## Taiwanese Journal of Mathematics

### POSITIVE SOLUTIONS FOR ELLIPTIC EQUATIONS IN TWO DIMENSIONS ARISING IN A THEORY OF THERMAL EXPLOSION

#### Abstract

In this paper we study a mathematical model of thermal explosion which is described by the boundary value problem $\begin{cases}-\Delta u = \lambda e^{u^{\alpha}}, &x \in \Omega, \\\mathbf{n} \cdot \nabla u + g(u) u = 0, &x \in \partial \Omega,\end{cases}$where the constant $\alpha \in (0,2],~ g:[0, \infty)\rightarrow (0, \infty)$ is an nondecreasing $C^1$ function,  $\Omega$ is a bounded domain in $\mathbb{R}^2$ with smooth boundary $\partial \Omega$ and $\lambda \gt 0$ is a bifurcation parameter.Using variational methodswe show that there exists $0\lt \Lambda \lt \infty$ such that the problem has at least two positive  solutions if $0 \lt \lambda \lt \Lambda,$ no solution if $\lambda \gt \Lambda$ and at least one positive solution when $\lambda =\Lambda.$

#### Article information

Source
Taiwanese J. Math., Volume 19, Number 6 (2015), 1759-1775.

Dates
First available in Project Euclid: 4 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499133738

Digital Object Identifier
doi:10.11650/tjm.19.2015.5968

Mathematical Reviews number (MathSciNet)
MR3434276

Zentralblatt MATH identifier
1357.35263

#### Citation

Ko, Eunkyung; Prashanth, S. POSITIVE SOLUTIONS FOR ELLIPTIC EQUATIONS IN TWO DIMENSIONS ARISING IN A THEORY OF THERMAL EXPLOSION. Taiwanese J. Math. 19 (2015), no. 6, 1759--1775. doi:10.11650/tjm.19.2015.5968. https://projecteuclid.org/euclid.twjm/1499133738

#### References

• Adimurthi, Existence of positive solutions of the semi linear Dirichlet problem with critical growth for the $n$-Laplacian, Ann. Della. Scuola. Norm. Sup. di Pisa, Serie IV XVII 3 (1990), 393–413.
• Adimurthi and S. L. Yadava, Critical exponent problem in $\mathbb{R}^2$ with Neumann boundary condition, Comm. Partial Differential Equations 15(4) (1990), 461–501.
• A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), 519–543.
• G. Azorero, I. Peral and J. D. Rossi, A convex-concave problem with a nonlinear boundary condition, J. Diff. Equations 198 (2004), 91–128.
• G. Azorero, J. Manfredi and I. Peral, Sobolev versus Holder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math. 2(3) (2000), 385–404.
• H. Brezis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers, C. R. Acad. Sci. Paris Ser. I Math. 317 (1993), 465–472.
• H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for $u_t -\Delta u = g(u)$ revisited, Adv. Diff. Eqns. 1 (1996), 73–90.
• P. Cherrier, Meilleures constantes dans des inegalités relatives aux espaces de Sobolev, Bull. Sci. Math. 108(2) (1984) 225–262.
• ––––, Problèmes de Neumann nonlinéaires sur les variétés Riemanniennes, J. Func. Anal. 57 (1984), 154–207.
• E. DiBenedetto, $C^{1, \alpha}$-local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), 827–850.
• D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbb{R}^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations 3(2) (1995), 139–153.
• N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points, Ann. Inst. H. Poincare Anal. Non Lineaire 6 (1989), 321–330.
• D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983.
• P. Gordon, E. Ko and R. Shivaji, Multiplicity and uniqueness of positive solutions for elliptic equations with nonlinear boundary conditions arising in a theory of thermal explosion, Nonlinear Anal. Real World Appl. 15 (2014), 51–57.
• Y. Y. Li and M. Zhu, Sharp Sobolev inequalities involving boundary terms, Geom. Funct. Anal. 8(1) (1998), 59–87.
• J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971), 1077–1092.
• W. M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. XLIV 8-9 (1991), 819–851.
• S. Prashanth and K. Sreenadh, Multiplicity of solutions to a nonhomogeneous elliptic equation in $\mathbb{R}^2$, Differ. Integral Equ. 18(6) (2005), 681–698.
• ––––, Existence of multiple positive solutions for $N$-Laplacian in a bounded domain in $\mathbb{R}^N$, Adv. Nonlinear Stud. 5 (2005), 13–22.
• ––––, Multiplicity positive solutions in $\mathbb{R}^2$ for a super linear elliptic problem with a sublinear Neumann boundary condition, Nonlinear Anal. 67 (2007), 1246–1254.
• M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.
• P. Tolksdorf, Regularity for more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126–150.
• N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–484.
• F. A. Williams, Combustion Theory, (reading, MA: Perseus Books), 1985.