## Taiwanese Journal of Mathematics

### EXPLICIT NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE OF NONNEGATIVE SOLUTIONS OF A p-LAPLACIAN BLOW-UP PROBLEM

#### Abstract

We establish explicit necessary and sufficient conditions for the existence of nonnegative solutions of the $p$-Laplacian boundary blow-up problem \begin{equation*} \left\{ \begin{array}{l} \left( \varphi _{p}(u^{\prime }(x))\right) ^{\prime }=\lambda f(u(x)), \, 0 \lt x \lt 1, \\ \lim\limits_{x\rightarrow 0^{+}}u(x)=\infty =\lim\limits_{x\rightarrow 1^{-}}u(x), \end{array} \right. \end{equation*} where $p>1$, $\varphi _{p}\left( y\right) =\left\vert y\right\vert ^{p-2}y$ and $\left( \varphi _{p}(u^{\prime })\right) ^{\prime }$ is the one-dimensional $p$-Laplacian, $\lambda$ is a positive bifurcation parameter and $f$ is a locally Lipschitz continuous function on $[0,\infty)$. The gap is extremely small between the explicit necessary condition and the explicit sufficient condition for the existence of nonnegative solutions. Our results improve and extend some main results of Anuradha, Brown and Shivaji [2] and of Wang [30] from $p=2$ to any $p\gt 1$.

#### Article information

Source
Taiwanese J. Math., Volume 13, Number 3 (2009), 1077-1093.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500405461

Digital Object Identifier
doi:10.11650/twjm/1500405461

Mathematical Reviews number (MathSciNet)
MR2526360

Zentralblatt MATH identifier
1194.34027

#### Citation

Huang, Pei-Yu; Shieh, Ming-Ting; Wang, Shin-Hwa. EXPLICIT NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE OF NONNEGATIVE SOLUTIONS OF A p-LAPLACIAN BLOW-UP PROBLEM. Taiwanese J. Math. 13 (2009), no. 3, 1077--1093. doi:10.11650/twjm/1500405461. https://projecteuclid.org/euclid.twjm/1500405461

#### References

• A. Aftalion and W. Reichel, Existence of two boundary blow-up solutions for semilinear elliptic equations, J. Differential Equations, 141 (1997), 400-421.
• V. Anuradha, C. Brown and R. Shivaji, Explosive nonnegative solutions to two point boundary value problems, Nonlinear Anal., 26 (1996), 613-630.
• C. Bandle and M. Marcus, Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior, J. Analyse Math., 58 (1992), 9-24.
• C. Bandle and M. Marcus, Asymptotic behaviour of solutions and their derivatives, for semilinear elliptic problems with blowup on the boundary, Ann. Inst. H. Poincaré Anal. Non Lineaire, 12 (1995), 155-171.
• C. Bandle and M. Marcus, On second-order effects in the boundary behaviour of large solutions of semilinear elliptic problems, Differential Integral Equations, 11 (1998), 23-34.
• L. Bieberbach, $\Delta u=e^{u}$ and die automorphen Funktionen, Math. Annln, 77 (1916), 173-212.
• Y. J. Cheng, Some surprising results on a one-dimensional elliptic boundary value blow-up problem, Z. Anal. Anwendungen, 18 (1999), 525-537.
• G. D\'\tiny laz and R. Letelier, Explosive solutions of quasilinear elliptic equations: existence and uniqueness, Nonlinear Anal., 20 (1993), 97-125.
• Y. Du and Z. M. Guo, Liouville type results and eventual flatness of positive solutions for $p$-Laplacian equations, Adv. Differential Equations, 7 (2002), 1479-1512.
• Y. Du and Z. M. Guo, Boundary blow-up solutions and their applications in quasilinear elliptic equations, J. Anal. Math., 89 (2003), 277-302.
• Y. Du and Z. M. Guo, Uniqueness and layer analysis for boundary blow-up solutions, J. Math. Pure Appl., 83 (2004), 739-763.
• Y. Du and S. Yan, Boundary blow-up solutions with a spike layer, J. Differential Equations, 205 (2004), 156-184.
• F. Gladiali and G. Porru, Estimates for explosive solutions to $p$-Laplace equations. Progress in partial differential equations, Vol. 1, $($Pont-à-Mousson, 1997), 117-127, Pitman Res. Notes Math. Ser., 383, Longman, Harlow, 1998.
• Z. M. Guo and J. R. L. Webb, Structure of boundary blow-up solutions for quasilinear elliptic problems. I. Large and small solutions, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 615-642.
• Z. M. Guo and J. R. L. Webb, Structure of boundary blow-up solutions for quasi-linear elliptic problems. II. Small and intermediate solutions, J. Differential Equations, 211 (2005), 187-217.
• Z. M. Guo and F. Zhou, Exact multiplicity for boundary blow-up solutions, J. Differential Equations, 228 (2006), 486-506.
• J. B. Keller, On solutions of $\Delta u=f(u)$, Comm. Pure Appl. Math., 10 (1957), 503- 510.
• A. V. Lair and A. W. Wood, Large solutions of semilinear elliptic problems, Nonlinear Anal., 37 (1999), 805-812.
• A. C. Lazer and P. J. McKenna, On a problem of Bieberbach and Rademacher, Nonlinear Anal., 21 (1993), 327-335.
• C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, in: Contributions to Analysis (A Collection of Paper Dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 245-272.
• M. Marcus and L. Véron, Uniqueness of solutions with blowup at the boundary for a class of nonlinear elliptic equations, C. R. Acad. Sci. Paris, 317 (1993), 559-563.
• M. Marcus and L. Véron, Existence and uniqueness results for large solutions of general nonlinear elliptic equations, J. Evolution Equations, 3 (2003), 637-652.
• J. Matero, Quasilinear elliptic equations with boundary blow-up, J. Analyse Math., 69 (1996), 229-247.
• P. J. McKenna, W. Reichel and W. Walter, Symmetry and multiplicity for nonlinear elliptic differential equations with boundary blow-up, Nonlinear Anal., 28 (1997), 1213-1225.
• A. Mohammed, Existence and asymptotic behavior of blow-up solutions to weighted quasilinear equations, J. Math. Anal. Appl., 298 (2004), 621-637.
• A. Olofsson, Apriori estimates of Osserman-Keller type, Differential Integral Equations, 16 (2003), 737-756.
• R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.
• H. Rademacher, Einige besondere problem partieller Differential gleichungen, in Die Differential- and Integralgleichungen, der Mechanik and Physik I, 2nd edition Rosenberg, New York, 1943, pp. 838-845.
• L. Véron, Singularities of solutions of second order quasilinear equations. Pitman Research Notes in Mathematics Series, 353, Longman, Harlow, 1996.
• S.-H. Wang, Existence and multiplicity of boundary blow-up nonnegative solutions to two point boundary value problems, Nonlinear Anal., 42 (2000), 139-162.
• S.-H. Wang, Y.-T. Liu and I-A. Cho, An explicit formula of the bifurcation curve for a boundary blow-up problem, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 431-446.
• S.-H. Wang and Y.-T. Liu, Shape and structure of the bifurcation curve of a boundary blow-up problem, Taiwanese J. Math., 9 (2005), 201-214.