Existence of positive radial solutions for nonlinear elliptic equations with gradient terms in an annulus

• Published in: Journal of elliptic and parabolic equations 2023-12, Vol.9 (2), p.807-829
• Main Author: Gou, Haide
• Format: Article
• Language: English
• Subjects: Mathematics; Mathematics and Statistics; Partial Differential Equations
• ISSN: 2296-9020; 2296-9039
• DOI: 10.1007/s41808-023-00224-w

In this paper, we concern with the existence of positive radial solutions of the elliptic equation with nonlinear gradient term - Δ u = f ( | u | , u , x | x | · ∇ u ) , x ∈ Ω , u | ∂ Ω = 0 , where Ω = { x ∈ R n : a < | x | < b } , 0 < a < b < ∞ , n ≥ 3 , f ∈ [ a , b ] × R + × R → R + is continuous. Under the conditions that the nonlinearity f ( r , u , η ) may be of superlinear or sublinear growth in u and η , existence results of positive radial solutions are obtained. For the superlinear case, the growth of f in η is restricted to quadratic growth. The superlinear and the sublinear growth of the nonlinearity of f are described by inequality conditions instead of the usual upper and lower limits conditions as well as the nonlinearity is related to derivative terms. The result is obtained basing on the fixed point index theory in cones.