Multiple finite energy solutions of critical semilinear field equations (Q1909999)

The authors obtain, by using minimax methods and exploring compactness arguments, existence and multiplicity results of solutions of the semilinear elliptic problem \[ -\Delta u=p_0|u|^{\tau-1}u+q(x)|u|^{\gamma-1}u\quad\text{in }\mathbb{R}^N,\quad u\not\equiv 0,\quad u\in D^{1,2}_0(\mathbb{R}^N),\tag{1} \] where \(N\geq 3\), \(\tau+1=2N/(N-2)\), \(1<\gamma<\tau\), \(p_0\) is a positive constant, and \(q\) is a nonnegative \(L^\infty_{\text{loc}}\) function. Qualitative properties such as the signs of the solutions of (1) and their asymptotic behaviour are studied.