In this paper, we deal with the Kirchhoff-type equationwhere λ > 0, V and Q are radial functions, which can be vanishing or coercive at infinity. With assumptions on f just in a neighborhood of the origin, existence and multiplicity of nontrivial radial solutions are obtained via variational methods.In particular, if f is sublinear and odd near the origin, we obtain infinitely many solutions of (P) λ for any λ > 0.Mathematics Subject Classification. 35J50 · 35J60 · 35J65.
In this paper, we deal with the fractional Laplacian equationswhere 0 < s < 1 < p < +∞, N ∈ N, N > 2s, ⊂ R N is a bounded domain with smooth boundary. Under local growth conditions of f (x, t), infinitely many solutions for problem (P) are obtained via variational methods.
In this paper, we give some properties about the (2, p)-Laplacian operator (p > 1, p = 2), and consider the existence of solutions to two kinds of partial differential equations related to the (2, p)-Laplacian operator by those properties. Specifically, we establish an existence result of positive solutions using fixed point index theory and an existence result of nodal solutions via the quantitative deformation lemma.
MSC: 35J05; 47H10; 47H11
In this paper, we deal with fractional p-Laplacian equations of the form\left\{\begin{aligned} \displaystyle(-\Delta)_{p}^{s}u&\displaystyle=\lambda f%
(x,u),&&\displaystyle x\in\Omega,\\
\displaystyle u(x)&\displaystyle=0,&&\displaystyle x\in\mathbb{R}^{N}\setminus%
\Omega,\end{aligned}\right.where {\lambda\in(0,+\infty)}, {0<s<1<p<+\infty} and {\Omega\subset\mathbb{R}^{N}}, {N\geqslant 2}, is a bounded domain with smooth boundary.
With assumptions on {f(x,t)} just in {\Omega\times(-\delta,\delta)}, where {\delta>0} is small, existence and multiplicity of nontrivial solutions are obtained via variational methods.
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