Electron. J. Differential Equations, Vol. 2024 (2024), No. 42, pp. 130.
Radial bounded solutions for modified Schrodinger equations
Federica Mennuni, Addolorata Salvatore
Abstract:
We study the quasilinear elliptic equation
$$
\hbox{div} (a(x,u,\nabla u)) +A_t(x,u,\nabla u) + u^{p2}u
=g(x,u) \quad \hbox{in }R^N,
$$
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 \(\mathcal{J}\) 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 CeramiPalaisSmale condition; minimum principle.
DOI: 10.58997/ejde.2024.42
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Federica Mennuni
Dipartimento di Matematica
Università di Bologna
Via Zamboni, 33, 40126 Bologna, Italy
email: federica.mennuni@unibo.it


Addolorata Salvatore
Dipartimento di Matematica
Università degli Studi di Bari Aldo Moro
Via E. Orabona 4, 70125 Bari, Italy
email: addolorata.salvatore@uniba.it

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