In this paper we generalize the nonexistence result of S. I. Pokhozhaev concerning the problem Δu+f(u)=0 in Ω,u=0 on ∂Ω [Soviet Math. Dokl. 6 (1965),1408-1411] to the general variational problem δ∫ΩF(x,u,Du)dx=0,u=0 on ∂Ω. The main result is the following theorem: Let Ω be a bounded star-shaped (with respect to the origin) domain and piFpi(x,0,p)−F(x,p)≥0,x∈∂Ω,p∈Rn; let there exist a real number n such that nF(x,p)+xiFxi(x,p)−auFu(x,p)−(a+1)piFpi(x,(x,p)∈Ω×R×Rn and let either u=0 or p=0 whenever equality holds. Then the above variational problem has no trivial solution u∈C2(Ω)∩C1(Ω¯). Pokhozhaev's result corresponds to the case F(x,p)=(|p|2/2)−F(u). The proof is based on an identity for the solution to the problem, generalizing a corresponding Pokhozhaev identity. For the special case F(x,p)=G(p)−F(x,u) a number of immediately ensuing applications are given. Using the identity we obtain extend a nonexistence result of P. H. Rabinowitz [J. Funct. Anal. 7 (1971),487-513] and H. Brezis and L. Nirenberg [Comm. Pure Appl. Math. 36 (1983),437-477] concerning the radial eigenvalue problem to bounded star-shaped domains and more general operators. The main result of the paper is extended to the case of vector-valued extremals and higher-order equations. In particular,new results are obtained for the system Δui+fi(u1,⋯,uN)=0,i=1,2,N,and for the semilinear poliharmonic equation (−Δ)Ku−f(u)=0,K≥2.

A general variational identity

PUCCI, Patrizia;
1986

Abstract

In this paper we generalize the nonexistence result of S. I. Pokhozhaev concerning the problem Δu+f(u)=0 in Ω,u=0 on ∂Ω [Soviet Math. Dokl. 6 (1965),1408-1411] to the general variational problem δ∫ΩF(x,u,Du)dx=0,u=0 on ∂Ω. The main result is the following theorem: Let Ω be a bounded star-shaped (with respect to the origin) domain and piFpi(x,0,p)−F(x,p)≥0,x∈∂Ω,p∈Rn; let there exist a real number n such that nF(x,p)+xiFxi(x,p)−auFu(x,p)−(a+1)piFpi(x,(x,p)∈Ω×R×Rn and let either u=0 or p=0 whenever equality holds. Then the above variational problem has no trivial solution u∈C2(Ω)∩C1(Ω¯). Pokhozhaev's result corresponds to the case F(x,p)=(|p|2/2)−F(u). The proof is based on an identity for the solution to the problem, generalizing a corresponding Pokhozhaev identity. For the special case F(x,p)=G(p)−F(x,u) a number of immediately ensuing applications are given. Using the identity we obtain extend a nonexistence result of P. H. Rabinowitz [J. Funct. Anal. 7 (1971),487-513] and H. Brezis and L. Nirenberg [Comm. Pure Appl. Math. 36 (1983),437-477] concerning the radial eigenvalue problem to bounded star-shaped domains and more general operators. The main result of the paper is extended to the case of vector-valued extremals and higher-order equations. In particular,new results are obtained for the system Δui+fi(u1,⋯,uN)=0,i=1,2,N,and for the semilinear poliharmonic equation (−Δ)Ku−f(u)=0,K≥2.
1986
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11391/117435
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