Existence of the Eigenvalues for the Cone Degenerate p-Laplacian

Abstract

The present paper is concerned with the eigenvalue problem for cone degenerate p-Laplacian. First the authors introduce the corresponding weighted Sobolev s-paces with important inequalities and embedding properties. Then by adapting Lusternik-Schnirelman theory, they prove the existence of infinity many eigenvalues and eigenfunctions. Finally, the asymptotic behavior of the eigenvalues is given.

Keywords

Quasi-linear / Degenerate operator / Variational methods

References

[1] Amann H. Lusternik-Schnirelman theory and nonlinear eigenvalue problems. Math. Ann., 1972, 199: 55-72

[2] Azorero J G, Alonso I P. Existence and nonuniqueness for the p-Laplacian: nonlinear eigenvalues. Commun. in PDE, 1987, 12: 1389-1430

[3] Brezis H. Functional Analysis, Sobolev Spaces, and Partial Differential Equations, 2011, New York: Springer-Verlag

[4] Cao D, Peng S, Yan S. Infinitely many solutions for p-Laplacian equation involving critical Sobolev growth. Journal of Functional Analysis, 2012, 262(6): 2861-2902

[5] Carl S, Motreanu D. Multiple and sign-changing solutions for the multivalued p-Laplacian equation. Mathematische Nachrichten, 2010, 283(7): 965-981

[6] Chen H, Liu X, Wei Y. Existence theorem for a class of semi-linear totally characteristic elliptic equations with Cr Y.itical cone sobolev exponents. Annals of Global Analysis and Geometry, 2011, 39: 27-43

[7] Chen H, Liu X, Wei Y. Cone Sobolev inequality and dirichlet problems for nonlinear elliptic equations on manifold with conical singularities. Calculus of variations and PDEs, 2012, 43(3): 463-484

[8] Chen H, Liu X, Wei Y. Multiple solutions for semilinear totally characteristic elliptic equations with subcritical or critical cone sobolev exponents. J. Differential Equations, 2012, 252: 4200-4228

[9] Chen, H. and Wei, Y., Multiple solutions for nonlinear cone degenerate p-Laplacian equations, preprint, 2018.

[10] Chen H, Wei Y, Zhou B. Existence of solutions for degenerate elliptic equations with singular potential on conical singular manifolds. Math. Nachrichten, 2012, 285(11–12): 1370-1384

[11] Coriasco S, Schrohe E, Seiler J. Realizations of differential operators on conic manifolds with boundary. Annals of Global Analysis and Geometry, 2007, 31: 223-285

[12] Drabek P. Resonance problems for the p-Laplacian. Journal of Functional Analysis, 1999, 169(1): 189-200

[13] Egorov J V, Schulze B-W. Pseudo-differential operators, singularities, applications. Operator Theory, Advances and Applications, 1997, Basel: Birkhäuser Verlag 93

[14] Evans L C. Partial Differential Equations, 2010 2nd ed. Providence, Rhode Island: R, American. Mathematical Society

[15] Melrose, R. B. and Mendoza, G. A., Elliptic operators of totally characteristic type, Preprint, Math. Sci. Res. Institute, MSRI 047–83, 1983.

[16] Rabinowitz P H. Some Aspects of nonlinear eigenvalue problems. Rocky Mountain J. of Math., 1973, 2(2): 70

[17] Rabinowitz P H. Minimax Methods in Critical Point Theory with Applications to Differential Equations, 1986, Providence, Rhode Island: A.M.S. CBMS Regional Conference, 65

[18] Schulze B-W. Boundary Value Problems and Singular Pseudo-differential Operators, 1998, Chichester: J. Wiley

[19] Schrohe E, Seiler J. Ellipticity and invertibility in the cone algebra on L _p-Sobolev spaces. Integral Equations Operator Theory, 2001, 41: 93-114