Mountain pass solutions for an entire semipositone problem involving the Grushin subelliptic operator - Malanchini, Paolo; Bisci, Giovanni Molica; Secchi, Simone (Topological Methods in Nonlinear Analysis, 2026)
For N3 we study the following semipositone problem equation* -Δ_γ u = g(z) f_a(u) in R^N, equation* where Δ_γ is the Grushin operator Δ_ γ u(z) = Δ_x u(z) + x^2γ Δ_y u (z) (γ0), % z=(x,y)R^N = R^m× R^, gL^1(R^N)L^∞(R^N) is a positive function, a> 0 is a parameter and f_a is a continuous function on R that coincides with f(t) -a for tR^+, where f is a continuous function with subcritical and Ambrosetti-Rabinowitz type growth and which satisfies f(0) = 0. Depending on the range of a, we obtain the existence of mountain pass solutions in a suitable Sobolev space associated to the Grushin operator, extending the results found in alves to the Grushin operator.