Elliptic equations with indefinite and unbounded potential and a nonlinear concave boundary condition
We consider an elliptic problem driven by the negative Laplacian plus an indefinite and unbounded potential and a superlinear reaction. The boundary condition is parametric, nonlinear and superlinear near zero. Thus, the problem is a new version of the classical "convex-concave" problem (problem with competing nonlinearities). First, we prove a bifurcation-type result describing the set of positive solutions as the parameter λ>0 varies. We also show the existence of a smallest positive solution ūλ and investigate the properties of the map λ→ūλ. Finally, by imposing bilateral conditions on the reaction we generate two more solutions, one of which is nodal.
nonlinear boundary condition, competing nonlinearity, bifurcation-type theorem, extremal constant sign solutions, nodal solutions, indefinite and unbounded potential
Hu, Shouchuan, and Nikolaos S. Papageorgiou. "Elliptic equations with indefinite and unbounded potential and a nonlinear concave boundary condition." Communications in Contemporary Mathematics 19, no. 01 (2017): 1550090.