BIFURCATION PROPERTIES FOR A CLASS OF CHOQUARD EQUATION IN WHOLE R3

ALVES, CLAUDIANOR O. ; LIMA, ROMILDO N. DE ; NÓBREGA, ALÂNNIO B.

Glasgow mathematical journal, Tome 62 (2020) no. 3, pp. 531-543

Résumé

This paper concerns the study of some bifurcation properties for the following class of Choquard-type equations:(P)$$\left\{ {\begin{array}{*{20}{l}}{ - \Delta u = \lambda f(x)\left[ {u + \left( {{I_\alpha }*f( \cdot )H(u)} \right)h(u)} \right],{\rm{ in }} \ {{\mathbb{R}}^3},}\\{{{\lim }_{|x| \to \infty }}u(x) = 0,\quad u(x) > 0,\quad x \in {{\mathbb{R}}^3},\quad u \in {D^{1,2}}({{\mathbb{R}}^3}),}\end{array}} \right.$$ where ${I_\alpha }(x) = 1/|x{|^\alpha },\,\alpha\in (0,3),\,\lambda> 0,\,f:{{\mathbb{R}}^3} \to {\mathbb{R}}$ is a positive continuous function and h : ${\mathbb{R}} \to {\mathbb{R}}$ is a bounded Hölder continuous function. The main tools used are Leray–Schauder degree theory and a global bifurcation result due to Rabinowitz.

Détail

DOI : 10.1017/S0017089519000260

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