Multiplicity Results of Solutions to the Double Phase Problems of Schrödinger–Kirchhoff Type with Concave–Convex Nonlinearities

Abstract

The present paper is devoted to establishing several existence results for infinitely many solutions to Schrödinger–Kirchhoff-type double phase problems with concave–convex nonlinearities. The first aim is to demonstrate the existence of a sequence of infinitely many large-energy solutions by applying the fountain theorem as the main tool. The second aim is to obtain that our problem admits a sequence of infinitely many small-energy solutions. To obtain these results, we utilize the dual fountain theorem. In addition, we prove the existence of a sequence of infinitely many weak solutions converging to 0 in L∞-space. To derive this result, we exploit the dual fountain theorem and the modified functional method.

Infinitely many small energy solutions to the $ p $-Laplacian problems of Kirchhoff type with Hardy potential

“…Our second main result on the existence of a sequence of small energy solutions converging to 0 in L ∞ -space was motivated by the works [20,24,31,53,54]. To this end, we derive this multiplicity result by employing the dual fountain theorem as in our first consequence.…”

“…Also, as far as we know, the L ∞ -bound for weak solutions to nonlinear elliptic problems with Hardy potentials has not been much studied, and we are only aware of the work of [30]. With the help of the regularity result in [30], we provide our second main result by combining the modified functional method with the dual fountain theorem as in [24,31]. For this reason, our approach differs from previously related studies [20,53,54] that used the global variational formulation given in [19].…”