| Abstract: |
| In this talk, we discuss some recent results on the existence and uniform boundedness of solutions for a general class of Dirichlet anisotropic elliptic problems, defined in open bounded subsets of $\mathbb{R}^N$ $(N\ge 2)$, and driven by the the so-called $\overrightarrow{p}$-Laplacian operator, where $\overrightarrow{p}=\left(p_1,p_2,\dots,p_N\right)$, with $N/p=\sum_{k=1}^N (1/p_k)>1$. The feature of this study is the inclusion of a possibly singular gradient-dependent term $\Psi(u,\nabla u)=\sum_{j=1}^N |u|^{\theta_j-2}u\, |\partial_j u|^{q_j}$, where $\theta_j>0$ and $0\leq q_j < p_j$ for $1\leq j\leq N$. We also investigate the case $q_j=p_i$ for every $1\leq j\leq N$ in the singularity-free case, and we prove the existence of a solution which a further regularity of exponential type. Based on joint works with Florica C. C\^irstea (The University of Sydney, Australia) and V. Ferone (Universit\`a degli Studi di Napoli Federico II). |
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