Summary
This paper treats nonlinear elliptic boundary value problems of the form
in the spaceL 2(Ω) by degree theoretic methods. Emphasis is placed on existence of multiplesolutions in the case, where the nonlinearity f crosses several eigenvalues of the correspondingeigenvalue problem Δθ+λθ = 0 with zero boundary values. No differentiability conditions(but Lipschitz type conditions) on f are assumed. A main tool is a new a priori bound forsolutions (Theorem 1). The method is not confined to the selfadjoint case. It applies alsoto some time-periodic parabolic and hyperbolic problems.
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McKenna, P.J., Redlinger, R. & Walter, W. Multiplicity results for asymptotically homogeneous semilinear boundary value problems. Annali di Matematica pura ed applicata 143, 247–257 (1986). https://doi.org/10.1007/BF01769219
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DOI: https://doi.org/10.1007/BF01769219