Infinitely many solutions of p-sublinear p-Laplacian equations

Multiple solutions for nonlinear cone degenerate elliptic equations

“…The existence and non-uniqueness for the p-Laplacian have been verified in [15]. More interesting and important results about the p-Laplacian can be found in [3,4,12,16] and references therein.…”

The existence of infinitely many solutions for nonlinear elliptic equations involving p-Laplace type operators in R^N

“…Since F(x, t) is continuous on R N × [−t 0 , t 0 ], there exists a positive constant M such that |F(x, t)| M, for all (x, t) ∈ R N × [−t 0 , t 0 ]. Therefore we can choose a real number M 0 such that F(x, t) M 0 , for all (x, t) ∈ R N × R, and thus 19) for all x ∈ R N and for all n ∈ N. By the relation (2.15), we know that |u n (x)| = | n (x)| u n X → ∞ as n → ∞ for all x ∈ Ω 1 . Furthermore, it follows from the condition (F3) that…”

“…Utilizing the argument in [33], Naimen [30] showed that nonlinear Neumann problems have infinitely many solutions whose L ∞ -norms converge to zero. In this direction, many authors considered the results for the nonlinear equations on a bounded domain in R N ; see [9,15,19,31]. To the best of our knowledge, such a result on the whole space R N is very rare.…”

Multiplicity of Small or Large Energy Solutions for Kirchhoff–Schrödinger-Type Equations Involving the Fractional p-Laplacian in ℝN

“…Utilizing the argument in [27], Guo [28] showed that the p-Laplacian equations with indefinite concave nonlinearities have infinitely many solutions. In this regard, lots of authors have considered the results for the elliptic equations with nonlinear terms on a bounded domain in R N ; see [29][30][31]. It is well known that the studies in [14,17,19,21,22,26,29,32,33] as well as our first primary result essentially demand some global conditions on f (x, t) for t, such as oddness and behavior at infinity, for applying the fountain theorem to allow an infinite number of solutions.…”

“…In contrast to these studies that yield large solutions in that they form an unbounded sequence, by modifying and extending the function f (x, t) to a adequate functionf (x, t), the authors in [27][28][29] investigated the existence of small energy solutions to equations of the elliptic type. A natural question is whether the results in [27][28][29][30][31] may be extended to Equation (1). As is known, such a result for Kirchhoff-Schrödinger-type equations involving the non-local fractional p-Laplacian on the whole space R N has not been much studied, although a given domain is bounded.…”