On eigenvalue problems governed by the \((p, q)\)-Laplacian
DOI:
https://doi.org/10.24193/subbmath.2023.1.05Keywords:
eigenvalue problem, \((p, q)\)-Laplacian, Sobolev space, Nehari manifold, Variational methods, Lagrange multipliersAbstract
This is a survey on recent results, mostly of the authors, regarding eigenvalue problems governed by the \((p, q)\)-Laplacian and related open problems.
References
Abreu, J., Madeira, G., Generalized eigenvalues of the (p,2)-Laplacian under a parametric boundary condition, Proc. Edinb. Math. Soc, 63(2020), no. 1, 287-303.
Anane, A., Tsouli, N., On the second eigenvalue of the p-Laplacian, in "Nonlinear Partial
Differential Equations (From a Conference in Fes, Maroc, 1994)" (Benkirane, A., Gossez,
J.-P., Eds.), Pitman Research Notes in Math. 343, Longman, 1996.
Arora, R., Shmarev, S., Double-phase parabolic equations with variable growth and non-linear sources, Adv. Nonlinear Anal., 12(2023), no. 1, 304-335.
Barbu, L., Eigenvalues for anisotropic p-Laplacian under a Steklov-like boundary condition, Stud. Univ. Babes-Bolyai Math., 66(2021), no. 1, 85-94.
Barbu, L., Burlacu, A., Morosanu, G., On a bulk-boundary eigenvalue problem involving the (p,q)-Laplacian (in preparation).
Barbu, L., Morosanu, G., Eigenvalues of the negative (p,q)-Laplacian under a Steklov-like boundary condition, Complex Var. Elliptic Equ., 64(2019), no. 4, 685-700.
Barbu, L., Morosanu, G., Full description of the eigenvalue set of the (p; q)-Laplacian with a Steklov-like boundary condition, J. Differential Equations, 290(2021), 1-16.
Barbu, L., Morosanu, G., On a Steklov eigenvalue problem associated with the (p,q)-Laplacian, Carpathian J. Math., 37(2021), 161-171.
Barbu, L., Morosanu, G., Eigenvalues of the (p,q,r)-Laplacian with a parametric boundary condition, Carpathian J. Math., 38(2022), no. 3, 547-561.
Barbu, L., Morosanu, G., On the eigenvalue set of the (p,q)-Laplacian with a Neumann-Steklov boundary condition, Differential Integral Equations, 36(2023), no. 5-6, 437-452.
Barbu, L., Morosanu, G., Full description of the spectrum of a Steklov-like eigenvalue problem involving the (p,q)-Laplacian, Ann. Acad. Rom. Sci., Ser. Math. Appl. (in press).
Bartnik, R. and Simon, L., Space-like hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., 87(1982), 131-152.
Belloni M., Kawohl B., Juutinen P., The p-Laplace eigenvalue problem as $ptoinfty$ in a Finsler metric, J. Eur. Math. Soc., 8(2006), 123-138.
Benci, V., D'Avenia, P., Fortunato, D., et al., Solitons in several space dimensions: Derrick's problem and in nitely many solutions, Arch. Ration. Mech. Anal., 154(2000), 297-324.
Benci, V., Fortunato, D., Pisani, L., Soliton like solutions of a Lorentz invariant equation in dimension 3, Rev. Math. Phys., 10(1998), 315-344.
Bonheure, D., Colasuonno, F., Foldes, J., On the Born-Infeld equation for electrostatic fields with a superposition of point charges, Ann. Mat. Pura Appl., 198(2019), 749-772.
Cheng, S.-Y., Yau, S.-T., Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math., 104(1976), 407-419.
Cherfils, L., Il'yasov, Y., On the stationary solutions of generalized reaction diffusion equations with p&q-Laplacian, Commun. Pure Appl. Anal., 4(2005), 9-22.
Costea, N., Morosanu, G., Steklov-type eigenvalues of Delta p+Delta q, Pure Appl. Funct. Anal., 3(2018), no. 1, 75-89.
Della Pietra, F., Gavitone, N., Faber-Krahn inequality for anisotropic eigenvalue problems with Robin boundary conditions, Potential Anal., 41(2014), 1147-1166.
Della Pietra F., Gavitone N., Sharp bounds for the rst eigenvalue and the torsional rigidity related to some anisotropic operators, Math. Nachr., 287(2014), 194-209.
Della Pietra F., Gavitone N., Piscitelli G., On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators, Bull. Sci. Math., 155(2019), 10-32.
Dr abek P., Robinson, S., Resonance problems for the p-Laplacian, J. Funct. Anal.,169(1999), 189-200.
Farcaseanu, M., Mihailescu, M., Stancu-Dumitru, D., On the set of eigenvalues of some PDEs with homogeneous Neumann boundary condition, Nonlinear Anal., 116(2015), 19-25.
Ferone, V., Kawohl, B., Remarks on Finsler-Laplacian, Proc. Amer. Math. Soc., 137(2008), no. 1, 247-253.
Fortunato, D., Orsina, L., Pisani, L., Born-Infeld type equations for electrostatic fields, J. Math. Phys., 43(2002), 5698-5706.
Francois, F., Spectral asymptotics stemming from parabolic equations under dynamical boundary conditions, Asymptot. Anal., 46(2006), no. 1, 43-52.
Garcia-Azorero, J.P., Peral, I., Existence and nonuniqueness for the p-Laplacian: non- linear eigenvalues, Comm. Partial Differential Equations, 12(1987), 1389-1430.
Gibbons, G.W., Born-Infeld particles and Dirichlet p-branes, Nuclear Phys. B, 514(1998), 603-639.
Gyulov, T., Morosanu, G., Eigenvalues of -(Delta_p+Delta_q) under a Robin-like boundary condition, Ann. Acad. Rom. Sci. Ser. Math. Appl., 8(2016), 114-131.
Huang, Z., The weak solutions of a nonlinear parabolic equation from two-phase problem, J. Inequal. Appl., 1(2021), 1-19.
Kawohl B., Novaga M., The p-Laplace eigenvalue problem as pto 1 and Cheeger sets in a Finsler metric, J. Convex Anal., 15(2008), 623-634.
Le, A., Eigenvalue problems for p-Laplacian, Nonlinear Anal., 64(2006), 1057-1099.
Lindqvist, P., On the equation $-div(|nabla u|^{p-2}nabla u)-lambda |u|^{p-2} u=0$, Proc. Amer. Math. Soc., 109(1990), 157-164.
Marcellini, P., A variational approach to parabolic equations under general and p, q-growth conditions, Nonlinear Anal., 194(2020), 111-456.
Mihailescu, M., An eigenvalue problem possesing a continuous family of eigenvalues plus an isolated eigenvale, Commun. Pure Appl. Anal., 10(2011), 701-708.
Mihailescu, M., Morosanu, G., Eigenvalues of -Delta_p-Delta_q under Neumann boundary condition, Canad. Math. Bull., 59(2016), no. 3, 606-616.
Papageorgiou, N.S., Vetro, C., Vetro, F., Continuous spectrum for a two phase eigenvalue problem with an indefinite and unbounded potential, J. Differential Equations, 268(2020), no. 8, 4102-4118.
Pohozaev, S.I., The bering method and its applications to nonlinear boundary value problem, Rend. Istit. Mat. Univ. Trieste, 31(1999), no. 1-2, 235-305.
Pomponio, A., Watanabe, T., Some quasilinear elliptic equations involving multiple P-Laplacians, Indiana Univ. Math. J., 67(2018), no. 6, 2199-2224.
Szulkin, A.,Weth, T., The Method of Nehary Manifold, Handbook of Nonconvex Analysis and Applications, Int. Press, Somerville, MA, 597-632, 2010.
Tanaka, M., Generalized eigenvalue problems for (p,q)-Laplacian with indefinite weight, J. Math. Anal. Appl., 419(2014), 1181-1192.
Von Below, J., Francois, G., Spectral asymptotics for the Laplacian under an eigenvalue dependent boundary condition, Bull. Belg. Math. Soc. Simon Stevin, 12(2005), no. 4, 505-519.
Wang G., Xia C., An optimal anisotropic Poincare inequality for convex domains. Pacific J. Math., 258(2012), 305-326.
Zhikov, V.V., Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50(1986), 675-710; English translation in Math. USSRIzv.,
(1987), 33-66.
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