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Multiple solutions of semilinear elliptic problems at resonance. (English) Zbl 0832.35048

The authors consider the problem \[ \Delta u+ \lambda_1 u+ g(u)= 0\quad\text{in }\Omega,\quad u= 0\quad\text{on }\partial\Omega, \] where \(g\in C^1\), \(g(0)= 0\) and \(\lambda_1\) is the principal eigenvalue of the fixed membrane problem on \(\Omega\). They assume that \(0\leq \liminf_{|s|\to \infty} (g(s)/ s)\leq \limsup_{|s|\to \infty} (g(s)/s)\leq \gamma\), and \(0< \gamma< \lambda_2- \lambda_1\). Their main result is that if \(g'(0)< 0\) or \(\lambda_m< \lambda_1+ g'(0)< \lambda_{m+ 1}\) for some \(m\geq 2\) and \(g\) satisfies a Landesman-Lazer type condition then there are at least two solutions.
Reviewer: R.Sperb (Zürich)

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
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[1] Rumbos, A., A semilinear elliptic boundary value problem at resonance where the nonlinearity may grow linearly, Nonlinear Analysis, 16, 1159-1168 (1991) · Zbl 0739.35023
[2] ROBINSON S., Double resonance in semilinear elliptic boundary value problems over bounded and unbounded domains, Nonlinear Analysis; ROBINSON S., Double resonance in semilinear elliptic boundary value problems over bounded and unbounded domains, Nonlinear Analysis · Zbl 0798.35055
[3] Ahmad, S., Multiple nontrivial solutions of resonant and nonresonant asymptotically linear problems, (Proc. Am. math. Soc., 96 (1986)), 405-409 · Zbl 0634.35029
[4] Rabinowitz, P., Minimax Methods in Critical Point Theory with Applications to Differential Equations, (CBMS Regional Conf. Ser. in Math. (1986), American Mathematical Society: American Mathematical Society Providence, R.I), No. 65
[5] Ambrosetti, A., Differential Equations with Multiple Solutions and Nonlinear Functional Analysis, Equadiff 82, (Lecture Notes in Mathematics, Vol. 1017 (1983), Springer: Springer Berlin)
[6] Hofer, H., A note on the topological degree at a critical point of mountain pass type, (Proc. Am. math. Soc., 90 (1984)), 309-315 · Zbl 0545.58015
[7] Struwe, M., Variational Methods (1990), Springer: Springer Berlin
[8] Costa, D. G.; Oliveira, A. S., Existence of solutions for a class of semilinear elliptic problems at double resonance, Boll. Bras. Mat., 19, 21-37 (1988) · Zbl 0704.35048
[9] Nirenberg, L., Topics in Nonlinear Functional Analysis (1974), Courant Inst. of Math. Sciences: Courant Inst. of Math. Sciences New York · Zbl 0286.47037
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