Event Details
Título: Mountain Pass solutions for an entire semipositone problem involving the Grushin operator.
Conferenciante: Paolo Malanchini (Università degli Studi di Milano-Bicocca).
Resumen:
For \(N\ge 3\) we study the following semipositone problem$$ -\Delta_\gamma u = g(z) (f(u)- a) \quad \hbox{in $\mathbb{R}^N$}, $$ where \(\Delta_\gamma\) is the Grushin operator $$ \Delta_ \gamma u(z) = \Delta_x u(z) + |x|^{2\gamma} \Delta_y u (z) \quad (\gamma\ge 0), $$ \(g\in L^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)\) is a positive function, \(a>0\) is a parameter and \(f\) 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. Finally we study the positivity of the solutions in the degeneracy set.
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