Multiple solutions of a semilinear elliptic equation in \(\mathbb{R}^ N\). (English) Zbl 0797.35039

Summary: We are concerned with the existence of multiple solutions of \[ -\Delta u + u = \lambda b(x) | u |^{p-1} u + c(x) | u |^{q-1} u \] where \(1<p\), \(q<(N+2)/(N-2)\) if \(N \geq 3\), \(1<p\), \(q<+\infty\) if \(N=2\), \(\lambda>0\). We obtain the existence of multiple solutions by using concentrations-compactness method and dual variational principle to establish the corresponding existence of critical points.


35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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