On semilinear elliptic equations involving concave and convex nonlinearities (Q2785226)

The paper deals with the semilinear elliptic problem \(-\Delta u+a(x)u=\lambda u^q+\mu u^p\) in \({\mathbb{R}}^N\), where \(0<q<1<p\) and \(\lambda\), \(\mu\) are positive parameters. The potential \(a(x)\) is assumed to be positive, bounded and locally Hölder continuous on \({\mathbb{R}}^N\). The authors establish the existence of positive classical solutions depending on the range of \(\lambda\) and \(\mu\). In the first part of the paper the construction is based on the method of successive approximations. Furthermore, in the case of purely concave nonlinearities the solution obtained by successive approximations is bounded from below by positive constants. For more general nonlinearities, the existence of a solution is established by standard comparison principles based on the sub and super solutions method. The solution obtained in this way has the following asymptotic behaviour: (i) if \(\lambda\rightarrow 0\) then the solution converges uniformly to 0 in \({\mathbb{R}}^N\); (ii) if \(\mu\rightarrow 0\) then there exists a nonzero limit which is also a solution of the above problem. In the last part of the paper the authors discuss the case where \(\mu =0\) and \(\lambda\) is replaced by a positive or negative function. In this situation there are applied variational techniques based on the study of a constrained minimization problem combined with a concentration-compactness principle at infinity.