∈ Ω : u ( x ) < t } such that ω meet the externalboundary γ E . This completes the proof of step 1.Step 2: When q ≥ . Since the number of connected sets of the level lines { x ∈ Ω : u ( x ) = t } together with x , · · · , x k increases one leading the number of connected components of { x ∈ Ω : u ( x ) < t } increases one. If all the critical points x , x , · · · , x k together with the level lines { x ∈ Ω : u ( x ) = t } clustering round these points form q connected sets. By the results of step 1, then wehave (cid:93) (cid:110) the simply connected components ω of the sub-level set { x ∈ Ω : u ( x ) < t } such that ω meet the external boundary γ E (cid:111) = k (cid:80) i =1 m i + ( q − . This completes the proof of case 1.(ii) Case 2: We divide the proof of case 2 into two steps.Step 1: When q = 1, by induction. When k = 1 , the result holds by Lemma 3.1 and Lemma3.2. Assume that 1 ≤ k ≤ n the connected set, which consists of k critical points and the connectedcomponents clustering round these points, contains exactly k (cid:80) i =1 m i + 1 components of the super-levelset { x ∈ Ω : u ( x ) > t } . Let k = n + 1 . Let A be the set which consists of the points x , x , · · · , x n +1 together with the respective components clustering round these points. We may assume that thepoints x , x , · · · , x n together with the respective components clustering round these points forma connected set, denotes by B. By Lemma 3.1 we know that A cannot surround a component of { x ∈ Ω : u ( x ) < t } . Up to renumbering, therefore there is only one component of { x ∈ Ω : u ( x ) > t } whose boundary γ contains both x n and x n +1 . Next we give the distribution for the level lines of { x ∈ Ω : u ( x ) = t } . Figure 9.
The distribution for the level lines of { x ∈ Ω : u ( x ) = t } . A and B are connected. By using Lemma 3.2 and the inductive assumption to B, thenwe know that A contains exactly( n (cid:88) i =1 m i + 1) + ( m n +1 + 1) − n +1 (cid:88) i =1 m i + 1connected components of the super-level set { x ∈ Ω : u ( x ) > t } .Step 2: The proof is similar to the case 2 of Lemma 2.4. When q ≥ . Since the theorem ofHartman and Wintner [13] shows that the interior critical points of solution u are isolated, so thenumber of connected sets of the level lines { x ∈ Ω : u ( x ) = t } together with x , · · · , x k increases oneleading the number of connected components of { x ∈ Ω : u ( x ) > t } or { x ∈ Ω : u ( x ) < t } increasesone. If all the critical points x , x , · · · , x k together with the level lines of { x ∈ Ω : u ( x ) = t } clustering round these points form q connected sets. By the results of step 1, then we have M ≥ k (cid:80) i =1 m i + 1 , M ≥ k (cid:80) i =1 m i + 1 , and M + M = 2( k (cid:80) i =1 m i + 1) + ( q −
1) = 2 k (cid:80) i =1 m i + q + 1 . This completes the proof of case 2.
Remark 3.6.
Note that if all critical values are equal (i.e., u ( x ) = · · · = u ( x k ) ≡ t ) and there existsa non-simply connected component ω of { x ∈ Ω : u ( x ) < t } for critical value t , where the externalboundary γ of ω is a simply closed curve in Ω as in Lemma 3.3, then u has at least one critical pointon γ . In fact, suppose by contradiction that u has no critical point on γ . Without loss of generality,we may assume that ψ ( x ) has only two local maximal points q , q on γ E and one critical point x in Ω \ ω such that u ( x ) = t and the multiplicity of x is one, we denote the non-simply connectedcomponent of { x ∈ Ω : u ( x ) > t } by A . The distribution for the level lines of { x ∈ Ω : u ( x ) = t } asfollows: Figure 10.
The distribution for the level lines of { x ∈ Ω : u ( x ) = t } . By using the method of step 3 of the proof of Theorem 1.2, this would imply that either: u ( x ) = u ( q ) in interior points of A , or: there exists two level lines intersect in A , i.e., there exists critical pointsin A . This is a contradiction. We are now prepare to prove Theorem 1.3.
Proof of Theorem 1.3. (i) Case 1: If u ( x ) = u ( x ) = · · · = u ( x k ) ≡ t for some t ∈ ( H, max γ E ψ ( x )) . By the results of case 1 of Lemma 3.5, we know that (cid:93) (cid:110) the simply connected components of the sub-level set { x ∈ Ω : u ( x ) < t } (cid:111) ≥ k (cid:80) i =1 m i . k (cid:80) i =1 m i connected components ω such that ω meet γ E . Using the strong maximum principle, we have that u exists at least k (cid:80) i =1 m i local minimalpoints on γ E . Hence, we have k (cid:88) i =1 m i ≤ N. (ii) Case 2: The values at critical points x , · · · , x k are not totally equal. Next we need dividethe proof of case 2 into two situations.(1) Situation 1: If there exists a non-simply connected component ω of { x ∈ Ω : u ( x ) < t } forsome t ∈ ( H, max γ E ψ ( x )) such that ω meets γ E . Without loss of generality, we may suppose that u ( x ) = · · · = u ( x j ) < u ( x j +1 ) = · · · = u ( x j ) < · · · < · · · < u ( x j n − +1 ) = · · · = u ( x j n ) , (3.3)where x , · · · , x j , · · · , x j , · · · , x j n are different critical points in Ω, j n = k and n ≥ . Now we put E j := (cid:110) ω : open set ω is a connected component of { x ∈ Ω : u ( x ) > u ( x j ) } (cid:111) ( j = j , j , · · · , j n ) , and G j n := (cid:110) ω : open set ω is a connected component of { x ∈ Ω : u ( x ) > u ( x j ) } ( j = j , j , · · · , j n )such that there does not exist interior critical point in ω (cid:111) . According to the definition, we know that G j n consists of disjoint components. Denotes | G j n | := (cid:93) { ω : ω is a connected component of G j n } . To illustrate G j n , let us consider an illustration for G j . Assume that u has only three criticalpoints x j , x j +1 , x j with respective multiplicity m j = 1 , m j +1 = 1 , m j = 2 in Ω and u ( x j )
The distribution of elements of G j . By the definition of G j n , we know | G j | = 6 . Now let us show that (cid:12)(cid:12) G j s (cid:12)(cid:12) ≥ j s (cid:80) i =1 m i + 1 by induction on the number s. When s = 1 , the resultholds by case 1. Assume that (cid:12)(cid:12) G j s (cid:12)(cid:12) ≥ j s (cid:80) i =1 m i + 1 for 1 ≤ s ≤ n − . Let s = n. Then, by (3.4) andthe definition of E j , we have { x j n − +1 , · · · , x j n } ⊂ (cid:91) ω ∈ E jn − ω, where ω is a connected component of { x ∈ Ω : u ( x ) > u ( x j n − ) } . { x j n − +1 , · · · , x j n } are contained in exactly q components ω , · · · , ω q . Then x j n − +1 , · · · , x j n together with the corresponding level lines of { x ∈ Ω : u ( x ) = u ( x j n ) } clusteringround these points at least form q connected sets. By the case 3 of Lemma 3.5, we have (cid:102) M := (cid:93) (cid:110) the connected components of { x ∈ Ω : u ( x ) > u ( x j n ) } in all ω j ( j = 1 , · · · , q ) (cid:111) ≥ j n (cid:80) i = j n − +1 m i + q. By using the definition of (cid:12)(cid:12) G j n (cid:12)(cid:12) and the inductive assumption to 1 ≤ s ≤ n − , since { x j n − +1 , · · · , x j n } are contained in exactly q components ω , · · · , ω q , so when we calculate the number of | G j n | , thenumber of | G j n − | will be reduced by q . Then we have (cid:12)(cid:12) G j n (cid:12)(cid:12) = (cid:12)(cid:12) G j n − (cid:12)(cid:12) + (cid:102) M − q ≥ (cid:12)(cid:12) G j n − (cid:12)(cid:12) + ( j n (cid:80) i = j n − +1 m i + q ) − q ≥ j n (cid:80) i =1 m i + 1 . By the strong maximum principle and Lemma 3.1, we have that u has at least k (cid:80) i =1 m i + 1 localmaximal points on γ E . Therefore, we obtain k (cid:88) i =1 m i + 1 ≤ N. (2) Situation 2: Suppose that there exists a non-simply connected component ω of { x ∈ Ω : u ( x ) < t } for some t ∈ ( H, max γ E ψ ( x )) and the external boundary γ of ω is a simply closed curvebetween γ I and γ E such that u has at least one critical point on γ . The idea of proof is essentiallysame as the case 2 of the proof of Theorem 1.1 and the situation 1 of the proof of Theorem 1.3. Herewe omit the proof. This completes the proof of case 2. In this subsection, we investigate the geometric structure of interior critical points of a solutionin a planar, bounded, smooth annular domain Ω with the interior boundary γ I and the externalboundary γ E for the case of ψ having only N global maximal points and N global minimal pointson γ E , where N ≥ . Next we show (1.8) or (1.7) by proving the three “just right”s.
Proof of Theorem 1.4.
We divide the proof into two cases.(1) Case 1: If there exists a non-simply connected component ω of { x ∈ Ω : u ( x ) < t } for some t ∈ ( H, max γ E ψ ( x )) such that ω meets γ E . Next we need divide the proof of case 1 into five steps.Step 1, we should show that u has at least one interior critical point in Ω . Suppose by contradictionthat |∇ u | > . Next, the idea of proof is essentially same as the step 1 in the proof of Theorem1.2. So we omit the proof.Step 2, the first “just right”: According to Lemma 3.4, we assume that the interior criticalpoints of u are x , x , · · · , x k . We show that u ( x ) = u ( x ) = · · · = u ( x k ) = t for some t ∈ R , i.e.,we exclude the case of case 2 in Theorem 1.3. According to the assumption of Theorem 1.4, let q , · · · , q N and p , · · · , p N be respectively the global maximal points and minimal points on γ E . Weset up the usual contradiction argument. We assume that the values at critical points x , · · · , x k arenot totally equal. Without loss of generality, we suppose that u ( x ) < u ( x ) and that m , m arethe respective multiplicities of x , x . Then, by Lemma 3.1, we know that any connected componentof { x ∈ Ω : u ( x ) > u ( x ) } has to meet the boundary γ E and u ( p ) < u ( x ), where p is the minimal15oint of some one connected component of { x ∈ Ω : u ( x ) < u ( x ) } on γ E . At the same time, weknow that any connected component C of { x ∈ Ω : u ( x ) < u ( x ) < u ( x ) } has to meet the boundary γ E and u ( x ) < u ( p ) < u ( x ), where p is the minimal point of connected component C on γ E (see Figure 12). Then u ( p ) (cid:54) = u ( p ), which contradicts with the assumption of Theorem 1.4. Thiscompletes the proof of step 2. Figure 12.
The distribution of the connected components.
Step 3, the second “just right”: we show that x , x , · · · , x k together with the corresponding levellines of { x ∈ Ω : u ( x ) = t } clustering round these points exactly form one connected set. Withoutloss of generality, we suppose by contradiction that x , x , · · · , x k together with the level lines of { x ∈ Ω : u ( x ) = t } clustering round these points form two connected sets. Therefore, there existsa connected components of { x ∈ Ω : u ( x ) < t } , which meets two parts γ , γ of γ E , denoting by A (see Figure 13). Figure 13.
The distribution of some level lines { x ∈ Ω : u ( x ) = (cid:101) t, H < (cid:101) t < t } . Note that u is monotonically decreasing on the connected components of boundary γ E from onemaximal point to the near minimal point. Therefore, { x ∈ A : u ( x ) = t − (cid:15) } exactly exists two levellines in A for any (cid:15) such that 0 < (cid:15) < t − H . This is impossible, because this would imply thateither: u ( x ) = min γ E ψ ( x ) = H in interior points of A , or: there exists two level lines intersect in A ,i.e., there exists critical points in A. This completes the proof of step 3.Step 4, the third “just right”: we show that every simply connected component of { x ∈ Ω : u ( x ) > t } has exactly one global maximal point on boundary γ E . In fact, we assume that somesimply connected component B of { x ∈ Ω : u ( x ) > t } exists two global maximal points on boundary γ E . According to ψ has only N global maximal points and N global minimal points on γ E , thenthere must exist a minimal point (cid:101) p between the two maximal points on γ E such that u ( (cid:101) p ) = H .Since u ( x ) > t > H in B , by the continuity of solution u , this contradicts with the definition ofconnected component B . This completes the proof of step 4.Step 5, By the results of step 3 and the results of case 2 in Lemma 3.5, we have (cid:93) (cid:110) the simply connected components of the super-level set { x ∈ Ω : u ( x ) > t } (cid:111) = k (cid:80) i =1 m i + 1 . (3.4)16n the other hand, using the results of step 4 and the strong maximum principle, therefore we obtain k (cid:80) i =1 m i + 1 = N. (3.5)(2) Case 2: Suppose that there exists a non-simply connected component ω of { x ∈ Ω : u ( x ) < t } for some t ∈ ( H, max γ E ψ ( x )) and the external boundary γ of ω is a simply closed curve between γ I and γ E such that u has at least one critical point on γ . The idea of proof is essentially same as theproof of case 1. Next we need divide the proof of case 2 into four steps.Step 1, the first “just right”: According to Lemma 3.4, we assume that the interior critical pointsof u are x , x , · · · , x k . We show that u ( x ) = u ( x ) = · · · = u ( x k ) = t for some t ∈ R , i.e., weexclude the case of case 2 in Theorem 1.3. The proof is same as the step 2 of the proof of case 1.Step 2, the second “just right”: we show that x , x , · · · , x k together with the correspondinglevel lines of { x ∈ Ω : u ( x ) = t } clustering round these points exactly form one connected set. Theproof is same as the step 3 of the proof of case 1.Step 3, the third “just right”: we show that every simply connected component ω of { x ∈ Ω : u ( x ) < t } has exactly one global minimal point on γ E , where ω meets the external boundary γ E . Infact, we assume that some simply connected component ω of { x ∈ Ω : u ( x ) < t } exists two globalminimal points on boundary γ E . According to ψ has only N global maximal points and N globalminimal points on γ E , then there must exist a maximal point p between the two minimal points on γ E such that u ( p ) = max γ E ψ ( x ). Since u ( x ) < t < max γ E ψ ( x ) in ω , by the continuity of solution u , this contradicts with the definition of connected component ω . This completes the proof of step3. Step 4, By the results of step 2 and the results of step 1 of case 1 in Lemma 3.5, we have (cid:93) (cid:110) the simply connected components ω of the sub-level set { x ∈ Ω : u ( x ) < t } such that ω meet the external boundary γ E (cid:111) = k (cid:80) i =1 m i . (3.6)On the other hand, using the results of step 3 and the strong maximum principle, therefore we obtain k (cid:80) i =1 m i = N. (3.7)This completes the proof of Theorem 1.4.Let N = 1, we have: Corollary 3.7.
Suppose that u is a non-constant solution of (1.5) and that ψ has exactly onemaximal point on γ E . Then u has at most one interior critical point p in Ω . If u has one interiorcritical point p , then the multiplicity of the interior critical point p is one. According to Theorem 1.3 and Theorem 1.4, we can easily have the following results.
Corollary 3.8.
Let Ω be a bounded smooth annular domain with the interior boundary γ I and theexternal boundary γ E in R . Suppose that ψ ( x ) ∈ C (Ω) , H is a given constant, ψ ( x ) ≤ H. Let u bea non-constant solution of (1.5). Then we have:(i) If ψ has N local minimal points on γ E , then u has finite interior critical points and inequality(1.6) holds;(ii) If ψ has only N global minimal points and N global maximal points on γ E , i.e., all themaximal and minimal points of ψ are global, then u has finite interior critical points and equality(1.7) or (1.8) holds; iii) If ψ has exactly one minimal point on γ E . Then u has at most one interior critical point p in Ω . If u has one interior critical point p , then the multiplicity of the interior critical point p isone. min γ E ψ ( x ) < H < max γ E ψ ( x ) In this subsection, we put z := min γ E ψ ( x ) , Z := max γ E ψ ( x ) . In order to prove Theorem 1.5, weneed the following basic lemmas.
Lemma 4.1.
Let u be a non-constant solution of (1.5), constant H satisfies z < H < Z. (1) For any t ∈ ( H, Z ) , then any connected component ω of { x ∈ Ω : u ( x ) > t } has to meet theexternal boundary γ E . (2) For any t ∈ ( z, H ] , then the connected component ω of { x ∈ Ω : u ( x ) > t } is simply connectedor non-simply connected. If ω is simply connected, which has to meet the external boundary γ E . Proof. (1) Since t ∈ ( H, Z ) , the proof is same as the proof of Lemma 3.1. The results of (2)naturally holds. In fact, by the strong maximum principle and u | γ I = H , any connected component ω of { x ∈ Ω : u ( x ) > H } or { x ∈ Ω : u ( x ) < H } has to meet the external boundary γ E . Lemma 4.2.
Suppose that x is an interior critical point of u in Ω and that m is the multiplicityof x . Then m + 1 distinct connected components of { x ∈ Ω : u ( x ) > u ( x ) } cluster around the point x .Proof. According to the results of Hartman and Wintner [13], in a neighborhood of x the level line { x ∈ Ω : u ( x ) = u ( x ) } consists of m + 1 simple arcs intersecting at x . By the results of Lemma4.1, there exist m + 1 distinct connected components of { x ∈ Ω : u ( x ) > u ( x ) } clustering aroundthe point x . This completes the proof. Lemma 4.3.
If there exists t ∈ ( z, H ) such that a connected component ω of { x ∈ Ω : u ( x ) > t } isnon-simply connected and the external boundary of ω is a simply closed curve between γ I and γ E .Then there does not exist any interior critical point in ω .Proof. The proof is same as the proof of Lemma 3.3, so we omit the proof.
Lemma 4.4.
Suppose that u is a non-constant solution to (1.5). Then u has finite interior criticalpoints in Ω .Proof. We set up the usual contradiction argument. Suppose that u has infinite interior criticalpoints in Ω , denoting by x , x , · · · . The results of Lemma 4.1 and Lemma 4.2 show that thereexists infinite simply connected components of { x ∈ Ω : u ( x ) > u ( x i ) } ( i = 1 , , · · · ), which meet theexternal boundary γ E . The strong maximum principle implies that there exists at least a maximumpoint on γ E for any simply connected component ω of { x ∈ Ω : u ( x ) > u ( x i ) } such that ω meet theexternal boundary γ E . Therefore there exists infinite maximal points on γ E , this contradicts withthe assumption. This completes the proof. Lemma 4.5.
Let x , x , · · · , x k be the interior critical points of u in Ω . Suppose that u ( x ) = u ( x ) = · · · = u ( x k ) ≡ t and that all the critical points x , x , · · · , x k together with the correspondinglevel lines of { x ∈ Ω : u ( x ) = t } clustering round these points form q connected sets, where q ≥ nd m , m , · · · , m k are the multiplicities of critical points x , x , · · · , x k respectively.Case 1: Suppose that there exists a non-simply connected component ω of { x ∈ Ω : u ( x ) > t } forsome t ∈ ( z, H ) and the external boundary γ of ω is a simply closed curve between γ I and γ E suchthat u has at least one critical point on γ , then (cid:93) (cid:110) the simply connected components ω of the super-level set { x ∈ Ω : u ( x ) > t } such that ω meet the external boundary γ E (cid:111) = k (cid:80) i =1 m i + q − . (4.1) Case 2: Suppose that t = H or that there exists a non-simply connected component ω of { x ∈ Ω : u ( x ) > t } for some t ∈ ( z, H ) such that ω meets γ E . In addition, we set M and M as thenumber of the connected components of the super-level set { x ∈ Ω : u ( x ) > t } and the sub-level set { x ∈ Ω : u ( x ) < t } , respectively. Then M ≥ k (cid:80) i =1 m i + 1 , M ≥ k (cid:80) i =1 m i + 1 , and M + M = 2 k (cid:80) i =1 m i + q + 1 . (4.2) Case 3: Suppose that t ∈ ( H, Z ) , the results see Lemma 3.5.Proof. (i) Case 1: The proof is same as the proof of case 1 of Lemma 3.5.(ii) Case 2: Lemma 2.4 and the case 2 of Lemma 3.5 implies case 2.We are now ready to prove Theorem 1.5. Proof of Theorem 1.5. (i) Case 1: If u ( x ) = u ( x ) = · · · = u ( x k ) ≡ t . By the results of Lemma 4.5,we know that (cid:93) (cid:110) the simply connected components ω of the super-level set { x ∈ Ω : u ( x ) > t } for t ∈ ( z, H ] such that ω meet the external boundary γ E (cid:111) ≥ k (cid:80) i =1 m i , or (cid:93) (cid:110) the simply connected components ω of the sub-level set { x ∈ Ω : u ( x ) < t } for t ∈ ( H, Z ) such that ω meet the external boundary γ E (cid:111) ≥ k (cid:80) i =1 m i , By the strong maximum principle, we have that u exists at least k (cid:80) i =1 m i local maximal points orminimal points on γ E . Hence, we have k (cid:88) i =1 m i ≤ N. (ii) Case 2: The values at critical points x , · · · , x k are not totally equal. The idea of proof isessentially same as the case 2 of the proof of Theorem 1.1 and Theorem 1.3. Here we omit theproof. In this subsection, we investigate the geometric structure of interior critical points of a solutionin a planar, bounded, smooth annular domain Ω with the interior boundary γ I and the externalboundary γ E for the case of min γ E ψ ( x ) < H < max ∂ Ω ψ ( x ) and ψ having only N global maximal pointsand N global minimal points on γ E , where N ≥
2. Next we show (1.11) or (1.10) by proving thethree “just right”s. 19 roof of Theorem 1.6.
We divide the proof into three cases.(1) Case 1: Suppose that there exists a non-simply connected component ω of { x ∈ Ω : u ( x ) > t } for some t ∈ ( z, H ) such that ω meets γ E or that there exists critical value H . The idea of proof isessentially same as the proof of Theorem 1.2 and the case 1 of Theorem 1.4. Then we have k (cid:80) i =1 m i + 1 = N, (4.3)where m , m , · · · , m k are the multiplicities of critical points x , x , · · · , x k respectively.(2) Case 2: Suppose that there exists a non-simply connected component ω of { x ∈ Ω : u ( x ) > t } for some t ∈ ( z, H ) and the external boundary γ of ω is a simply closed curve between γ I and γ E such that u has at least one critical point on γ . We deduce (1.10) by proving the three “just right”s.We should divide the proof of case 2 into four steps.Step 1, the first “just right”: According to Lemma 4.4, we assume that the interior critical pointsof u are x , x , · · · , x k . We show that u ( x ) = u ( x ) = · · · = u ( x k ) ≡ t, i.e., we exclude the case ofcase 2 in Theorem 1.5. The proof is same as the step 2 of the proof of case 1 of Theorem 1.4.Step 2, the second “just right”: we show that x , x , · · · , x k together with the correspondinglevel lines of { x ∈ Ω : u ( x ) = t } clustering round these points exactly form one connected set. Theproof is same as the step 3 of the proof of case 1 of Theorem 1.4.Step 3, the third “just right”: we show that every simply connected component ω of { x ∈ Ω : u ( x ) > t } has exactly one global maximal point on boundary γ E , where ω meets the externalboundary γ E . In fact, we assume that some simply connected component B of { x ∈ Ω : u ( x ) > t } exists two global maximal points on boundary γ E . According to ψ has only N global maximal pointsand N global minimal points on γ E , then there must exist a minimal point p between the twomaximal points on γ E such that u ( p ) = z . Since u ( x ) > t > z in B , by the continuity of solution u ,this contradicts with the definition of connected component B . This completes the proof of step 3.Step 4, By the results of step 2 and case 1 of Lemma 4.5, we have (cid:93) (cid:110) the simply connected components ω of the super-level set { x ∈ Ω : u ( x ) > t } such that ω meet the external boundary γ E (cid:111) = k (cid:80) i =1 m i . (4.4)On the other hand, using the results of step 3 and the strong maximum principle, therefore we obtain k (cid:80) i =1 m i = N. (4.5)Case 3: For t ∈ ( H, Z ), the results see Theorem 1.4. This completes the proof of Theorem 1.6.Let N = 1, we have: Corollary 4.6.
Suppose that min γ E ψ ( x ) < H < max ∂ Ω ψ ( x ) and that ψ has exactly one maximal pointon γ E , Let u be a non-constant solution of (1.5). Then u has at most one interior critical point p in Ω . If u has one interior critical point p , then the multiplicity of the interior critical point p is one. Acknowledgement.