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With suitable assumptions, uniqueness of nontrivial nonnegative solution is established for the following nonlinear integral equation φ(x)=∫DE(x,y)G(y,φ(y))dy. This result is applied to branching d...

We are concerned with determining values of γ, for which there exist radial nodal solutions of the boundary-value problem where B is the unit open ball in ℝ N with N ≥ 1. The proofs of our main results are based upon bifurcation techniques. In particular, we establish the global bifurcation theory for a second-order weighted p-Laplacian eigenvalue problem in B.

In this paper, we shall establish a Dancer-type unilateral global bifurcation result for a class of quasilinear elliptic problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that is a bifurcation point of the above problems and there are two distinct unbounded continua, and , consisting of the bifurcation branch from , where is the th positive or negative eigenvalue of the linear problem corresponding to the above problems, . As applications of the above unilateral global bifurcation result, we study the existence of nodal solutions for a class of quasilinear elliptic problems with sign-changing weight. Moreover, based on the bifurcation result of Drabek and Huang (1997) Dai G, Ma R. Unilateral global bifurcation phenomena and nodal solutions for -Laplacian. J. Differ. Equ. 2012;252:2448–2468., we study the existence of one-sign solutions for a class of high-dimensional quasilinear elliptic problems with sign-changing weight.

In this article, we consider a nonlocal problem involving the p(x)-Laplacian of the type Applying a three critical points theorem from Bonanno [A critical points theorem and nonlinear differential problems, J. Global Optim. 28 (2004), pp. 249–258], we obtain the existence of two intervals of positive real parameters λ for which the above problem admits three weak solutions in , whose norms are uniformly bounded with respect to λ belonging to one of the two open intervals. In particular, we also prove some properties of p(x)-Kirchhoff–Laplace operator.