Asymptotically linear elliptic problems at resonance

Summary In this paper we study the existence of solutions for some semilinear, elliptic equation with homogeneous Dirichelet boundary conditions and a non linear term which is asymptotically linear «at resonance».In memoria di Paolo Bartolo     

Asymptotic behavior of positive solutions of some quasilinear elliptic problems

We discuss the asymptotic behavior of positive solutions ofthe quasilinear elliptic problem –_pu = a u^p–1–b(x)u^q, u| = 0, as q p – 1 + 0 and as q , via a scale argument.Here _p is the p-Laplacian with 1 < p and q > p –1. If p = 2, such problems arise in population dynamics. Ourmain results generalize the results for p = 2, but some technicaldifficulties arising from the nonlinear degenerate operator–_p are successfully overcome. As a by-product, we cansolve a free boundary problem for a nonlinear p-Laplacian equation.     

Multiplicity results for asymptotically linear elliptic problems at resonance

In this paper several new multiplicity results for asymptotically linear elliptic problem at resonance are obtained via Morse theory and minimax methods. Some new observations on the critical groups of a local linking-type critical point are used to deal with the resonance case at 0.     

Existence and multiplicity results for nonlinear elliptic problems with linear part at resonance the case of the simple eigenvalue

We prove existence and multiplicity theorems for nonlinear elliptic boundary value problems with noninvertible linear parts. It is considered the case in which the kernel of the linear part is unidimensional and the nonlinearity is taken either bounded or sublinear at infinity. Applications to nonlinear elasticity are given.     

Applications of local linking to asymptotically linear elliptic problems at resonance

We prove the existence of nontrivial solutions for asymptotically linear elliptic problems at resonance without assuming that the linearized equations at zero and infinity are different. The proof is based on a penalization technique and Morse index estimates for critical points produced by local linking. Received October 10, 1997     

Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian

In this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional Laplacians and the critical exponents. This work can be seen as a nonlocal analog of the results of Han (1991) [24] and Rey (1990) [35].     

Asymptotic behavior of solutions to multiplicative control problems for elliptic equations

The problems of optimal multiplicative control for the Helmholtz equation and the diffusion equation are studied. The control function is included multiplicatively in a mixed-type boundary condition specified on the entire domain boundary or its part. For each of the models under study, an iterative method for determining an approximate solution is constructed and theoretically substantiated for sufficiently large values of the regularization parameter.     

Asymptotic behavior for elliptic problems with singular coefficient and nearly critical Sobolev growth

Let be a ball centered at the origin with radius R. We investigate the asymptotic behavior of positive solutions for the Dirichlet problem in on ∂B_R when ɛ→⁺ for suitable positive numbers μ Mathematics Subject Classification (2000) 35J60, 35B33     

Semilinear degenerate elliptic boundary value problems at resonance

The purpose of this paper is to study a class of semilinear degenerate elliptic boundary value problems at resonance which include as particular cases the Dirichlet and Robin problems. The approach here is based on the global inversion theorems between Banach spaces, and is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. By making use of the Lyapunov–Schmidt procedure and the global inversion theorem, we prove existence and uniqueness theorems for our problem. The results here extend an earlier theorem due to Landesman and Lazer to the degenerate case.     

Nontrivial solutions of elliptic boundary value problems with resonance at zero

Summary We consider a general nonlinear parabolic BVP (P) on a bounded and smooth domain R_n, the nonlinearity being given by a functionf: . We impose various hypotheses on f: « nonresonance » (with respect to the linearized BVP) at infinity, « nonresonance » or «resonance» at zero. Using an extension of Conley's index theory to noncompact spaces, we prove the existence of equilibria of (P) (i.e. solutions of a corresponding elliptic equation), as well as trajectories joining some of these equilibria. The results obtained generalize earlier results of Amann and Zehnder (who were the first to apply the Conley index to elliptic equations), of Peitgen and Schmitt, and of this author.Dedicated to Professor Jack K. Hale on his 55-th birthdayThis research was supported, in part, by a grant from the Deutsche Forschungsgemeinschaft (D.F.G.).     

Asymptotic behavior of large solution for boundary blowup problems with non-linear gradient terms

Let Ω⊂R^N(N?3) be a bounded domain with smooth boundary. We show the asymptotic behavior of boundary blowup solutions to non-linear elliptic equation Δu±|∇u|^q=b(x)f(u) in Ω, subject to the singular boundary condition u(x)=∞ as dist(x,∂Ω)→0,f is Γ-varying at ∞. Our analysis is based on the Karamata regular variation theory combined with the method of lower and supper solution.     

Asymptotic behavior of the fundamental solution of an elliptic equation with respect to a complex parameter

Maslov's canonical operator method is used for constructing the asymptotic behavior with respect to a complex parameter of the fundamental solution of a secondorder elliptic equation with smooth finite coefficients. The asymptotic form is constructed on the assumption that all trajectories of the corresponding Hamiltonian system depart to infinity. The asymptotic form is used for investigating the analytic properties of the fundamental solution.Translated from Matematicheskie Zametki, Vol. 21, No. 3, pp. 377–390, March, 1977.The author thanks V. V. Kucherenko for suggesting the problem and continued interest, V. P. Maslov, V. P. Mikhailov, and L. A. Muravei for valuable advice and comments, and M. V. Karasev for his interest.     

Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters, Part III

We study the asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters. Thanks to an additional (fixed) parameter, we show that two different critical exponents play a crucial role in the asymptotic analysis, giving an explanation of the phenomena discovered in Gazzola et al. (Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters, Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear) and Gazzola and Serrin (Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002) 477).