Perhaps I should mention what I did in my thesis. I studied the non-homogeneous case, that is $\Delta u=\lambda u^2+f$, where $f=f(x)$ is the data. It is particularly interesting to consider the dependence of the solution in terms of the parameter $\lambda$. I worked in a bounded domain $\Omega$, with Dirichlet boundary condition $u=0$ on $\partial\Omega$.

Let me call a solution stable if the linear operator $-\Delta+2u$ is positive, that is if $w\mapsto\int_\Omega(|\nabla w|^2+2uw^2)dx$ is equivalently to the (square) norm of $H^1_0(\Omega)$.

Theorem : the boundary value problem admits a stable solution if and only if $\lambda\in(\lambda_-(f),\lambda_+(f))$. Moreover, this stable solution is unique. And there exists at least a non-stable solution.

It happens that $\lambda_+(f)=+\infty$ iff the solution of the loinear equation $\Delta U=f$ is non-positive. At the extremities of the interval, the solution still exists but is marginally stable.

Even more interesting is:

Theorem: the function $\lambda\mapsto u(\lambda)$ is analytic and extends to the upper and lower halves of the complex plane. This extension is continuous up to the real line. When $\lambda\in{\mathbb R}\setminus\lambda\in(\lambda_-(f),\lambda_+(f))$, this extension provides a (non-real) complex solution which has a stability property, and it is unique in this class.

See: Prolongement analytique et nombre de solutions d'une équation aux dérivées partielles elliptique non linéaire paramétrée, C. R. Acad. Sci. Paris Sér. A (1978) 287, pp 1021-1023.

Perhaps I should mention what I did in my PhD thesis. I studied the non-homogeneous case, that is $\Delta u=\lambda u^2+f$, where $f=f(x)$ is the data. It is particularly interesting to consider the dependence of the solution in terms of the parameter $\lambda$. I worked in a bounded domain $\Omega$, with Dirichlet boundary condition $u=0$ on $\partial\Omega$.

Let me call a solution stable if the linear operator $-\Delta+2u$ is positive, that is if $w\mapsto\int_\Omega(|\nabla w|^2+2uw^2)dx$ is equivalently to the (square) norm of $H^1_0(\Omega)$.

Theorem : There exist two numbers $-\infty\le\lambda_-(f)<\lambda_+(f)\le+\infty$ such that the boundary value problem admits a stable solution if and only if $\lambda\in(\lambda_-(f),\lambda_+(f))$. Moreover, this stable solution is unique (denote it $u_\lambda$). For such parameters, there exists at least one non-stable solution. The map $\lambda\mapsto u_\lambda$ is increasing.

It happens that $\lambda_+(f)=+\infty$ iff the solution of the linear equation $\Delta U=f$ is non-positive. At a finite extremity of the interval, the solution still exists (it is the limit of $u_\lambda$) but is marginally stable.

Even more interesting is:

Theorem: The function $\lambda\mapsto u(\lambda)$ is analytic and extends to the upper and lower halves of the complex plane. This extension is continuous up to the real line. When $\lambda\in{\mathbb R}\setminus[\lambda_-(f),\lambda_+(f)]$, this extension provides a (non-real) complex solution which has a stability property, and it is unique in this class, up to complex conjugacy.

See: Prolongement analytique et nombre de solutions d'une équation aux dérivées partielles elliptique non linéaire paramétrée, C. R. Acad. Sci. Paris Sér. A (1978) 287, pp 1021-1023.

Perhaps I should mention what I did in my thesis. I studied the non-homogeneous case, that is $\Delta u=\lambda u^2+f$, where $f=f(x)$ is the data. It is particularly interesting to consider the dependence of the solution in terms of the parameter $\lambda$. I worked in a bounded domain $\Omega$, with Dirichlet boundary condition $u=0$ on $\partial\Omega$.

Let me call a solution stable if the linear operator $-\Delta+2u$ is positive, that is if $w\mapsto\int_\Omega(|\nabla w|^2+2uw^2)dx$ is equivalently to the (square) norm of $H^1_0(\Omega)$.

Theorem : the boundary value problem admits a stable solution if and only if $\lambda\in(\lambda_-(f),\lambda_+(f))$. Moreover, this stable solution is unique. And there exists at least a non-stable solution.

It happens that $\lambda_+(f)=+\infty$ iff the solution of the loinear equation $\Delta U=f$ is non-positive. At the extremities of the interval, the solution still exists but is marginally stable.

Even more interesting is:

Theorem: the function $\lambda\mapsto u(\lambda)$ is analytic and extends to the upper and lower halves of the complex plane. This extension is continuous up to the real line. When $\lambda\in{\mathbb R}\setminus\lambda\in(\lambda_-(f),\lambda_+(f))$, this extension provides a (non-real) complex solution which has a stability property, and it is unique in this class.

Perhaps I should mention what I did in my thesis. I studied the non-homogeneous case, that is $\Delta u=\lambda u^2+f$, where $f=f(x)$ is the data. It is particularly interesting to consider the dependence of the solution in terms of the parameter $\lambda$. I worked in a bounded domain $\Omega$, with Dirichlet boundary condition $u=0$ on $\partial\Omega$.

Let me call a solution stable if the linear operator $-\Delta+2u$ is positive, that is if $w\mapsto\int_\Omega(|\nabla w|^2+2uw^2)dx$ is equivalently to the (square) norm of $H^1_0(\Omega)$.

Theorem : the boundary value problem admits a stable solution if and only if $\lambda\in(\lambda_-(f),\lambda_+(f))$. Moreover, this stable solution is unique. And there exists at least a non-stable solution.

It happens that $\lambda_+(f)=+\infty$ iff the solution of the loinear equation $\Delta U=f$ is non-positive. At the extremities of the interval, the solution still exists but is marginally stable.

Even more interesting is:

Theorem: the function $\lambda\mapsto u(\lambda)$ is analytic and extends to the upper and lower halves of the complex plane. This extension is continuous up to the real line. When $\lambda\in{\mathbb R}\setminus\lambda\in(\lambda_-(f),\lambda_+(f))$, this extension provides a (non-real) complex solution which has a stability property, and it is unique in this class.

See: Prolongement analytique et nombre de solutions d'une équation aux dérivées partielles elliptique non linéaire paramétrée, C. R. Acad. Sci. Paris Sér. A (1978) 287, pp 1021-1023.