Multiplicity results for an inhomogeneous nonlinear elliptic problem. (English) Zbl 1042.35012

The problem \[ u \in H_0^1(\Omega ),\;-\Delta u + \mu u = Q(x)| u| ^{p-2}u + h(x) \] is considered with \(N \geq 3,\;2<p<\frac {2N}{N-2}, \;\mu >0,\;Q \in C(\overline \Omega ),\;h \in L^2(\Omega ),\;h\not \equiv 0,\;\Omega \) a bounded domain in \(\mathbb R^n\). It is proved that if the maximum of \(Q\) is achieved at exactly \(k\) different points of \(\Omega \) then for \(\mu \) large enough this problem has at least \(k+1\) positive solutions and \(k\) nodal solutions (i.e. solutions changing signs). The solutions are obtained as local minima of the corresponding functional constrained to suitably constructed closed subsets of \(H_0^1(\Omega )\).


35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs