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

Ratnasingham Shivaji, Byungjae Son

DOI: http://dx.doi.org/10.12775/TMNA.2016.036

#### 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 $\Delta_{m}u:=\mbox{div}(|\nabla u|^{m-2}\nabla u)$, $m> 1$, 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.

#### Keywords

Positive solutions; bifurcation; existence; multiplicity

#### Full Text:

PREVIEW FULL TEXT

#### References

J. Ali, K.J. Brown and R. Shivaji, Positive solutions for n × 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 pLaplace 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.

### Refbacks

• There are currently no refbacks.