Volume 5, Issue 1

In this paper, we study the existence of solutions for the Kirchhoff problem M(∫ℝ3|∇u|2dx+∫ℝ3(λa(x)+1)u2dx)(-Δu+(λa(x)+1)u)=f(u)$M\Biggl (\int _{\mathbb {R}^{3}}|\nabla u|^{2}\, dx + \int _{\mathbb {R}^{3}} (\lambda a(x)+1)u^{2}\, dx\Biggl ) (- \Delta u + (\lambda a(x)+1)u) = f(u)$ in ℝ 3 , u ∈ H 1 (ℝ 3 ) Assuming that the nonnegative function a ( x ) has a potential well with int ( a  -1 ({0})) consisting of k disjoint components Ω 1 ,Ω 2 ,...,Ω k and the nonlinearity f ( t ) has a subcritical growth, we are able to establish the existence of positive multi-bump solutions by using variational methods.

The purpose of this paper is mainly to investigate the existence of entire solutions of the stationary Kirchhoff type equations driven by the fractional p -Laplacian operator in ℝ N . By using variational methods and topological degree theory, we prove multiplicity results depending on a real parameter λ and under suitable general integrability properties of the ratio between some powers of the weights. Finally, existence of infinitely many pair of entire solutions is obtained by genus theory. Last but not least, the paper covers a main feature of Kirchhoff problems which is the fact that the Kirchhoff function M can be zero at zero. The results of this paper are new even for the standard stationary Kirchhoff equation involving the Laplace operator.

We study the existence of positive solutions for fractional elliptic equations of the type (-Δ) 1/2 u = h ( u ), u > 0 in (-1,1), u = 0 in ℝ∖(-1,1) where h is a real valued function that behaves like e u 2 as u → ∞ . Here (-Δ) 1/2 is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near t = 0. In case h is concave near t = 0, we show the existence of multiple solutions for suitable range of λ by analyzing the fibering maps and the corresponding Nehari manifold.

Some existence results for a parametric Dirichlet problem defined on the Sierpiński fractal are proved. More precisely, a critical point result for differentiable functionals is exploited in order to prove the existence of a well-determined open interval of positive eigenvalues for which the problem admits at least one non-trivial weak solution.

By means of variational methods we investigate existence, nonexistence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.