American Institute of Mathematical Sciences

$-( | u' | ^{p-2}u')' + \varepsilon |u|^{p-2}u= \nabla F(t,u), $ in $(0,T)$,

$((|u'|^{p-2}u')(0), $ $ -(|u'|^{p-2}u')(T))$ $\in \partial j(u(0), u(T)),$

where $\varepsilon \geq 0$, $p \in (1, \infty)$ are fixed, the convex function $j:\mathbb R^N \times \mathbb R^N \to (- \infty , +\infty ]$ is proper, even, lower semicontinuous and $F:(0,T) \times \mathbb R^N \to \mathbb R $ is a Carathéodory mapping, continuously differentiable and even with respect to the second variable.

$ - \Delta U = AU + (u^p_+, v^p_+)+ F$ in $\Omega$

$ U = 0 $ on $ \partial\Omega,$

where $\Omega\subset \mathbb R^{N}$ is a bounded smooth domain; $U=(u,v), p=2^\star -1$, with $2^\star=\frac{2N}{N-2}, N \geq 3$; ${w_+}=$ max{ $w,0$} and $F \in L^s(\Omega)\times L^s(\Omega)$ for some $s>N$.
Using variational methods, we prove the existence of at least two solutions. The first is obtained explicitly by a direct calculation and the second via the Mountain Pass Theorem for the case $0< \mu_1 \leq \mu_2< \lambda_1$ or Linking Theorem if $\lambda_k < \mu_1 \leq \mu_2 < \lambda_{k+1}$, where $\mu_1, \mu_2$ are eigenvalues of symmetric matrix $A$ and $\lambda_j$ are eigenvalues of $(-\Delta, H_0^1(\Omega))$.

$\beta (k,\alpha)$: $=$ min{ $2^{3/2+1/2\alpha}\cdot\delta( k,\alpha )a(0) a'(0)^{-1}:k\in (0,1)$},

where $a(0)=\int_\Omegau_{0}(x)^{2}dx,$ $a'(0)=2\int_\Omega u_{0}( x) u_1(x) dx$ and $\delta(k,\alpha )$ is given by

$\delta(k,\alpha)$ :$=\frac{1}{k}(\frac{k^2}{1-k^2})^{\frac{\alpha }{1+2\alpha}}$ $(1-(1+(\frac{1}{ k^2}-1)^{\frac{\alpha}{1+2\alpha}})^{\frac{-1}{2\alpha} }).

$ -\Delta u= g(x,u) + f(x,u)\quad x\in \Omega $

$u=0\quad x\in \partial \Omega$

where $g(x,-\xi )=-g(x,\xi)$ and $g$ has subcritical exponential growth in $\mathbb R^2$. Using the method developed by Bolle, we prove that this problem has infinitely many solutions under suitable conditions on the growth of $g(u)$ and $f(u)$.