Concentration of ground state solutions for supercritical zero-mass \((N, q)\)-equations of Choquard reaction (Q6635468)

The authors propose a delicate study of the existence and concentration of ground state solutions to a class of singularly perturbed \((N,q)\)-Laplacian \(\varepsilon\)-parametric equations of the form: \N\[\N-\varepsilon^N \Delta_N u-\varepsilon^q \Delta_q u=\varepsilon^{\mu-N}\left( \int_{\mathbb{R}^N}\frac{K(y)F(u(y))}{|x-y|^\mu}dy\right) K(x)f(u), \, x \in \mathbb{R}^N, \, \mu \in (0,N), \, q\in (1,N), \, \varepsilon > 0.\N\]\NThis is a Choquard type equation where \(\Delta_N u\) and \(\Delta_q u\) denote, respectively, the \(N\)-Laplace differential operator and the \(q\)-Laplace differential operator, with \(u\) element of the appropriate abstract space. All the involved functions, namely, \(f\), \(F\) and \(K\), are properly defined and fulfill several properties and technical conditions, for example, the function \(f\) exhibits a supercritical growth in the sense of Trudinger-Moser inequality. Hence, the authors point out the following features:\N\begin{itemize}\N\item[1.] Two distinct operators originate a double phase associated energy.\N\item[2.] There is a lack of compactness property for the variational functional associated to the equation.\N\item[3.] The right-hand side shows the combined effects of a convolution type term and a nonlinearity involving supercritical exponential growth together with the potential.\N\end{itemize}\N\NThe main results are given in Theorems 1.3--1.5 and Theorems 1.8--1.9, additional results are stated in Corollary 1.7. The proofs combine several analysis methods, based on regular theory and topological techniques.