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A nonlinear elliptic system involving the p-Laplacian is considered in the whole R. Existence of nontrivial solutions is obtained by applying critical point theory; also a regularity result is established.

In this paper, we study the spectrum for the following eigenvalue problem with the p-biharmonic operator involving the Hardy term: ${\Delta \left(\right|\Delta u|}^{p-2}{\Delta u)=\lambda (\left|u\right|}^{p-2}u)/\left(\delta {\left(x\right)}^{2p}\right)$ in Ω, $u\in W{₀}^{2,p}\left(\Omega \right)$. By using the variational technique and the Hardy-Rellich inequality, we prove that the above problem has at least one increasing sequence of positive eigenvalues.

We prove an approximation theorem in generalized Sobolev spaces with variable exponent $W^{1,p\left(·\right)}\left(\Omega \right)$ and we give an application of this approximation result to a necessary condition in the calculus of variations.

We study the existence of solutions for a p-biharmonic problem with a critical Sobolev exponent and Navier boundary conditions, using variational arguments. We establish the existence of a precise interval of parameters for which our problem admits a nontrivial solution.

We consider the following quasilinear Neumann boundary-value problem of the type $$ \begin {cases} -\displaystyle \sum _{i=1}^{N}\frac {\partial }{\partial x_{i}}a_{i}\Big (x,\frac {\partial u}{\partial x_{i}}\Big ) + b(x)|u|^{p_{0}(x)-2}u = f(x,u)+ g(x,u) &\text {in} \ \Omega , \\ \quad \dfrac {\partial u}{\partial \gamma } = 0 &\text {on} \ \partial \Omega . \end {cases} $$ We prove the existence of infinitely many weak solutions for our equation in the anisotropic variable exponent Sobolev...

The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both $p(·)$-Harmonic and $p(·)$-biharmonic operators $$\begin{array}{c}{\Delta }_{p\left(x\right)}^{2}u-{\Delta }_{p\left(x\right)}u=\lambda w\left(x\right){\left|u\right|}^{q\left(x\right)-2}u\text{in}\Omega ,\\ u\in W^{2,p(·)}\left(\Omega \right)\cap W_0^{1,p(·)}\left(\Omega \right),\end{array}$$ is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces $L^{p(·)}\left(\Omega \right)$ and $W^{m,p(·)}\left(\Omega \right)$.

We prove the existence of solutions to nonlinear parabolic problems of the following type: $$\left\{\begin{array}{cc}{\displaystyle \frac{\partial b\left(u\right)}{\partial t}}+A\left(u\right)=f+\mathrm{div}\left(\Theta (x;t;u)\right)\hfill & \text{in}Q,\hfill \\ u(x;t)=0\hfill & \text{on}\partial \Omega \times [0;T],\hfill \\ b\left(u\right)(t=0)=b\left(u_0\right)\hfill & \text{on}\Omega ,\hfill \end{array}\right.$$ where $b:ℝ\to ℝ$ is a strictly increasing function of class ${𝒞}^1$, the term $$A\left(u\right)=-\mathrm{div}\left(a\right(x,t,u,\nabla u\left)\right)$$ is an operator of Leray-Lions type which satisfies the classical Leray-Lions assumptions of Musielak type, $\Theta :\Omega \times [0;T]\times ℝ\to ℝ$ is a Carathéodory, noncoercive function which satisfies the following condition: ${sup}_{\left|s\right|\le k}\left|\Theta (·,·,s)\right|\in E_{\psi }\left(Q\right)$ for all $k>0$, where $\psi$ is the Musielak complementary function of $\Theta$, and the second term $f$ belongs to $L^1\left(Q\right)$.