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

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 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 [G. Autuori and P. Pucci, Nonlinear Differential Equations Appl. NoDEA 20 (2013), 977–1009] 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 [S. Alama and G. Tarantello, J. Funct. Anal. 41 (1996), 159–215] 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.