Existence of entire solutions for a class of quasilinear elliptic equations

Abstract

Abstract. The paper deals with the existence of entire solutions for a quasilinear equation (E) λ in R N , depending on a real parameter λ, which involves a general elliptic operator in divergence form A and two main nonlinearities. The competing nonlinear terms combine each other, being the first subcritical and the latter supercritical. We prove the existence of a critical value λ * > 0 with the property that (E) λ admits nontrivial non-negative entire solutions if and only if λ ≥ λ * . Furthermore, when λ > λ ≥ λ * , the existence of a second independent nontrivial non-negative entire solution of (E) λ is proved under a further natural assumption on A.Mathematics Subject Classification (2010). Primary 35J62, 35J70; Secondary 35J20.

Existence of solutions for perturbed fractional p-Laplacian equations

“…By using variational methods, the authors obtained the existence and multiplicity of entire solutions of (1.7). Similarly, two open problems mentioned above in [6] can be applied to the fractional setting. Recently, Pucci and Zhang [29] solved the above open problems for a class of quasilinear elliptic equations in the setting of variable exponents.…”

Density properties for fractional Sobolev spaces

“…Motivations. Recently, in the literature a deep interest was shown for nonlocal operators, thanks to their intriguing analytical structure and in view of several applications in a wide range of contexts, such as the thin obstacle problem, optimization, finance, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes, flame propagation, conservation laws, ultra-relativistic limits of quantum mechanics, quasi-geostrophic flows, multiple scattering, minimal surfaces, materials science and water waves: see for instance [3,4,5,6,8,9,10,11,14,15,16,20,21,22,23,24,27,29,30,31,32,33] and references therein. One of the typical models considered is the equation may be defined as (1.2) −(−∆) s u(x) =ˆR n u(x + y) + u(x − y) − 2u(x) |y| n+2s dy for x ∈ R n (see [12,28] and references therein for further details on the fractional Laplacian) and the right-hand side f is a function satisfying suitable regularity and growth conditions.…”

Combined concave–convex effects in anisotropic elliptic equations with variable exponent

“…This result was extended by Autuori and Pucci [4] to the quasilinear case and by Autuori and Pucci [5] and Brändle, Colorado, de Pablo, and Sánchez [6] to elliptic equations involving the fractional Laplacian.…”