Radial bounded solutions for modified Schrodinger equations

We study the quasilinear elliptic equation -div (a(x,u,∇ u)) +Aₜ(x,u,∇ u) + |u|ᵖ⁻²u =g(x,u) in RN, with \(N≥ 2\) and \(p > 1\). Here, \(A : R^N × R× R^N → R\) is a given \(C^1\)-Caratheodory function that grows as \(|ξ|^p\) with \(A_t(x,t,ξ) = ∂ A/∂ t(x,t,ξ)\), \(a(x,t,ξ) = ∇_ξ A(x,t,ξ)\) and \(g(x,t)\) is a given Caratheodory function on \(R^N × R\) which grows as \(|ξ|^q\) with \(1<q<p\). Suitable assumptions on \(A(x,t,ξ)\) and \(g(x,t)\) set off the variational structure of above problem and its related functional \(J\) is \(C^1\) on the Banach space \(X = W1,p(R^N) ∩ L^∞(R^N)\). 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 \(W1,p(R^N)\), 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