A value function and applications to translation invariant semilinear elliptic equations on unbounded domains
The nonlinear elliptic boundary value problem \(- \Delta u + u = \lambda f(x,u)\) in \(G\), \(u = 0\) on \(\partial G\), \(\lim_{| x | \to \infty} u(x) = 0\) is studied in this paper. Here \(G\) is an unbounded cylindrical domain, \(f\) is a continuous function with periodic dependence on \(x\) in the unbounded direction and such that \[ \lim_{s \to 0} {f(x,s) \over s} = 0, \quad \lim_{| s | \to \infty} {f(x,s) \over s^ r} = 0, \] where \(r = {N + 2 \over N - 2}\) \((N\) is the space dimension). In particular, it is proved that there is a weak solution (in the Sobolev space \(H^ 1_ 0 (G))\) of this problem for a dense set of values of the real parameter \(\lambda\). Moreover, if \(G\) contains a plane and \(f\) is autonomous, then there is a solution if \(\lambda\) is large enough. Proofs are given by combining properties of critical value functions and the concentration-compactness method of P. L. Lions. Existence for \(\lambda\) large when \(G\) contains a plane follows from a result of O. Kavian.
scientific article; zbMATH DE number 716355
scientific article