Abstract
We study the quasilinear elliptic equation $$ -\operatorname{div} (a(x,u,\nabla u)) +A_t(x,u,\nabla u) + |u|^{p-2}u =g(x,u) \quad \hbox{in }R^N, $$ 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 Caratheodory 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 \(J\) 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.
For more information see https://ejde.math.txstate.edu/Volumes/2024/42/abstr.html