On the Fredholm-type theorems and sign properties of solutions for $(p,q)$-Laplace equations with two parameters

Abstract

We consider the Dirichlet problem for the nonhomogeneous equation $-{\Delta }_{p}u-{\Delta }_{q}u=\alpha |u{|}^{p-2}u+\beta |u{|}^{q-2}u+f(x)$ in a bounded domain, where $p\ne q$, and $\alpha ,\beta \in \mathbb{R}$ are parameters. We explore assumptions on $\alpha$ and $\beta$ that guarantee the resolvability of the considered problem. Moreover, we introduce several curves on the $(\alpha ,\beta )$-plane allocating sets of parameters for which the problem has or does not have positive or sign-changing solutions, provided $f$ is of a constant sign.

24 pages, 2 figures. Minor improvements according to referee's suggestions. Accepted to Annali di Matematica Pura ed Applicata (1923 -)