Existence of entire solutions for a class of variable exponent elliptic equations

Abstract

The paper deals with the existence of entire solutions for a quasilinear equation (Eλ) in RN, depending on a real parameter λ, which involves a general variable exponent elliptic operator A in divergence form and two main nonlinearities. The competing nonlinear terms combine each other. Under some conditions, we prove the existence of a critical value λ⁎>0 with the property that (Eλ) admits nontrivial nonnegative entire solutions if and only if λ≥λ⁎. Furthermore, under the further assumption that the potential A of A is uniform convex, we give the existence of a second independent nontrivial nonnegative entire solution of (Eλ), when λ>λ⁎. Our results extend the previous work of Autuori and Pucci (2013) [6] from the case of constant exponents p, q and r to the case of variable exponents. More interesting, we weaken the condition max{2,p}<q<min{r,p⁎} to the simple request that 1≪q≪r. Furthermore, we extend the previous work of Alama and Tarantello (1996) [2] from Dirichlet Laplacian problems in bounded domains of RN to the case of a general variable exponent differential equation in the entire RN, and also remove the assumption q>2. Hence the results of this paper are new even in the canonical case p(⋅)≡2.