Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 42, pp. 1-30.

Title: Radial bounded solutions for modified Schrodinger equations

Authors: Federica Mennuni (Univ. di Bologna, Italy)
         Addolorata Salvatore (Univ. degli Studi di Bari Aldo Moro, Bari, Italy)

Abstract: 
We study the quasilinear elliptic equation 
$$
-\hbox{div} (a(x,u,\nabla u)) +A_t(x,u,\nabla u) + |u|^{p-2}u
=g(x,u) \quad \hbox{in }R^N,
$$
with $N\ge 2$ and $p > 1$.  Here, $A : R^N \times 
R\times R^N \to R$ is a given 
${C}^1$-Caratheodory function that grows as $|\xi|^p$ with 
$A_t(x,t,\xi) = \frac{\partial A}{\partial t}(x,t,\xi)$, 
$a(x,t,\xi) = \nabla_\xi A(x,t,\xi)$ and $g(x,t)$ is a given  
Carath\'eodory function on $R^N \times R$ which 
grows as $|\xi|^q$ with $1<q<p$.

Suitable assumptions on $A(x,t,\xi)$ and $g(x,t)$ 
set off the variational structure of above problem and its 
related functional $\mathcal{J}$ is $C^1$ on the Banach space 
$X = W^{1,p}(R^N) \cap L^\infty(R^N)$.
To overcome the lack of compactness, we assume
that the problem has radial symmetry, then we look for 
critical points of $J$ restricted to $X_r$, subspace of the radial 
functions in $X$.

Following an approach that exploits the interaction between
the intersection norm in $X$ and the norm in $W^{1,p}(R^N)$, 
we prove the existence of at least two weak bounded radial solutions,  
one positive and one negative. For this, we apply a generalized version
of the Minimum Principle.


Submitted April 30, 2024. Published July 31, 2024.
Math Subject Classifications: 35J20, 35J92, 35Q55, 58E05
Key Words: Quasilinear elliptic equation; modified Schrodinger equation; positive radial bounded  solution; weak Cerami-Palais-Smale condition; minimum principle.