Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient (Q512641)

The authors are concerned with the study of the problem \[ -\Delta u=\lambda c(x)u+\mu(x)|\nabla u|^2+h(x),\quad u\in H_0^1(\Omega)\cap L^\infty(\Omega), \] where \(\Omega \subset{\mathbb R}^N (N\geq 3)\) is a smooth and bounded domain, \(c,h\in L^p(\Omega)\), for some \(p>N\), \(\mu\in L^\infty(\Omega)\), \(\mu\) is bounded away from zero. Under various assumption on the data, the authors investigate the structure of the solution set to the above problem; particularly, bifurcation and order of solutions with respect to the parameter \(\lambda\) are obtained.