Electron. J. Differential Equations, Vol. 2017 (2017), No. 251, pp. 1-10.

Existence of solutions to superlinear p-Laplace equations without Ambrosetti-Rabinowizt condition

Duong Minh Duc

Abstract:
We study the existence of non-trivial weak solutions in $W^{1,p}_{0}(\Omega)$ of the super-linear Dirichlet problem
$$\displaylines{ - \hbox{div}(|\nabla u|^{p-2}\nabla u)=f(x,u) \quad \text{in }\Omega,\cr u=0 \quad \text{on }\partial\Omega, }$$
where f satisfies the condition
$$ |f(x,t)|\leq |\omega(x)t|^{r-1} + b(x)\quad \forall (x,t) \in \Omega\times \mathbb{R}, $$
where $r\in (p,\frac{Np}{N-p})$, $b\in L^{\frac{r}{r-1}}(\Omega)$ and $|\omega|^{r-1}$ may be non-integrable on $\Omega$.

Submitted May 8, 2017. Published October 10, 2017.
Math Subject Classifications: 46E35, 35J20.
Key Words: Nemytskii operators; p-Laplacian; multiplicity of solutions; mountain-pass theorem.

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Duong Minh Duc
University of Sciences
Vietnam National University
227 Nguyen Van Cu Q5
Hochiminh City, Vietnam
email: dmduc@hcmus.edu.vn

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