Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity

Abstract

In our previous paper [K. Ishige, S. Okabe, and T. Sato, A supercritical scalar field equation with a forcing term, J. Math. Pures Appl. 128 (2019), pp. 183–212], we proved the existence of a threshold κ∗>0{\kappa }^{\ast }\gt 0 such that the elliptic problem for an inhomogeneous elliptic equation −Δu+u=up+κμ-\Delta u+u={u}^{p}+\kappa \mu in RN{{\bf{R}}}^{N} possesses a positive minimal solution decaying at the space infinity if and only if 01p\gt 1 is in the Joseph-Lundgren subcritical case. In this article, we prove the existence of nonminimal positive solutions to the elliptic problem. Our arguments are also applicable to inhomogeneous semilinear elliptic equations with exponential nonlinearity.

Keywords

• scalar field equation
• the joseph-lundgren exponent
• multiple positive solutions
• 35b09
• 35j61
• 35b32