Existence of nontrivial weak solutions for a quasilinear Choquard equation

Résumé

We are concerned with the following quasilinear Choquard equation: $$ -\Delta_{p} u+V(x)|u|^{p-2}u=\lambda\bigl(I_{\alpha} \ast F(u)\bigr)f(u) \quad \text{in } \mathbb {R}^{N}, \qquad F(t)= \int_{0}^{t}f(s) \,ds, $$ where $1< p<\infty$ , $\Delta_{p} u=\nabla\cdot(|\nabla u|^{p-2}\nabla u)$ is the p-Laplacian operator, the potential function $V:\mathbb {R}^{N}\to(0,\infty)$ is continuous and $F \in C^{1}(\mathbb {R}, \mathbb {R})$ . Here, $I_{\alpha}: {\mathbb {R}}^{N}\rightarrow {\mathbb {R}}$ is the Riesz potential of order $\alpha\in(0,p)$ . We study the existence of weak solutions for the problem above via the mountain pass theorem and the fountain theorem. Furthermore, we address the behavior of weak solutions to the problem near the origin under suitable assumptions for the nonlinear term f.

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