Existence of multiple radial solutions for nonlinear equation involving the mean curvature operator in the Lorentz-Minkowski space
DOI:
https://doi.org/10.12775/TMNA.2025.030Słowa kluczowe
Mean curvature operator, radial solutions, concave-convex nonlinearity, variational methods, non-smooth functionalsAbstrakt
We prove the existence of multiple radial solutions for a class of nonlinear equations - involving the mean curvature operator in the Lorentz-Minkowski space - of the form \begin{equation*} -\dive \bigg(\frac{\nabla u}{\sqrt{1 - \abs{\nabla u}^{2}}}\bigg) = \lambda b(\abs{x})\abs{u}^{q-2}u + f(\abs{x}, u) \quad \text{in } B_{R}, \end{equation*} with Dirichlet boundary conditions in case $q \in (1,2)$ and $f(\abs{x}, s)$ is superlinear in $s$. Solutions are found using Szulkin's critical point theory for non-smooth functionals. Multiplicity results are also given for some cases in which $f$ depends also on (the norm of) the gradient of $u$.Bibliografia
A. Ambrosetti, J.G. Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996), no. 1, 219–242.
A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.
C. Bereanu, P. Jebelean and J. Mawhin, Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces, Proc. Amer. Math. Soc. 137 (2009), no. 1, 161–169.
C. Bereanu, P. Jebelean and J. Mawhin, The Dirichlet problem with mean curvature operator in Minkowski space – a variational approach, Adv. Nonlinear Stud. 14 (2014), no. 2, 315–326.
C. Bereanu, P. Jebelean and J. Mawhin, Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowski spaces, Math. Nachr. 283 (2010), no. 3, 379–391.
C. Bereanu, P. Jebelean and J. Mawhin, Multiple solutions for Neumann and periodic problems with singular φ-Laplacian, J. Funct. Anal. 261 (2011), no. 11, 3226–3246.
C. Bereanu, P. Jebelean and J. Mawhin, Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities, Calc. Var. Partial Differ. Equ. 46 (2013), no. 1–2, 113–122.
C. Bereanu, P. Jebelean and P.J. Torres, Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space, J. Funct. Anal. 265 (2013), no. 4, 644–659.
C. Bereanu, P. Jebelean and P.J. Torres, Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space, J. Funct. Anal. 264 (2013), no. 1, 270–287.
D. Bonheure, J.M. Gomes and L. Sanchez, Positive solutions of a second-order singular ordinary differential equation, Nonlinear Anal. 61 (2005), no. 8, 1383–1399.
D. Bonheure, P. d’Avenia and A. Pomponio, On the electrostatic Born–Infeld equation with extended charges, Commun. Math. Phys. 346 (2016), no. 3, 877–906.
H. Brézis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations 23 (2010), no. 9–10, 801–810.
S.Y. Cheng and S.T. Yau, Maximal space-like hypersurfaces in the Lorentz–Minkowski spaces, Ann. of Math. (2) 104 (1976), no. 3, 407–419.
I. Coelho, C. Corsato and S. Rivetti, Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball, Topol. Methods Nonlinear Anal. 44 (2014), no. 1, 23–39.
D. De Figueiredo, M. Girardi and M. Matzeu, Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques, Differential Integral Equations 17 (2004), no. 1–2, 119–126.
F.J. Flaherty, The boundary value problem for maximal hypersurfaces, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 10, 4765–4767.
A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), 77–109.
Pobrania
Opublikowane
Jak cytować
Numer
Dział
Statystyki
Liczba wyświetleń i pobrań: 0
Liczba cytowań: 0
