### An inverse boundary-value problem for semilinear elliptic equations.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

We consider the semilinear Lane–Emden problem $$ where $p>1$ and $\Omega $ is a smooth bounded domain of ${\mathbb{R}}^{2}$. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of ${(}_{p})$, as $p\to +\infty $. Among other results we show, under some symmetry assumptions on $\Omega $, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as $p\to +\infty $, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of the Liouville...

We study the leading order behaviour of positive solutions of the equation $-\Delta u+\u03f5u-{\left|u\right|}^{p-2}u+{\left|u\right|}^{q-2}u=0,\phantom{\rule{2.0em}{0ex}}x\in {\mathbb{R}}^{N}$, where $N\ge 3$, $q>p>2$ and when $\u03f5>0$ is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of $p$, $q$ and $N$. The behavior of solutions depends sensitively on whether $p$ is less, equal or bigger than the critical Sobolev exponent ${2}^{*}=\frac{2N}{N-2}$. For $p<{2}^{*}$ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p>{2}^{*}$ the solution asymptotically coincides...

Let $\Omega $ be a bounded domain in ${\mathbb{R}}^{n}$ with smooth boundary $\partial \Omega $. We consider the equation ${d}^{2}\Delta u-u+{u}^{\frac{n-k+2}{n-k-2}}=0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4.0pt}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\Omega $, under zero Neumann boundary conditions, where $\Omega $ is open, smooth and bounded and $d$ is a small positive parameter. We assume that there is a $k$-dimensional closed, embedded minimal submanifold $K$ of $\partial \Omega $, which is non-degenerate, and certain weighted average of sectional curvatures of $\partial \Omega $ is positive along $K$. Then we prove the existence of a sequence $d={d}_{j}\to 0$ and a positive solution ${u}_{d}$ such that ${d}^{2}{\left|\nabla {u}_{d}\right|}^{2}\phantom{\rule{0.166667em}{0ex}}\rightharpoonup \phantom{\rule{0.166667em}{0ex}}S\phantom{\rule{0.166667em}{0ex}}{\delta}_{K}\phantom{\rule{0.222222em}{0ex}}\mathrm{as}\phantom{\rule{0.222222em}{0ex}}d\to 0$ in the sense of measures, where ${\delta}_{K}$...

We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations $\Delta u-u+f\left(u\right)=0$ in ${\mathbb{R}}^{N}$, $u\in {H}^{1}\left({\mathbb{R}}^{N}\right)$, where $N\ge 2$. Under natural conditions on the nonlinearity $f$, we prove the existence of $\mathrm{\mathit{i}\mathit{n}\mathit{f}\mathit{i}\mathit{n}\mathit{i}\mathit{t}\mathit{e}\mathit{l}\mathit{y}\mathit{m}\mathit{a}\mathit{n}\mathit{y}\mathit{n}\mathit{o}\mathit{n}\mathit{r}\mathit{a}\mathit{d}\mathit{i}\mathit{a}\mathit{l}\mathit{s}\mathit{o}\mathit{l}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathit{s}}$ in any dimension $N\ge 2$. Our result complements earlier works of Bartsch and Willem $(N=4\mathrm{\U0001d698\U0001d69b}N\ge 6)$ and Lorca-Ubilla $(N=5)$ where solutions invariant under the action of $O\left(2\right)\times O(N-2)$ are constructed. In contrast, the solutions we construct are invariant under the action of ${D}_{k}\times O(N-2)$ where ${D}_{k}\subset O\left(2\right)$ denotes the dihedral group...

Let $\Omega \subset {\mathbb{R}}^{n}$, $n\ge 2$, be a bounded connected domain of the class ${C}^{1,\theta}$ for some $\theta \in (0,1]$. Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem $$u\in {W}^{1}{L}^{\Phi}\left(\Omega \right),\phantom{\rule{1.0em}{0ex}}-div\left({\Phi}^{\text{'}}\left(\right|\nabla u\left|\right)\frac{\nabla u}{\left|\nabla u\right|}\right)+V\left(x\right){\Phi}^{\text{'}}\left(\right|u\left|\right)\frac{u}{\left|u\right|}=f(x,u)+\mu h\left(x\right)\phantom{\rule{1.0em}{0ex}}\text{in}\Omega ,\frac{\partial u}{\partial \mathbf{n}}=0\phantom{\rule{1.0em}{0ex}}\text{on}\partial \Omega ,$$ where $\Phi $ is a Young function such that the space ${W}^{1}{L}^{\Phi}\left(\Omega \right)$ is embedded into exponential or multiple exponential Orlicz space, the nonlinearity $f(x,t)$ has the corresponding critical growth, $V\left(x\right)$ is a continuous potential,...

We consider a singularly perturbed elliptic equation ${\u03f5}^{2}\Delta u-V\left(x\right)u+f\left(u\right)=0,u\left(x\right)>0$ on ${\mathbb{R}}^{N}$, ${\mathrm{\U0001d695\U0001d692\U0001d696}}_{\left|x\right|\to \infty}u\left(x\right)=0$, where $V\left(x\right)>0$ for any $x\in {\mathbb{R}}^{N}$. The singularly perturbed problem has corresponding limiting problems $\Delta U-cU+f\left(U\right)=0,U\left(x\right)>0$ on ${\mathbb{R}}^{N}$, ${\mathrm{\U0001d695\U0001d692\U0001d696}}_{\left|x\right|\to \infty}U\left(x\right)=0,c>0$. Berestycki-Lions found almost necessary and sufficient conditions on nonlinearity $f$ for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of potential $V$ under possibly general conditions on $f$. In...

We consider a singularly perturbed elliptic equation with superlinear nonlinearity on an annulus in ${\mathbb{R}}^{4}$, and look for solutions which are invariant under a fixed point free 1-parameter group action. We show that this problem can be reduced to a nonhomogeneous equation on a related annulus in dimension 3. The ground state solutions of this equation are single peak solutions which concentrate near the inner boundary. Transforming back, these solutions produce a family of solutions which concentrate...