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We deal with some Dirichlet problems involving a nonlocal term. The existence of two nonzero, nonnegative solutions is achieved by applying a recent result by Ricceri.
Two nontrivial solutions are obtained for nonhomogeneous semilinear Schrödinger equations.
Let Ω be a bounded domain in with smooth boundary. Consider the following elliptic system:
in Ω,
in Ω,
u = 0, v = 0 in ∂Ω. (ES)
We assume that H is an even "-"-type Hamiltonian function whose first order partial derivatives satisfy appropriate growth conditions. We show that if (0,0) is a hyperbolic solution of (ES), then (ES) has at least 2|μ| nontrivial solutions, where μ = μ(0,0) is the renormalized Morse index of (0,0). This proves a conjecture by Angenent and van der Vorst.
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