We investigate the multiplicity of positive solutions for the superlinear singular [Formula: see text]-Laplacian equation [Formula: see text] on a weighted locally finite graph [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] is a parameter. The potential functions [Formula: see text] and [Formula: see text] satisfy some suitable conditions. The function [Formula: see text], [Formula: see text]. Besides, the nonlinear term [Formula: see text] exhibits a more general superlinear growth condition that is weaker than the Ambrosetti–Rabinowitz type condition. We establish the existence of two distinct solutions through variational methods combined with perturbation techniques. More precisely, the first solution is obtained via a local minimization argument. Furthermore, the existence of the second solution is established by employing the Mountain Pass Lemma. We overcome some difficulties which are caused by: [Formula: see text] the singular term; [Formula: see text] the lack of compactness of Sobolev embedding; [Formula: see text] the weakened Ambrosetti–Rabinowitz condition.
In this paper, we consider a degenerate Dirichlet problem with a reaction which can be resonant as [Formula: see text] with respect to the principal eigenvalue of the Dirichlet weighted [Formula: see text]-Laplacian [Formula: see text]. By using variational tools, critical groups (to overcome the fact that there is no global regularity theory for our problem) and the spectral properties of [Formula: see text], we show that problem has at least three nontrivial, bounded, ordered solutions. Two are of constant sign (positive and negative) and the third (between them) is nodal (sign-changing).
This paper investigates the following Dirac–Bopp–Podolsky system: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are [Formula: see text] Pauli–Dirac matrices. Under some suitable conditions for the external potential [Formula: see text] and the nonlinearity [Formula: see text], we establish the existence and multiplicity of steady-state solutions by using variational method and nonlocal analysis techniques. Moreover, we regard [Formula: see text] as a parameter and study the asymptotic behaviors of solutions as [Formula: see text], which reflect some relationship between [Formula: see text] and [Formula: see text].
In this paper, we consider the multiplicity of solutions to the following fractional Kirchhoff equation (a + b integral(R3) |(-Delta)(s/2) u|(2)dx) (-Delta)(s)u + V (epsilon x)u = lambda u + f (u) in R-3, with a prescribed mass integral(3)(R) |u|(2)dx = m(2), where s is an element of (3/4, 1), & varepsilon;,a,b,m > 0, lambda is an element of R is an unknown parameter that appears as a Lagrange multiplier, the potential V : R-3 -> (-infinity, 0] is a C-1 function and the nonlinearity f satisfies a general assumption of mass supercritical growth. Under some suitable assumptions on V and f, we relate the number of normalized solutions with the topology of the set where the potential V attains its minimum value. Our main results improve and complement the studies of He and Zou [Multiplicity of concentrating solutions for a class of fractional Kirchhoff equation, Manuscr. Math. 158(1-2) (2019) 159-203], in the abstract setting of normalized solutions, and seem to be the first contribution regarding existence of multiple normalized ground state solutions concentrating at a potential well, for the fractional Kirchhoff problems with L-2-supercritical perturbations.
We consider a nonlinear, nonhomogeneous singular elliptic problem with (p,q)-growth (unbalanced growth). Using a combination of variational tools with critical groups, we establish the existence of at least two bounded solutions.
