n-Kirchhoff–Choquard equations with exponential nonlinearity

Abstract

This article deals with the study of the following Kirchhoff equation with exponential nonlinearity of Choquard type (see (KC) below). We use the variational method in the light of Moser-Trudinger inequality to show the existence of weak solutions to (KC). Moreover, analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to convex-concave problem (P λ,M ) below.

Polyharmonic Kirchhoff problems involving exponential non-linearity of Choquard type with singular weights

“…Now J λ,M (t 1 (u λ )u λ ) < J λ,M (u λ ) ≤ θ which yields a contradiction, since t 1 (u λ )u λ ∈ N λ,M . The proof for u λ being a local minimum for J λ,M in W m,2 0 (Ω) follows exactly as the proof of Theorem 4.12 in [5].…”

“…implies that v k ≤ C. Thus we get S(v k ) * ≤ C 1 and from (3.4) we have ξ − k (0) * ≤ C 2 . Now the rest of the proof follows as in the proof of Theorem 3.9 with the help of Lemma 3.7 (refer Theorem 4.13 of [5]).…”

Adams–Moser–Trudinger inequality in the Cartesian product of Sobolev spaces and its applications

“…The proof of the above Lemma follows from similar arguments as in Lemma 3.5 and Lemma 3.6 in [5]. Proof.…”

“…It shows that the presence of nonlinear coefficient m is meaningful to be considered. We cite [3,4,5,14] and there references within for further considerations.…”

“…1: The class of system (KCS) can be extended to the following fractional Kirchhoff-Choquard system involving singular weights: Using Theorem 4.1, doubly weighted Hardy-Littlewood-Sobolev inequality, we can prove the existence and multiplicity of solutions for the problem (F ) (see [5,6]).…”