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We are concerned with the following elliptic equations: \begin{equation*} \begin{cases} (-\Delta)_p^su=\lambda f(x,u) \quad \textmd{in} \ \ \Omega,\\ u= 0\quad \text{on}\ \ \mathbb{R}^N\backslash\Omega, \end{cases} \end{equation*} where $\lambda$ are real parameters, $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator, $0<s<1<p<+\infty$, $sp<N$, and $f:\Omega\times\mathbb R \to \mathbb R$ satisfies a Carath\'eodory condition. By applying abstract critical point results, we establish an estimate of the positive interval of the parameters $\lambda$ for which our problem admits at least one or two nontrivial weak solutions when the nonlinearity $f$ has the subcritical growth condition. In addition, under adequate conditions, we establish an apriori estimate in $L^{\infty}(\Omega)$ of any possible weak solution by applying the bootstrap argument.
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참고문헌 (36)
참고문헌 신청
R. A. Adams / 2003 / Pure and Applied Mathematics (Amsterdam), 140 / Elsevier/Academic Press
A. Ambrosetti / 1973 / Dual variational methods in critical point theory and applications / J. Functional Analysis 14:349~381
J. -H. Bae / 2019 / Critical points theorems via the generalized Ekeland variational principle and its application to equations of p(x)-Laplace type in RN / Taiwanese J. Math 23(1):193~229
G. Barletta / 2016 / Existence results for a Neumann problem involving the p(x)-Laplacian with discontinuous nonlinearities / Nonlinear Anal. Real World Appl 27:312~325
B. Barrios / 2012 / On some critical problems for the fractional Laplacian operator / J. Dierential Equations 252(11):6133~6162
A. Ambrosetti / 1973 / Dual variational methods in critical point theory and applications / J. Functional Analysis 14:349~381
J. -H. Bae / 2019 / Critical points theorems via the generalized Ekeland variational principle and its application to equations of p(x)-Laplace type in RN / Taiwanese J. Math 23(1):193~229
G. Barletta / 2016 / Existence results for a Neumann problem involving the p(x)-Laplacian with discontinuous nonlinearities / Nonlinear Anal. Real World Appl 27:312~325
B. Barrios / 2012 / On some critical problems for the fractional Laplacian operator / J. Dierential Equations 252(11):6133~6162
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