Local analytic solutions to some nonhomogeneous problems with $p$-Laplacian

• Main Author: Gabriella Bognár
• Format: Article
• Language: English
• Published: University of Szeged 2008-07-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=319

Applying the Briot-Bouquet theorem we show that there exists an unique analytic solution to the equation $\left( t^{n-1}\Phi _{p}\left( y^{\prime}\right) \right) ^{\prime }+(-1)^{i}t^{n-1}\Phi _{q}(y)=0$, on $(0,~a)$, where $\Phi _{r}(y):=\left\vert y\right\vert ^{r-1}y$, $0<r,p,q\in R^{+}$, $i=0,1$, $1\leq n\in N,\ a$ is a small positive real number. The initial conditions to be added to the equation are $y(0)=A\neq 0$, $y^{\prime}(0)=0$, for any real number $A$. We present a method how the solution can be expanded in a power series for near zero.