Fractal and Fractional (Jul 2024)

Multiple Solutions to the Fractional <i>p</i>-Laplacian Equations of Schrödinger–Hardy-Type Involving Concave–Convex Nonlinearities

  • Yun-Ho Kim

DOI
https://doi.org/10.3390/fractalfract8070426
Journal volume & issue
Vol. 8, no. 7
p. 426

Abstract

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This paper is concerned with nonlocal fractional p-Laplacian Schrödinger–Hardy-type equations involving concave–convex nonlinearities. The first aim is to demonstrate the L∞-bound for any possible weak solution to our problem. As far as we know, the global a priori bound for weak solutions to nonlinear elliptic problems involving a singular nonlinear term such as Hardy potentials has not been studied extensively. To overcome this, we utilize a truncated energy technique and the De Giorgi iteration method. As its application, we demonstrate that the problem above has at least two distinct nontrivial solutions by exploiting a variant of Ekeland’s variational principle and the classical mountain pass theorem as the key tools. Furthermore, we prove the existence of a sequence of infinitely many weak solutions that converges to zero in the L∞-norm. To derive this result, we employ the modified functional method and the dual fountain theorem.

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