227 research outputs found
Uniqueness of ground states for quasilinear elliptic equations in the exponential case
No full textThe boundary value problem and notation are as in previous paper of Pucci and Serrin [Indiana Univ. Math. J. 47 (1998), no. 2, 501-528]. A uniqueness theorem for positive solutions is now proved for nonlinearities f of exponential type. In order that the main theorem of the authors' paper can be applied, the only difficulty is verification of the hypothesis (F/f)′≥0. This is very delicate. Also, under the stronger regularity condition on the elliptic part, ground states with connected support are shown to be radially symmetric in the plane, and hence unique up to translations by the main theorem of Pucci and Serrin. Related results were obtained recently by Adimurthi [Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), no. 5, 895-906]
Global asymptotic stability for strongly nonlinear second order systems
No full textThe global asymptotic stability of the rest point for nonlinear equations has been studied by Levin & Nohel, by Artstein & Infante, We extended this studies to extremals of scalar variational problems in Section 5 of our paper [P. Pucci and J. Serrin, Continuation and limit properties for solutions of strongly nonlinear second order differential equations, Asymptotic Anal. 4 (1991), 97–160], where the Euler-Lagrange equation exhibits even stronger nonlinearities. In this paper we treat the nontrivial case when the extremals are vectorial. We show that the results of our previous papers carry over to the vector case in a surprisingly close way, enough even to suggest that it is the variational character of the nonlinear system, more than anything else, which produces the desired asymptotic stability
Some remarks on the global nonexistence problem for nonautonomous abstract evolution equations
No full textIn this paper we consider the problem of nonexistence of solutions for abstract evolution equations of the type Putt+Q(t)ut+A(t,u)=F(t,u), 0≤t<∞, where P and Q(t) are linear selfadjoint operators. The first theorem is an extension of earlier works of Levine [cf. SIAM J. Math. Anal. 5 (1974), 138-146], where the case where Q(t) is independent of t is treated. Secondly, the authors consider the case where Q(t,ut) is nonlinear and show that under certain assumptions the problem is essentially reduced to the case where A(t,u)=A(u) and F(t,u)=F(u), which has been treated by Levine and Serrin [Arch. Rational Mech. Anal. 137 (1997), 341-361]
A mountain pass theorem
No full textSome variants of one of the main results in critical point theory, namely the mountain pass lemma by A. Ambrosetti and P. H. Rabinowitz [J. Funct. Anal. 14 (1973), 349-381], are proved in this paper. In this paper we prove that if X is finite-dimensional, then the mountain may have "zero'' altitude. In the infinite-dimensional case, provided that the mountain should have nonzero "thickness''. The main tool for the proofs is a variant of a well-known lemma by Clarke. Some applications to periodic functions and periodic solutions of the forced pendulum equation are included
A general variational identity
No full textIn 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
Asymptotic stability for intermittently controlled nonlinear oscillators
No full textIn this paper we obtain sharp conditions for the uniform asymptotic stability of u=0 with respect to an equation having the form (ΔL(t,u,u′))′−ΔuL(t,u,u′)=Q(t,u,u′) with L=G(u,u′)−F(t,u). This form includes a variety of oscillatory-type equations with damping. The sought for conditions are assumed to hold on time intervals In, n=1,2,⋯, which are arbitrarily spaced in [0,∞); the coefficients may be arbitrary otherwise, e.g., no damping at all or unbounded damping which amounts to freezing the state, etc. Thus the damping on the intervals should imply the asymptotic stability. Several more concrete examples illustrate and motivate the general results, which are at times surprising. For instance, in the particular case u′′+A(t,u,u′)u′+f(u)=0, under the standard conditions and with A(t,u,u′) continuous and uniformly positive definite on the union of In, the asymptotic stability is guaranteed by ∑|In|3=∞, and the exponent 3 is the best possible. Other interesting examples are provided
Global Nonexistence for Abstract Evolution Equations with Positive Initial Energy
No full textAbstractIn this paper we consider the problem of noncontinuation of solutions of the initial value problem for abstract evolution equations of the formPutt+Q(t)ut+A(t,u)=F(t,u),t∈J=[0,∞),wherePandQare linear self-adjoint operators, andA(t,u) andF(t,u) are respectively a linear operator inu(typically of differential type) and a nonlinear driving force. Our principal concern is the noncontinuation (or blow-up) of solutions when the initial energy is positive, but appropriately bounded
Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters
No full textA global nonexistence theorem for quasilinear evolution equations with dissipation
No full textLevine, H.A.; Serrin, J.. (1995). A global nonexistence theorem for quasilinear evolution equations with dissipation. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/2801
Asymptotic properties for solutions of strongly nonlinear second order differential equations
No full textIn this paper we consider equations of the form (A(u′)u′)′+δ(r)A(u′)u′+f(r,u)=0 on the real line, where A=A(p) determines the nonlinearity of the equation in the derivative u′, and f is a continuous function containing the nonlinearity in u. Examples are the Bessel equation, the Lane-Emden equation of astrophysics, the Emden-Fowler equation, the Haraux-Weissler equation, and radial versions of divergence-type quasilinear partial differential equations. The results include generalizations of theorems of Fowler, Levin and Nohel, and Artstein and Infante; we also show the detailed asymptotic behavior of solutions as r→∞ when A, δ and f are of asymptotically algebraic type in their arguments
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