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Multiple solutions of a nonlinear elliptic equation involving Neumann conditions and a critical Sobolev exponent. (English) Zbl 1115.35042
Let \(2^*=2N/(N-2)\) denote the critical Sobolev exponent, where \(N\geq 3\) is an integer. Assume that \(\Omega\) is a bounded domain with smooth boundary in \(\mathbb R^N\), \(\lambda\) is a positive parameter not belonging to the spectrum of \((-\Delta)\), \(f\in L^r(\Omega)\) with \(r>N\), and \(Q\) is a smooth positive potential on \(\overline\Omega\). The authors are concerned with the existence of multiple solutions to the nonlinear elliptic equation \(-\Delta u=\lambda u+Q(x)u_+^{2^*-1}+f(x)\) in \(\Omega\), under the Neumann boundary condition \(\partial u/\partial\nu =0\) on \(\partial\Omega\). By means of variational methods the authors establish the existence of at least two distinct solutions. The proof strongly depends on a careful analysis of Palais-Smale sequences of the associated energy functional. An important role is also played by the common effect of the mean curvature of the boundary and the shape of the graph of the potential \(Q\).

MSC:
35J60 Nonlinear elliptic equations
35B33 Critical exponents in context of PDEs
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
47J30 Variational methods involving nonlinear operators
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