In this paper, we study the existence of nontrivial solution to a quasi-linear problem (-Delta)(s)(p)u + V (x) vertical bar u vertical bar(p-2)u = Q(x) f (x, u), x is an element of R-N, where (-Delta)(s)(p)u(x) = 2 lim is an element of -> 0 integral R-N backslash B epsilon(x) vertical bar u(x) - u(y) vertical bar (p-2) (u(x) - u(y)) / vertical bar x - y vertical bar (N+sp) dy, x epsilon R-N is a nonlocal and nonlinear operator and p is an element of (1, infinity), s is an element of (0, 1). We study two cases: if f (x, u) is sublinear, then we get infinitely many solutions for (P) by Clark's theorem; if f (x, u) is superlinear, we obtain infinitely many solutions of the problem (P) by symmetric mountain pass theorem.

Infinitely many solutions for a class of elliptic problems involving the fractional Laplacian / Ferrara, Massimiliano; Ge, Bin; Sun, L.; Cui, Y-X; Zhao, T-T. - In: REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS, FÍSICAS Y NATURALES. SERIE A, MATEMÁTICAS. - ISSN 1578-7303. - 113:2(2019), pp. 657-673. [10.1007/s13398-018-0498-8]

Infinitely many solutions for a class of elliptic problems involving the fractional Laplacian

FERRARA, Massimiliano
Supervision
;
2019-01-01

Abstract

In this paper, we study the existence of nontrivial solution to a quasi-linear problem (-Delta)(s)(p)u + V (x) vertical bar u vertical bar(p-2)u = Q(x) f (x, u), x is an element of R-N, where (-Delta)(s)(p)u(x) = 2 lim is an element of -> 0 integral R-N backslash B epsilon(x) vertical bar u(x) - u(y) vertical bar (p-2) (u(x) - u(y)) / vertical bar x - y vertical bar (N+sp) dy, x epsilon R-N is a nonlocal and nonlinear operator and p is an element of (1, infinity), s is an element of (0, 1). We study two cases: if f (x, u) is sublinear, then we get infinitely many solutions for (P) by Clark's theorem; if f (x, u) is superlinear, we obtain infinitely many solutions of the problem (P) by symmetric mountain pass theorem.
2019
Fractional p-Laplacian; Without the (AR) condition; Fountain theorem; Clark's theorem; Nontrivial solution
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12318/3039
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