Multiple solutions for a Neumann problem in an exterior domain. (English) Zbl 0803.35049

From the introduction: We prove the existence of multiple solutions for the Neumann boundary value problem \[ - \Delta u + u = Q(x) | u |^{p - 1} u \quad \text{in} \quad \mathbb{R}^ N \backslash \Omega, \qquad {\partial u \over \partial n} = 0 \quad \text{on} \quad \partial \Omega \tag{1} \] where \(\Omega\) is a smooth bounded domain of \(\mathbb{R}^ N\), \(N \geq 3\), \(1<p<(N + 2)/(N - 2)\), \(Q(x) \in C (\mathbb{R}^ N)\), \(Q(x)>0\) in \(\mathbb{R}^ N \backslash \Omega\). We always assume that \(Q(x) \to \overline Q>0\) as \(| x | \to \infty\). In the present paper, for general domains \(\Omega\), with the help of the concentration-compactness argument, we first obtain the “ground state solution” and then combine it with some ideas of G. Cerami, S. Solimini and M. Struwe [J. Funct. Anal. 69, 289-306 (1986; Zbl 0614.35035)] to prove the existence of another solution which changes sign.


35J65 Nonlinear boundary value problems for linear elliptic equations


Zbl 0614.35035
Full Text: DOI


[1] A. Bahri and P.L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, preprint. · Zbl 0883.35045
[2] Benci V., Positive solution of semilinear elliptic equation in exterior domains 99 pp 283– (1987)
[3] Cerami G., Some existence results for superlinear elliptic boundary value problems involving critical exponents 69 pp 289– (1986) · Zbl 0614.35035
[4] Esteban, M.J. 1989.Rupture de symétrie pour des problèmes de Neumann surlinéaires dans des extérieurs, I Vol. 308, 281–286. Note C.R.A.S.
[5] Esteban M.J., Nonsymmetric ground states of symmetric variational problems pp 259– (1991) · Zbl 0826.49002
[6] M.J. Esteban and V. Coti Zelati, Symmetry breaking and multiple solutions for a Neumann problem in an exterior domain, preprint. · Zbl 0748.35012
[7] Lions, P.L. 1984.The concentration-compactness principle in the Calculus of Variations. The locally compact case. I and II, Vol. 1, 109–145. Ann. I.H.P. Anal. Non Lin. · Zbl 0541.49009
[8] Lions P.L., Solutions of Hartree-Fock equations for Coulomb systems 109 pp 33– (1987) · Zbl 0618.35111
[9] Lions P.L., On positive solutions of semilinear elliptic equation in unbounded domains (1988) · Zbl 0685.35039
[10] Lions P.L., Lagrange multipliers, Morse indices and compactness (1990)
[11] Miranda C., Un’ osservazione sul teorema di Brouwer pp 5– (1940)
[12] Zhu X.P., Multiple entire solutions of semilinear elliptic equation 12 pp 1297– (1988) · Zbl 0671.35023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.