Frail and strong solutions for a p-Laplace boundary problem with infinitely many discontinuities


We consider the problem (P) −∆_p u(x) = h(x)  f (u(x)) + q(x), x ∈ Ω, with u(x) = 0, x ∈ ∂Ω, where p > 1, Ω ⊆ R^N is a bounded domain with smooth boundary, q ∈ L ^p (Ω), 1/p + 1/p′= 1, h ∈ L∞ (Ω) \ {0}. We assume that f has a countable set of upward and downward discontinuities, D ⊆ R, and verifies | f (s)| ≤ C_1 + C_2 |s|^α, s ∈ R, where α, C 1, C2 > 0 and 1 +α ∈ [p, p ∗], p ∗ = pN/(N − p). Since the standard functional, I, associated to (P) is not Fréchet differentiable but locally Lipschitz continuous on W ^{1, p}_0 (Ω), we apply the variational tools developed by Chang and Clarke. We characterize a frail solution of (P), one that verifies a.e. a condition involving an appropriate multivalued function, as a generalized critical point of I. Given u, a frail solution of (P), we find sufficient conditions for u^ −1 (D) to have zero measure; this is enough for u to become a strong solution of (P): it satisfies (P) a.e. We show conditions for the existence of local-extremum strong solutions of (P). Finally we prove that if f verifies a growing condition involving the first eigenvalue of −∆ , then (P) has a ground state, i.e., a strong solution which globaly minimizes I


Ambrosetti, A., & Badiale, M. (1989). The dual variational principle and elliptic problems with discontinuous nonlinearities. Journal of Mathematical Analysis and Applications, 140(2), 363-373. doi: 10.1016/0022-247X(89)90070-X

Ambrosetti, A., & Struwe, M. (1989). Existence of Steady Vortex Rings in an Ideal Fluid. Archive for Rational Mechanics and Analysis, 108(2), 97-109. doi: 10.1007/BF01053458.

Ambrosetti, A., & Turner, R. (1988). Some discontinuous variational problems. Differential and Integral Equations, 1(3), 341-349. Retrieved from

Arcoya, D., & Calahorrano, M. (1994). Some Discontinuous Problems with a Quasilinear Operator. Journal of Mathematical Analysis and Applications, 187(3), 1059-1072. doi: 10.1006/jmaa.1994.1406

Brezis, H. (2010). Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer. doi: 10.1007/978-0-387-70914-7.

Calahorrano, M., & Mayorga-Zambrano, J. (2001). Un problema discontinuo con operador cuasilineal. Revista Colombiana de Matemáticas, 35(1), 1-11. doi: 10.15446/recolma.

Chandrasekar, S. (1985). An introduction to the study of stellar structure. Ed. Dover Publishing Inc.

Chang, K. (1981). Variational methods for non-differentiable functionals and their applications to partial differential equations. Journal of Mathematical Analysis and Applications, 80(1), 102-129. doi: 10.1016/0022-247X(81)90095-0.

Cimatti, G. (1979). A nonlinear elliptic eigenvalue problem for the Elenbaas equation. Bollettino della Unione Matematica Italiana B, 16(2), 555-565.

Clarke, F. (1990). Optimization and Nonsmooth Analysis. Society for Industrial and Applied Mathematics. doi: 10.1137/1.9781611971309.

Frank-Kamenetskii, D. (1969). Diffusion and heat transfer in chemical kinetics. Ed. Plenum Press.

Giaquinta, M. (1983). Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press.

Lindqvist, P. (2019). Notes on the stationary p-Laplace equation. Springer. doi: 10.1007/978-3-030-14501-9.

Morrey, C. (2008). Multiple integrals in the calculus of variations. Springer. doi: 10.1007/978-3-540-69952-1

Pavlenko, V., & Potapov, D. (2018). Elenbaas Problem of Electric Arc Discharge. Mathematical Notes, 103(1), 89-95. doi: 10.1134/S0001434618010108

Potapov, D. (2014). On One Problem of Electrophysics with Discontinuous Nonlinearity. Differential Equations, 50(3), 419-422. doi: 10.1134/S0012266114030173

Struwe, M. (2008). Variational methods: Applications to nonlinear partial differential equations and Hamiltonian systems. Springer-Verlag. doi: 10.1007/978-3-540-74013-1

Vásquez, J. (2006). Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Oxford University Press. doi: 10.1093/acprof:oso/9780199202973.001.0001