Multiple solutions of nonhomogeneous elliptic equation with critical nonlinearity (Q1386290)

The paper deals with existence of multiple solutions to the problem \[ -\Delta u = Q(x){|}u{|}^{p-2}u + \varepsilon h(x) \quad \text{in } \Omega , \] with \(u=0\) on \(\partial \Omega \), where \(\Omega \) is a bounded smooth domain in \({\mathbb R}^N\), \(p=2N/(N-2)\) the critical Sobolev exponent, \(\varepsilon >0\), \(h\in L^2(\Omega )\), \(h\geq 0\) nontrivial, and \(Q\in C(\bar \Omega )\) a positive function satisfying some growth conditions for \(Q(x)\) in neighbourhoods of its \(k\) strict local maxima. For small \(\varepsilon \) the problem is proved to have, besides a minimum positive solution also \(k\) positive solutions \(u_{k,\varepsilon }\) converging as \(\varepsilon \to 0\) to a multiple of the Dirac measure. A similar property is proved for the problem \[ -\Delta u = Q(x){|}u{|}^{p-2}u \quad \text{in } \Omega , \] with boundary condition \(u(x)=\varepsilon g(x)\). The results generalize that of \textit{J. Escobar} [Commun. Pure Appl. Math. 40, 623-657 (1987; Zbl 0635.35033)]. Using the assumption that \(Q\) has \(k\) local maxima, the problem is transformed to the minimization of a variational functional on suitably chosen subsets. The minimizing sequences satisfying the Palais-Smale condition are constructed by applying the Ekeland variational principle.