In this paper, we investigate the existence of nontrivial weak solutions to a class of elliptic equations involving a general nonlocal integrodifferential operator LAK, two real parameters, and two weight functions, which can be sign-changing. Considering different situations concerning the growth of the nonlinearities involved in the problem (P), we prove the existence of two nontrivial distinct solutions and the existence of a continuous family of eigenvalues. The proofs of the main results are based on ground state solutions using the Nehari method, Ekeland’s variational principle, and the direct method of the calculus of variations. The difficulties arise from the fact that the operator LAK is nonhomogeneous and the nonlinear term is undefined.
In this paper we prove the compactness of the embeddings of the space of radially symmetric functions of $BL(\mathbb{R}^N)$ into some Lebesgue spaces. In order to do so we prove a regularity result for solutions of the Poisson equation with measure data in $\mathbb{R}^N$, as well as a version of the Radial Lemma of Strauss to the space $BL(\mathbb{R}^N)$. An application is presented involving a quasilinear elliptic problem of higher-order, where variational methods are used to find the solutions.
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