A Variational Approach to the Fredholm Alternative for the p-Laplacian near the First Eigenvalue

We are concerned with the existence of a weak solution to the degenerate quasi-linear Dirichlet boundary value problem It is assumed that 1  <  p  <  ∞, p  ≠  2, Ω is a bounded domain in is a given function, and λ stands for the (real) spectral parameter near the first (smallest) eigenvalue λ₁ of the positive p-Laplacian  − Δ _p , where. Eigenvalue λ₁ being simple, let φ₁ denote the eigenfunction associated with it. We show the existence of a solution for problem (P) when f “nearly” satisfies the orthogonality condition ∫_Ω f φ₁  dx  =  0 and λ  ≤  λ₁  +  δ (with δ >  0 small enough). Moreover, we obtain at least three distinct solutions if either p < 2 and λ₁  −  δ ≤  λ  <  λ₁, or else p > 2 and λ₁  <  λ  ≤  λ₁  +  δ. The proofs use a minimax principle for the corresponding energy functional performed in the orthogonal decomposition induced by the inner product in L ²(Ω). First, the global minimum is taken over, and then either a local minimum or a local maximum over lin {φ₁}. If the latter is a local minimum, the local minimizer in thus obtained provides a solution to problem (P). On the other hand, if it is a local maximum, one gets only a pair of sub- and supersolutions to problem (P), which is then used to obtain a solution by a topological degree argument.