Multiple solutions of a semilinear elliptic equation in \(\mathbb{R}^ N\). (English) Zbl 0797.35039

Summary: We are concerned with the existence of multiple solutions of \[ -\Delta u + u = \lambda b(x) | u |^{p-1} u + c(x) | u |^{q-1} u \] where \(1<p\), \(q<(N+2)/(N-2)\) if \(N \geq 3\), \(1<p\), \(q<+\infty\) if \(N=2\), \(\lambda>0\). We obtain the existence of multiple solutions by using concentrations-compactness method and dual variational principle to establish the corresponding existence of critical points.


35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
Full Text: DOI Numdam EuDML


[1] Ambrosetti, A.; Rabinowitz, P., Dual variational methods in critical point theory and applications, J. Fund. Anal., Vol. 14, 327-381, (1973) · Zbl 0273.49063
[2] A. Bahri and P. L. Lions, On the Existence of a Positive Solution of Semilinear Elliptic Equations in Unbounded Domains, preprint. · Zbl 0883.35045
[3] Benci, V.; Cerami, G., Positive solutions of semilinear elliptic problems in exterior domains, Arch. Rat. Mech. Anal., Vol. 99, 283-300, (1987) · Zbl 0635.35036
[4] Berestycki, H.; Lions, P. L., Nonlinear scalar field equations, I and II, Arch. Rat. Mech. Anal., Vol. 82, 313-376, (1983)
[5] Ding, W. Y.; Ni, W. M., On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rat. Mech. Anal., Vol. 91, 288-308, (1986)
[6] Ekeland, I., Nonconvex minimization problems, Bull. Amer. Math. Soc., Vol. 1, 443-474, (1979) · Zbl 0441.49011
[7] Gidas, B.; Ni, W. M.; Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in ℝ^{n}, Advances in Math., Supplementary Studies, Vol. 7, 369-402, (1981)
[8] Li, Y. Y., On second order nonlinear elliptic equations, Dissertation, (1988), New York Univ.
[9] Lions, P. L., The concentration-compactness principle in the calculus of variations. the locally compact case, I and II, Vol. 1, 109-145, (1984), and 223-283 · Zbl 0541.49009
[10] Lions, P. L., On positive solution of semilinear elliptic equation in unbounded domains, Nonlinear Diffusion Equations and Their Equilibrium States, (1988), Springer New York · Zbl 0685.35039
[11] Kwong, M. K., Uniqueness of positive solution of ∆uu + u^{p} = 0 in R^{n}, Arch. Rat. Mech. Anal., Vol. 105, 243-266, (1989) · Zbl 0676.35032
[12] Strauss, W., Existence of solitary waves in higher dimensions, Comm. Math. Phys., Vol. 55, 109-162, (1977) · Zbl 0356.35028
[13] Zhu, X. P., Multiplie entire solutions of semilinear elliptic equations, Nonlinear Anal., Vol. 12, 1297-1316, (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.