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 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.
In this paper, Morse theory is used to study the existence and multiplicity of nontrivial solutions for the following class of quasilinear elliptic equations:where
In this paper, we study the following nonlinear Klein-Gordon-Maxwell system:where ω and λ are positive constants, V is a continuous function with negativeis a positive potential function. Under the classic Ambrosetti-Rabinowitz condition, nontrivial solutions are obtained via the symmetric mountain pass theorem and the mountain pass theorem. In our paper, the nonlinearity F can also change sign and does not need to satisfy any 4-superlinear condition. We extend and improve some existing results to some extent.
MSC: 35J10; 35J60; 35J65
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