As key problems we consider semilinear elliptic equations $\triangle u + \lambda f(u)=0\, \quad u|_{\partial {\Omega}} =0$ whose nonlinearities have supercritical growth near $u=\infty$ with $f(0)\ge 0$. For examples of these problems on ball domains it is known that there is a $\lambda^*>0$ such that there exists a unique small nonnegative solution for $0 <\lambda <\lambda^*$. This is in contrast to the situation where $f$ has superlinear but subcritical growth , in which case always also a large unstable solution can be found. We show that if $\Omega\subset{{\Bbb R}}^n$ is starshaped, $f(s)>0$ as $s\to\infty$ and $\lim\sup_{s\to\infty} \frac{F(s)}{sf(s)} <\frac{n-2}{2n}$ that a $\lambda^*$ as above always exists. We do not have to assume anything on the behavior of $f$ between zero and infinity. More generally we can assign to any domain $\Omega$ a number $M(\Omega)$ which in the case that $M(\Omega) < \frac12$ allows the same conclusion if in the above growth condition $\frac{n-2}{2n}$ is replaced by $\frac12-M(\Omega)$. There exist nonstarshaped domains in ${{\Bbb R}}^n$, $n \ge 3$ and topologically nontrivial ones in ${{\Bbb R}}^n$, $n\ge 4$ for which $M(\Omega) < \frac 12$. This indicates that for such domains a generalized critical exponent $\le\frac{1+2M(\Omega)}{ 1-2M(\Omega)}$ exists. Moreover, for starshaped domains $M(\Omega)=\frac 1 n$, so in this case the expression gives the usual critical exponent $\frac {n+2}{n-2}$.

We prove a few existence results of a solution for a class of nonlinear systems in thermoviscoelasticity in which the mechanical dissipation is not linearized. Under a natural assumption on the growth of the thermal stress with respect to the temperature, we establish an existence result of small solutions. Under a stronger assumption on these stresses for (relative) nonpositive temperature, we prove an existence result for arbitrary data. The techniques of renormalized solutions for a parabolic equation with $L^1$ data are used to handle the energy conservation law.

In this paper, we consider the Gaussian curvature equation \begin{equation}{\label{eq:01}} {\Delta} u+K(x)e^{2u}=0 \quad \mbox{in} \ {{\bf R}}^2 , \end{equation} where ${\Delta}=\sum_{i=1}^{2}\frac{\partial^2}{\partial x_i^2}$ is the Laplace operator in ${{\bf R}}^2$. $K(x)$ is always assumed to be of one sign for $|x|$ large. For each $K$, we introduce $\alpha_1=\alpha_1(K)$ by \begin{equation} \alpha_1=\sup \{\alpha \in {{\bf R}} : \int_{{{\bf R}}^2} |K(x)|(1+|x|^2)^{\alpha}\,dx <+{\infty} \} . \tag*{(0.1)} \end{equation} Suppose that $\alpha_1>0$, $K(x)$ is positive somewhere in ${{\bf R}}^2$ and satisfies \begin{equation} \int_{{{\bf R}}^2}K(x)\,dx <0 . \tag*{(0.2)} \end{equation} We prove that there exists $0 <\alpha_0\leq\alpha_1$ such that for any given $\alpha \in (0,\alpha_0)$, there exist at least two solutions of (0.1) with $-2\pi \alpha$ as their total curvature.

Let $p > 1,$ and $\psi_p : {{\mathbb R}}^N \to {{\mathbb R}}, \; v \mapsto |v|^{p-2}v,$ with $|v|$ the Euclidian norm of $v.$ This paper is devoted to the study of the corresponding eigenvalue problem $$\left(\psi_p(u')\right)' + \lambda \psi_p(u) = 0,$$ under the Dirichlet, Neumann and periodic boundary conditions. The eigenvalues in the Dirichlet and Neumann cases are the same when $N = 1$ and $N \geq 2,$ but not in the periodic case, where the exact nature of the set of eigenvalues is still open. We provide some information about this set. Variational characterizations of the first positive eigenvalue are obtained in the case of all three boundary conditions, as well as the corresponding generalized Poincaré's or Wirtinger's inequalities. Applications are given to forced Liénard-type systems and to systems with growth of order $p-1.$

We examine the autonomous reaction-diffusion system $u_t=\lambda_1u_{xx}+f(u,v)u-v$, $v_t=\lambda_2v_{xx}+f(u,v)v+u$ for $t>0$ with Dirichlet boundary conditions on $I=(0,1)$, where $\lambda_1,\lambda_2$ are positive constants. If $\lambda_1$ and $\lambda_2$ coincide, then the zero solution is the only stationary solution. If the zero solution is unstable and $f$ satisfies some monotonicity condition, then periodic motion can occur. In this case, we actually construct a periodic solution and show, in addition, that it has certain properties. If $\lambda_1\ne\lambda_2$, nontrivial stationary solutions might exist---we prove the existence for some values of $\lambda_1,\lambda_2$. On the other hand, we show that no such nontrivial stationary solution exists if the ratio $\lambda_1/\lambda_2$ is contained in an interval around $1$ which we will state explicitly.

In this paper, we prove, following [1], existence and uniqueness of the solutions of convection-diffusion equations on an open subset of $\mathbb R^N$, with a measure as data and different boundary conditions: mixed, Neumann or Fourier. The first part is devoted to the proof of regularity results for solutions of convection-diffusion equations with these boundary conditions and data in $(W^{1,q}(\Omega))'$, when $q <N/(N-1)$. The second part transforms, thanks to a duality trick, these regularity results into existence and uniqueness results when the data are measures.

We study some perturbed semilinear problems with Dirichlet or Neumann boundary conditions, $$ \begin{cases} -{\varepsilon}^2\Delta u+u=u^p & \mbox{ in $\Omega$}\cr u>0 & \mbox{ in $\Omega$}\cr u=0\ \ \mbox{ or }\ \ {{\partial u}\over{\partial\nu}}=0 & \mbox{ in $\partial\Omega,$} \end{cases} $$ where $\Omega$ is a bounded, smooth domain of $\mathbb R^N$, $N\ge2$, ${\varepsilon}>0$, $1 <p <{{N+2}\over{N-2}}$ if $N\ge3$ or $p>1$ if $N=2$ and $\nu$ is the unit outward normal at the boundary of $\Omega$. We show that any ``suitable" critical point $x_0$ of the distance function generates a family of single interior spike solutions, whose local maximum point tends to $x_0$ as ${\varepsilon}$ tends to zero.

We study the well-posedness for the Cauchy problem of the KP II equation. We prove the local well-posedness in the anisotropic Sobolev spaces $H_{x,y}^{-1/4+\epsilon,0}$ and in the anisotropic homogeneous Sobolev spaces $H_{x,y}^{-1/2+4\epsilon,0}\cap\dot{H}_{x,y}^{-1/2+\epsilon,0}$. The first result is an improvement of the result in $L^2$ obtained by J. Bourgain [2].

Integral and supremum estimates are proven for local solutions of doubly degenerate quasilinear parabolic equations. This is done with the aid of a local energy inequality and estimates in $L^{\text{weak}}_q$ spaces.

Generalized (entropy) solutions of degenerate second-order quasilinear parabolic and elliptic equations are considered. In classes of functions under consideration it is proved that a function is a generalized solution if and only if it is a weak solution and satisfies discontinuity conditions. Uniqueness and stability of the generalized solution of the Cauchy problem for a degenerate parabolic equation is proved.

We prove the existence of infinitely many positive, radially symmetric and decreasing solutions of the quasilinear elliptic problem $$ \begin{cases} -{\hbox{div} (a(|\nabla u|^2) \, \nabla u )} = f(|\hbox{x}|,u), & \hbox{ in }\, B_R,\\ \quad u = 0, & \hbox{ on }\, \partial B_R, \end{cases} $$ where $B_R$ is a ball in ${\Bbb R}^N$. The nonlinear differential operator is rather general and includes, as a particular case, the $m$--Laplacian, with $m>1$. The assumptions on $f$ are placed only at $+\infty$ and require a sort of oscillatory behaviour of the primitive $\int_0^s f(|\hbox{x}|,\xi)\, d\xi$. The proof is based on time--mapping estimates, the upper and lower solutions method and a maximum--like principle. The results are new even in the case of the Laplacian $\Delta$. A counterexample is also constructed in order to show that the assumptions cannot be relaxed.