Multiplicity of Positive Solutions for Critical Fractional Equation Involving Concave–Convex Nonlinearities and Sign-Changing Weight Functions

Abstract

This paper is devoted to study a class of fractional equations with critical exponent, concave nonlinearity and sign-changing weight functions. By means of variational methods, the multiplicity of the positive solutions to this problem is obtained.Mathematics Subject Classification. 35J20, 35S15, 47J30, 47G20.

Fractional weighted problems with a general nonlinearity or with concave‐convex nonlinearities

“…For related results, see also previous work 5 for the spectral fractional Laplacian, recalling that such an operator is quite different from the one considered here, see previous work 6, Section 2.3 for a detailed discussion on this fact. We also mention, 7 where a probem like () with pure powers and with f having critical growth has been studied in presence of continuous and sign changing coefficients, showing the existence of two positive solutions for λ small enough. We conclude recalling that many other concave–convex problems have been studied in different situations, for instance, in previous works 8–11 …”

The global existence and time-decay for the solutions of the fractional pseudo-parabolic equation

“…Since the fractional dissipation operator (−∆) α is nonlocal and can be regarded as the infinitesimal generators of Levy stable diffusion processes, many scientists have found that it describes some physical phenomena more exact than integral differential equations( refer to [14], [15], [16], [17], [18]). More and more work has been devoted to the investigation of fractional differential equations( [14], [15], [19], [20]). Motivated by these results, we will mainly study the fractional pseudo-parabolic equations (1.1).…”