One-Sided Resonance Problems for Quasilinear Elliptic Operators

Robinson, Stephen B.; Rumbos, Adolfo J.; Shapiro, Victor L.

doi:10.1006/jmaa.2000.7333

Cited by 6 publications

(2 citation statements)

References 16 publications

(12 reference statements)

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“…The first inequality follows from the ellipticity condition and a Poincaretype inequality. For the details of a more general estimate see Lemma 3.2 in [8]. The second inequality follows from the growth condition on A.…”

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confidence: 99%

On The Second Eigenvalue For Nonhomogeneous Quasi-linear Operators

Robinson¹

2004

SIAM J. Math. Anal.

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In this paper we study a nonlinear eigenvalue problem and associated perturbations of the problem. More specifically, we generalize a variational characterization of the second eigenvalue for homogenous quasi-linear elliptic operators, such as the p-Laplacian, to a class of nonhomogeneous quasi-linear elliptic operators. Neumann boundary data is assumed throughout the paper. To demonstrate the utility of this characterization we use it to prove a generalized Fredholm alternative for nonresonant perturbations of the given eigenvalue problem.

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On The Second Eigenvalue For Nonhomogeneous Quasi-linear Operators

Robinson¹

2004

SIAM J. Math. Anal.

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“…These results were obtained via various methods, such as minimax method, the degree theory, Morse theory, Leray-Schauder degree theory, saddle point theorem, Galerkin-type techniques, Brouwer's fixed point theorem and so on. One can refer to [2][3][4][5][6][7][8], etc. However, there seem to be relatively few papers that consider the singular quasilinear elliptic equations in weighted Sobolev space.…”

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Existence Results in Weighted Sobolev Spaces for Some Singular Quasilinear Elliptic Equations

Jia

Zhao

2008

Acta Appl Math

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In this paper, we obtain the existence of a nontrivial solution for a class of singular quasilinear elliptic equations in weighted Sobolev spaces. The proofs rely on Galerkin-type techniques, Brouwer fixed point theorem, and a new weighted compact Sobolev-type embedding theorem established by Shapiro. The equation is one of the most useful sets of Navier-Stokes equations, which describe the motion of viscous fluid substances such as liquids, gases and so on.

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