Existence of constant sign solutions for the p-Laplacian problems in the resonant case with respect to Fučík spectrum
Mieko Tanaka · SUT Journal of Mathematics · 2009
We consider the following the p-Laplacian equation in a bounded domain Ω: { −Δpu=f(x,u)in Ω,u=0on ∂Ω. We treat the case of nonlinearity term f satisfying the following conditions f(x,t)={ a0t+p−1−b0t−p−1+o(|t|p−1)at 0,at+p−1−bt−p−1+o(|t|p−1)at ∞, for constants a0,b0,a and b. We prove the existence of a positive solution or a negative solution in the case of (a0−λ1)(a−λ1)=0 or (b0−λ1)(b−λ1)=0 respectively, where λ1 is the first eigenvalue of −Δp.