Existence and bifurcation theorems for nonlinear elliptic eigenvalue problems on unbounded domains

We consider nonlinear elliptic eigenvalue problems on unbounded domains G?$\text{R}$ⁿ. Using an extended Ljusternik-Schnirelman theory we prove the existence of infinitely many eigenfunctions on every sphere in L²(G). Moreover, we establish that the infimum λ1 of the spectrum of the linearized problem L is always a bifurcation point. In addition, there is an infinity of branches emanating at λ1 from the trivial line of solutions if λ1 belongs to the essential spectrum of L.     

Nonlinear eigenvalue problems for quasilinear equations in unbounded domains

We prove the existence of a solution of the nonlinear equation in IR^N and in exterior domains, respectively. We concentrate to the case when p ≥ N and the nonlinearity f(x, · ) is “superlinear” and “subcritical”.     

Nonlinear eigenvalue problems for quasilinear operators on unbounded domains

We prove several existence results for eigenvalue problems involving the p-Laplacian and a nonlinear boundary condition on unbounded domains. We treat the non-degenerate subcritical case and the solutions are found in an appropriate weighted Sobolev space. Received May 2000     

On a class of quasilinear eigenvalue problems on unbounded domains

We prove several existence and non-existence results for a quasilinear eigenvalue problem with nonlinear boundary condition on unbounded domain. Our paper extends previous results obtained in Chabrowski [1] and Pflüger [4].     

Multiplicity results for a class of quasilinear eigenvalue problems on unbounded domains

Using a recent result of Ricceri [10] we prove a multiplicity result for a class of quasilinear eigenvalue problems with nonlinear boundary conditions on an unbounded domain. Our paper completes previous results obtained by Carstea and Rădulescu [4], Chabrowski [1], [2], Kandilakis and Lyberopoulos [6] and Pflüger [7]. Received: 17 April 2007     

An existence result for a class of quasilinear elliptic eigenvalue problems in unbounded domains

We consider a nonlinear eigenvalue problem under Robin boundary conditions in a domain with (possibly noncompact) smooth boundary. The problem involves a weighted p–Laplacian operator and subcritical nonlinearities satisfying Ambrosetti–Rabinowitz type conditions. Using Morse theory and a cohomological local splitting as in Degiovanni et al. (Commun Contemp Math 12:475–486, 2010), we prove the existence of a nontrivial weak solution for all (real) values of the eigenvalue parameter. Our result is new even in the semilinear case p = 2 and complements some recent results obtained in Autuori et al. (Adv Anal Equ 18:1–48, 2013).     

THE SINGULARLY PERTURBED NONLINEAR ELLIPTIC SYSTEMS IN UNBOUNDED DOMAINS

Abstract. The singularly perturbed problems for elliptic systems in unbounded domains are considered. Under suitable conditions and by using the comparison theorem the existence and asymptotic behavior of solution for the boundary value problems studied,     

Semilinear elliptic resonant problems at higher eigenvalue with unbounded nonlinear terms

In this paper we study the existence of nontrivial solutions of a class of asymptotically linear elliptic resonant problems at higher eigenvalues with the nonlinear term which may be unbounded by making use of the Morse theory for aC ²-function at both isolated critical point and infinity.     

Semilinear problems in unbounded domains

As formulated by Silva [E.A. de B.e. Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Anal. 16 (1991) 455-477] and Schechter [M. Schechter, A generalization of the saddle point method with applications, Ann. Polon. Math. 57 (3) (1992) 269-281; M. Schechter, New saddle point theorems, in: Generalized Functions and Their Applications, Varanasi, 1991, Plenum, New York, 1993, pp. 213-219], the sandwich theorem has become a very useful tool in finding critical points of functionals leading to solutions of partial differential equations. In the present paper, this theorem is strengthened to apply to more general situations. We present some applications.     

Existence of positive solutions of some nonlinear elliptic problems in unbounded domains

In this paper, we study the existence of positive solutions of some nonlinear elliptic problems in unbounded domains. The existence is affected by the properties of the geometry and the topology of the domain.     

Elliptic and parabolic problems in unbounded domains

We consider elliptic and parabolic problems in unbounded domains. We give general existence and regularity results in Besov spaces and semi‐explicit representation formulas via operator‐valued fundamental solutions which turn out to be a powerful tool to derive a series of qualitative results about the solutions. We give a sample of possible applications including asymptotic behavior in the large, singular perturbations, exact boundary conditions on artificial boundaries and validity of maximum principles. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Attractors for nonlinear reaction-diffusion systems in unbounded domains via the method of short trajectories

We consider a nonlinear reaction-diffusion equation on the whole space Rd. We prove the well-posedness of the corresponding Cauchy problem in a general functional setting, namely, when the initial datum is uniformly locally bounded in L² only. Then we adapt the short trajectory method to establish the existence of the global attractor and, if d?3, we find an upper bound of its Kolmogorov's ε-entropy.     

Eigenfunctions for singular fully nonlinear equations in unbounded domains

In this paper, we prove the existence of a generalized eigenvalue and a corresponding eigenfunction for fully nonlinear elliptic operators singular or degenerate, homogeneous of degree 1+α, α > −1 in unbounded domains of IR ^N . The main tool will be Harnack’s inequality.     

Structured pseudospectra for nonlinear eigenvalue problems

In this paper we introduce structured pseudospectra for nonlinear eigenvalue problems and derive computable formulae. The results are applied to the sensitivity analysis of the eigenvalues of a second-order system arising from structural dynamics and of a time-delay system arising from laser physics. In the former case, a comparison is made with the results obtained in the framework of random eigenvalue problems.