Uniqueness of positive and compacton-type solutions for a resonant quasilinear problem

Giovanni Anello, Luca Vilasi


We study a one-dimensional $p$-Laplacian resonant problem with $p$-sublinear terms and depending on a positive parameter. By using quadrature methods we provide the exact number of positive solutions with respect to $\mu\in\mathopen{]}0,+\infty\mathclose[$. Specifically, we prove the existence of a critical value $\mu_1> 0$ such that the problem under examination admits: no positive solutions and a continuum of nonnegative solutions compactly supported in $[0,1]$ for $\mu\in\mathopen{]}0,\mu_1\mathclose[$; a unique positive solution of compacton-type for $\mu=\mu_1$; a unique positive solution satisfying Hopf's boundary condition for $\mu\in\mathopen{]}\mu_1,+\infty\mathclose[$.


Quasilinear problem; resonant problem; positive solution; compacton-type solution; uniqueness

Full Text:



G. Anello and L. Vilasi, Positive and compacton-type solutions for a quasilinear twopoint boundary value problem, J. Math. Anal. Appl. 431 (2015), 429–439.

J. Benedikt, P. Girg and P. Takáč, On the Fredholm alternative for the p-Laplacian at higher eigenvalues (in one dimension), Nonlinear Anal. 72 (2010), 3091–3107.

J. Benedikt, P. Girg and P. Takáč, Perturbation of the p-Laplacian by vanishing nonlinearities (in one dimension), Nonlinear Anal. 75 (2012), 3691–3703.

J. Bouchala, Strong resonance problems for the one-dimensional p-Laplacian, Elect. J. Differential Equations 8 (2005), 1–10.

J. Cheng and Y. Shao, The positive solutions of boundary value problems for a class of one-dimensional p-Laplacians, Nonlinear Anal. 68 (2008), 883–891.

J.I. Dı́az and J. Hernández, Global bifurcation and continua of nonnegative solutions for a quasilinear elliptic problem, C. R. Acad. Sci. Paris, Sér. I 329 (1999), 587–592.

J.I. Dı́az, J. Hernández and F.J. Mancebo, Branches of positive and free boundary solutions for some singular quasilinear elliptic problems, J. Math. Anal. Appl. 352 (2009), 449–474.

P. Drábek and S.B. Robinson, Resonance problems for the one-dimensional p-Laplacian, Proc. Amer. Math. Soc. 128 (1999), 755–765.

M. Feng, X. Zhang and W. Ge, Exact number of solutions for a class of two-point boundary value problems with one-dimensional p-Laplacian, J. Math. Anal. Appl. 338 (2008), 784–792.

L. Gasiński, Positive solutions for resonant boundary value problems with the scalar p-Laplacian and nonsmooth potential, Discrete Contin. Dyn. Syst. Ser. A 17 (2007), 143–158.

A. Lakmeche and A. Hammoudi, Multiple positive solutions of the one-dimensional pLaplacian, J. Math. Anal. Appl. 317 (2006), 43–49.


  • There are currently no refbacks.

Partnerzy platformy czasopism