Pohozaev-type identities for differential operators driven by homogeneous vector fields
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Nonlinear Differential Equations and Applications NoDEA
Pohozaev-type identities for differential operators driven by homogeneous vector fields Stefano Biagi, Andrea Pinamonti and Eugenio Vecchi Abstract. We prove Pohozaev-type identities for smooth solutions of EulerLagrange equations of second and fourth order that arise from functional a depending on homogeneous H¨ ormander vector fields. We then exploit such integral identities to prove non-existence results for the associated boundary value problems. Keywords. sub-elliptic semilinear equations, Non-existence results, Pohozaevtype identities, Geometric methods for boundary-value problems.
1. Introduction In [33] Pohozaev proved an integral identity for solutions of the following elliptic boundary value problem: −Δu = f (u) in Ω, (1.1) u = 0 on ∂Ω, where Ω ⊂ Rn is an open and bounded set with smooth boundary ∂Ω. In particular, under the geometric assumption on the set Ω of being star-shaped, he was able to show that (1.1) does not admit non-trivial solutions in C 2 (Ω) ∩ C 1 (Ω). In 1986 in [34]. Pucci and Serrin extended the approach of Pohozaev proving integral identities for solutions of a large class of variational PDEs coming from functionals possibly depending on the Hessian. The authors are members of INdAM. S. Biagi is partially supported by the INdAMGNAMPA 2020 project Metodi topologici per problemi al contorno associati a certe classi di equazioni alle derivate parziali. A. Pinamonti and E. Vecchi are partially supported by the INdAM-GNAMPA 2020 project Convergenze variazionali per funzionali e operatori dipendenti da campi vettoriali. 0123456789().: V,-vol
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NoDEA
The starting idea of Pohozaev goes actually back to Rellich and Nehari [30,35] and can be summarized as follows: given a sufficiently smooth solution of (1.1), it suffices to multiply the equation −Δu = f (u) by x · ∇u, integrate over Ω and apply the Divergence Theorem. This will lead to the celebrated Pohozaev identity 2 ∂u 1 n−2 2 x, ν dx = 0, ∇u dx − n F (u) dx + (1.2) 2 2 ∂Ω ∂ν Ω Ω where F is a primitive of f and ν denotes the unit outward normal of ∂Ω. From (1.2) it is pretty easy to get nonexistence results (of non-trivial solutions) under appropriate assumptions on the nonlinearity f , and therefore on F itself. Assume, e.g., that F (u) ≤ 0. Then 2 ∂u n−2 1 x, ν dx. 0≥n F (u) dx = ∇u2 dx + 2 2 ∂Ω ∂ν Ω Ω Since the first integral on the right hand side (r.h.s., in short) is non-negative, it is now clear that the entire sign of the r.h.s. depends on x · ν; the latter is a purely geometric quantity, only depending on the set Ω and not on the solution u. Therefore, assuming for example that Ω is star-shaped, one can achieve the famous non-existence result of Pohozaev. Since the paper of Pucci and Serrin [34], the intimate connection between integral identities of Pohozaev-type and non-existence results has been the object of study of many papers, which have extended the ideas previously recalled to cover an alway
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