# American Institute of Mathematical Sciences

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January  2022, 42(1): 137-162. doi: 10.3934/dcds.2021110

## A semilinear problem with a gradient term in the nonlinearity

 Universidad de Santiago de Chile (USACH), Facultad de Ciencia, Departamento de Matemática y Ciencia de la Computación, Las Sophoras 173, Estación Central, Santiago, Chile

Received  February 2021 Revised  June 2021 Published  January 2022 Early access  August 2021

Fund Project: Ignacio Guerra was supported by Proyecto Fondecyt Regular 1180628 and by Proyecto DICYT 061733GB, Universidad de Santiago de Chile

We consider the following semilinear problem with a gradient term in the nonlinearity
 \begin{align*} -\Delta u = \lambda \frac{(1+|\nabla u|^q)}{(1-u)^p}\quad\text{in}\quad\Omega,\quad u>0\quad \text{in}\quad \Omega, \quad u = 0\quad\text{on}\quad \partial \Omega. \end{align*}
where
 $\lambda,p,q>0$
and
 $\Omega$
be a bounded, smooth domain in
 ${\mathbb R}^N$
. We prove that when
 $\Omega$
is a unit ball and
 $p = 1$
for
 $q\in (0,q^*(N))$
with
 $q^*(N)\in (1,2)$
, we have infinitely many radial solutions for
 $2\leq N<2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1$
and
 $\lambda = \tilde \lambda$
. On the other hand, for
 $N>2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1$
there exists a unique radial solution for
 $0<\lambda<\tilde \lambda$
.
Citation: Ignacio Guerra. A semilinear problem with a gradient term in the nonlinearity. Discrete & Continuous Dynamical Systems, 2022, 42 (1) : 137-162. doi: 10.3934/dcds.2021110
##### References:

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##### References:
Bifurcation diagram for problem (1) in the case $N = 3$ and $q = 1$
Bifurcation diagram for problem (1) in the case $N = 22$ and $q = 1$
Phase diagram for problem in the case $N = 3$ and $q = 1.3$. Thicker lines describe the trapping region
Phase diagram for problem in the case $N = 7$ and $q = 1.4$. Thicker lines describe the trapping region
Diagram of curves $q = N/(N-1)$, $q^*(N),$ $\tilde q(N)$, and $\hat q(N)$.
Bifurcation diagram for problem in the case $N = 3$ and $q = 2.3$. Numerically we observe that for $1.126\ldots<\lambda<1.127\ldots$ we have six solutions, four solutions to the left of $\lambda = 1.126\ldots$ and two solutions to the right of $\lambda = 1.127\ldots$. This is due to a global bifurcation occurs at $\lambda = 1.127\ldots$. We find infinitely many solutions for $\lambda = \tilde \lambda = 1.$
Diagram of curves for $0\leq q\leq 1$, $\hat q(N) = 2\frac{N-2-2\sqrt{N-1}}{N-1}$, $\bar q(N) = \frac 12\left((N-2)\sqrt{\frac{N-9}{N-1}}+8-N\right)$, and $q = \frac{N-2-2\sqrt{N-1}}{N-1}$ which is equivalent to $\sqrt{N-1} = \frac{1+\sqrt{2-q}}{1-q},$ see (25)
The shape of the function $\tilde f(q),$ with $k = 16$ and $\hat b = 0.9$
Phase diagram for the case $N = 9$ and $q = 0.1$. The isoclines are the dashed lines. Here the trapping region is given by the isocline $y' = 0$, the two straight lines and $y = 0$. We plot for $b = 1$, $k = 16$ and $a$ given by (30) the positive root
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