Equações de Choquard côncava-convexas: soluções via quociente de Rayleigh não linear

Abstract

The principal reference for this work i s [21], in which the authors investigate the existence and multiplicity of solutions for semilinear elliptic problems obtained from the Choquard equation, under specific hypotheses, in the form 8<:􀀀Δu+V(x)u = (Iα jujp)jujp􀀀2u+λjujq􀀀2u; em RN;u 2 H1(RN)where λ > 0, N 3, α 2 (0;N). The potential V is a continuous function and Iα denotes the standard Riesz potential. Assume also that 1 < q < 2, 2α < p < 2 α, where 2α =N +αN; and 2 α =N +αN 􀀀2: The primary aim of this work is to study the existence and multiplicity of solutions for problems with concave-convex nonlinearities. To achieve this, we consider a functional Eλ whose critical points correspond to weak solutions of the principal equation. Employing the nonlinear Rayleigh quotient, we establish the existence of a parameter λ > 0, and subsequently, using the Nehari manifold method, we demonstrate that the problem admits at least two solutions for each λ 2 (0;λ ]. The fundamental approach involves ensuring that the fiber t 7! Eλ(tu) admits at least two critical points for each λ 2 (0;λ ). Additionally, through a sequence-based procedure, we analyze the behavior of these solutions as the parameter λ approaches zero or λ . We also develop regularity results for the main problem, specifically showing that any weak solution belongs to C1;β loc (RN) for some β 2 (0;1).