## Topological Methods in Nonlinear Analysis

### Bifurcation and multiplicity results for classes of $p,q$-Laplacian systems

#### Abstract

We study positive solutions to boundary value problems of the form \begin{equation*} \begin{cases} -\Delta_{p} u = \lambda \{u^{p-1-\alpha}+f(v)\} & \mbox{in } \Omega,\\ -\Delta_{q} v = \lambda \{v^{q-1-\beta}+g(u)\} & \mbox{in } \Omega,\\ u = 0=v & \mbox{on }\partial\Omega, \end{cases} \end{equation*} where $\Delta_{m}u:={\rm div}(|\nabla u|^{m-2}\nabla u)$, $m> 1$, is the $m$-Laplacian operator of $u$, $\lambda> 0$, $p,q> 1$, $\alpha\in(0,p-1)$, $\beta\in(0,q-1)$ and $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $N\geq 1$, with smooth boundary $\partial \Omega$. Here $f,g\colon [0,\infty)\rightarrow \mathbb{R}$ are nondecreasing continuous functions with $f(0)=0=g(0)$. We first establish that for $\lambda\approx 0$ there exist positive solutions bifurcating from the trivial branch $(\lambda,u\equiv 0,v\equiv 0)$ at $(0,0,0)$. We further discuss an existence result for all $\lambda > 0$ and a multiplicity result for a certain range of $\lambda$ under additional assumptions on $f$ and $g$. We employ the method of sub-super solutions to establish our results.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 48, Number 1 (2016), 103-114.

Dates
First available in Project Euclid: 30 September 2016

https://projecteuclid.org/euclid.tmna/1475266373

Digital Object Identifier
doi:10.12775/TMNA.2016.036

Mathematical Reviews number (MathSciNet)
MR3561424

Zentralblatt MATH identifier
1376.35084

#### Citation

Shivaji, Ratnasingham; Son, Byungjae. Bifurcation and multiplicity results for classes of $p,q$-Laplacian systems. Topol. Methods Nonlinear Anal. 48 (2016), no. 1, 103--114. doi:10.12775/TMNA.2016.036. https://projecteuclid.org/euclid.tmna/1475266373

#### References

• J. Ali, K.J. Brown and R. Shivaji, Positive solutions for $n \times n$ elliptic systems with combined nonlinear effects, Differential Integral Equations 24 (2011), No. 3–4, 307–324.
• J. Ali and R. Shivaji, Multiple positive solutions for a class of p-q-Laplacian systems with multiple parameters and combined nonlinear effects, Differential Integral Equations 22 (2009), No. 7–8, 669–678.
• H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620–709.
• M. Belloni and B. Kawohl, A direct uniqueness proof for equations involving the p-Laplace operator, Manuscripta Math. 109 (2002), 229–231.
• K.J. Brown, M.M.A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, Nonlinear Anal. 5 (1981), No. 5, 475–486.
• C. Maya, S. Oruganti and R. Shivaji, Positive solutions for classes of p-Laplacian equations, Differential Integral Equations 16 (2003), No. 6, 757–768.
• M. Ramaswamy and R. Shivaji, Multiple positive solutions for classes of p-Laplacian equations, Differential Integral Equations 17 (2004), No. 11–12, 1255–1261.
• R. Shivaji, A remark on the existence of three solutions via sub-super solutions, Nonlinear Analysis and Application, Lecture Notes in Pure and Applied Mathematics (V. Lakshmikantham, ed.) 109 (1987), 561–566.