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We are concerned with the following elliptic equations: \begin{equation*} (-\Delta)_p^su+V(x)|u|^{p-2}u=\lambda g(x,u) \quad \textmd{in} \quad \mathbb R^{N}, \end{equation*} where $(-\Delta)^s_p$ is the fractional $p$-Laplacian operator with $0<s<1<p<+\infty$, $sp<N$, the potential function $V:\mathbb R^{N}\to(0,\infty)$ is a continuous potential function, and $g:\mathbb R^{N}\times\mathbb R \to \mathbb R$ satisfies a Carath\'eodory condition. We show the existence of at least one weak solution for the problem above without the Ambrosetti and Rabinowitz condition. Moreover, we give a positive interval of the parameter $\lambda$ for which the problem admits at least one nontrivial weak solution when the nonlinearity $g$ has the subcritical growth condition.
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참고문헌 (50)
참고문헌 신청
R. A. Adams / 2003 / Sobolev Spaces / Elsevier/Academic Press
Claudianor O. Alves / 2010 / On superlinear -Laplacian equations in / Nonlinear Analysis: Theory, Methods & Applications 73(8):2566~2579
A. Ambrosetti / 1973 / Dual variational methods in critical point theory and applications / J. Functional Analysis 14:349~381
Giuseppina Autuori / 2013 / Elliptic problems involving the fractional Laplacian in RN / Journal of Differential Equations 255(8):2340~2362
B. Barrios / 2012 / On some critical problems for the fractional Laplacian operator / Journal of Differential Equations 252(11):6133~6162
Claudianor O. Alves / 2010 / On superlinear -Laplacian equations in / Nonlinear Analysis: Theory, Methods & Applications 73(8):2566~2579
A. Ambrosetti / 1973 / Dual variational methods in critical point theory and applications / J. Functional Analysis 14:349~381
Giuseppina Autuori / 2013 / Elliptic problems involving the fractional Laplacian in RN / Journal of Differential Equations 255(8):2340~2362
B. Barrios / 2012 / On some critical problems for the fractional Laplacian operator / Journal of Differential Equations 252(11):6133~6162
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