Consider the Banach normalized weighted spaces

with norm

For $q=2,$ we endow $L_{\rho }^2$ with the inner product and norm

Consider the linear operator in $L_{\rho ,}^2$ defined by

with the domain

where $C_0^2(\mathrm{\Omega })$ is the space of all functions in $C^2(\mathrm{\Omega })$ with compact support included in $\mathrm{\Omega }$. For any $u\in C_0^2\left(\mathrm{\Omega }\right),$ $\mathrm{\Delta }u\in C_0\left(\mathrm{\Omega }\right),$ and since $H\in C^1\left(\mathrm{\Omega }\right),$ one has $\nabla H\in C(\mathrm{\Omega },{ℝ}^d).$ Hence $ℒu\in C_0\left(\mathrm{\Omega }\right)\subset L_{\rho }^2,$ that is $ℒ$ is well-defined. Also $D\left(ℒ\right)$ is dense in $L_{\rho }^2.$ Indeed, if $u\in L_{\rho }^2,$ then $v:=\sqrt{\rho }u\in L^2\left(\mathrm{\Omega }\right)$ and in view of the density of $C_0^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ into $L^2\left(\mathrm{\Omega }\right),$ there exists in $C_0^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ a sequence $\left(v_k\right)$ with $v_k\to v$ in $L^2\left(\mathrm{\Omega }\right).$ Let ${\phi }_k\in C_0^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ be such that ${\phi }_k\to 1$ in $L^2\left(\mathrm{\Omega }\right),$ and let

where $c_k={\int }_{\mathrm{\Omega }}\sqrt{\rho }{v}_k/{\int }_{\mathrm{\Omega }}\sqrt{\rho }{\phi }_k,$ Clearly $u_k\in D\left(ℒ\right).$ Also

Hence $u_k\to u$ in $L_{\rho }^2$ if $c_k\to 0.$ To show this, first note that

whence we have that the sequence $\left({\int }_{\mathrm{\Omega }}\sqrt{\rho }{\phi }_k\right)$ is bounded from below by a positive number $C.$ Then $c_k\le \left(1/C\right){\int }_{\mathrm{\Omega }}\sqrt{\rho }{v}_k.$ Next, in view of ${\int }_{\mathrm{\Omega }}\sqrt{\rho }v=0,$ one has

Hence $c_k\to 0$ and from (2.1) it follows that $u_k\to u$ in $L_{\rho }^2.$ Therefore $ℒ$ is densely defined on $L_{\rho }^2.$

The operator $ℒ$ is symmetric. Indeed, since $\nabla \rho =\epsilon \rho \nabla H,$ we have

Finally note that

for every $u\in D\left(ℒ\right)\setminus \left\{0\right\},$ that is the operator $ℒ$ is strictly positive.

We may endow $D\left(ℒ\right)$ with two inner products

and the corresponding norms

Let $E_{ℒ}$ (called the energetic space of $ℒ$) be the completion of the prehilbertian space $(D\left(ℒ\right),\parallel \cdot \parallel )$ and let us use the same notations $⟨\cdot ,\cdot ⟩,$ $[\cdot ,\cdot ],$ $\parallel \cdot \parallel$ and $[\cdot ]$ for the corresponding maps extended by density to $E_{ℒ}.$ Since ${\left|u\right|}_{\rho }\le \Vert u\Vert$ for all $u\in D\left(ℒ\right),$ we have $D\left(ℒ\right)\subset E_{ℒ}\subset L_{\rho }^2$ with dense and continuous embeddings. Recall that, from the construction of the completion, any element $u$ of $E_{ℒ}$ can be seen as the limit in $L_{\rho }^2$ of a sequence of functions from $D\left(ℒ\right)$ which is fundamental with respect the norm $\parallel \cdot \parallel ,$ and that this limit is common for all such sequences $\left(u_k\right),\left(v_k\right)$ which are equivalent in the sense that $\Vert u_k-v_k\Vert \to 0.$ If $\left(u_k\right)$ is a fundamental sequence in $D\left(ℒ\right),$ then there exist $v,v_i\in L^2\left(\mathrm{\Omega }\right),$ $i=1,\cdot \cdot ,d$ such that

Thus, if we denote

then we may say that for every $u,v\in E_{ℒ},$

Notice that the functional $[\cdot ]$ is only a semi-norm on $E_{ℒ}.$ To make it a norm, equivalent to the norm $\parallel \cdot \parallel$ on $E_{ℒ},$ we need a compactness assumption. To this aim, we state the following condition:

(C_q)

The embedding $D\left(ℒ\right)\subset L_{\rho }^q$ is compact, i.e., any sequence of functions in $D\left(ℒ\right)$ which is bounded with respect to the norm $\parallel \cdot \parallel$ has a subsequence that converges in $L_{\rho }^q.$

Clearly condition (C_q) implies that the embedding $E_{ℒ}\subset L_{\rho }^q$ is also compact.

The next condition (H) gives an exact representation of the space $E_{ℒ},$ and consequently, it is sufficient for (C_q) to hold for some values of $q.$

(H)

There exists a constant $c>0$ such that

$$\left|\nabla H\right|{\delta }_{\mathrm{\Omega }}\le c\text{in }\mathrm{\Omega },$$

where ${\delta }_{\mathrm{\Omega }}$ gives the distance to the boundary $\partial \mathrm{\Omega },$ i.e.

$${\delta }_{\mathrm{\Omega }}\left(x\right)=\underset{y\in \partial \mathrm{\Omega }}{\min }\left|x-y\right|\left(x\in \mathrm{\Omega }\right).$$

Assume that condition (C₂) holds. The space $E_{ℒ}$ being reflexive (as a Hilbert space), one deduces from a result in paper [12] that

and the infimum is reached. From this, we have the Poincaré inequality

which ensures that $[\cdot ]$ is a norm on $E_{ℒ},$ equivalent to the norm $\parallel \cdot \parallel .$ Let $E_{ℒ}^{\prime }$ be the dual of $(E_{ℒ},[\cdot ]).$ If we identify $L_{\rho }^2$ to its dual, then we have

where the last embedding is compact too. For $f\in E_{ℒ}^{\prime }$ and $u\in E_{ℒ},$ let $(f,u)$ be the value of the linear functional $f$ at $u.$ In case that $f\in L_{\rho }^2,$ one has $(f,u)={(f,u)}_{\rho }.$

Throughout the paper we assume that condition (C₂) holds.

Returning to the operator $ℒ,$ for a fixed $f\in E_{ℒ}^{\prime },$ we define the weak solution of the stationary problem

as being a function $u\in E_{ℒ}$ such that for every $v\in E_{ℒ},$ one has that $[u,v]=(f,v).$ In particular, if $f\in L_{\rho }^2,$ this identity becomes

From Riesz’s representation theorem, since $(f,\cdot )$ is a continuous linear functional on $(E_{ℒ},[\cdot ]),$ it follows that problem (2.7) has a unique weak solution $u_f.$ Thus we may define the solution operator

Recall that under condition (C₂), the operator $ℒ$ has a sequence of eigenvalues $\left({\lambda }_k\right)$ with ${\lambda }_k\to +\mathrm{\infty },$ and correspondingly a sequence $\left({\varphi }_k\right)$ of eigenfunctions, which is orthonormal and complete in $L_{ℒ}^2.$ Also the sequence $\left({\varphi }_k/\sqrt{{\lambda }_k}\right)$ is orthonormal and complete in and $(E_{ℒ},[\cdot ]).$ This yields the Fourier representation of the solution operator:

where the series converges in $E_{ℒ}$ and $L_{\rho }^2.$

Also note that ${ℒ}^{-1}$ is an isometry between $E_{ℒ}^{\prime }$ and $E_{ℒ},$ i.e., $\left[{ℒ}^{-1}f\right]={\left|f\right|}_{E_{ℒ}^{\prime }}$ for every $f\in E_{ℒ}^{\prime },$ and that the exact Poincaré inequality

is accompanied by the Poincaré inequality for the dual, namely

Indeed, if $f\in L_{\rho }^2,$ then using (2.8) we have

that is (2.9).

According to the variational theory of positive-define symmetric linear operators, for each fixed $f\in E_{ℒ}^{\prime },$ the functional $J:E_{ℒ}\to ℝ,$

is $C^1$ and $J^{\prime }u=ℒu-f,$ more exactly

Therefore the weak solution of problem (2.7) is the critical point of the energy functional $J.$

Our first result is about the existence and uniqueness of the solution to the semilinear problem (1.2) and consequently to (1.1).

Let $j_0$ and $j$ be the canonical injections of the embeddings $E_{ℒ}\subset L_{\rho }^2$ and $L_{\rho }^2\subset E_{ℒ}^{\prime },$ respectively.

Notice that problem (1.2) is equivalent with the fixed point equation $u={ℒ}^{-1}\mathrm{\Psi }\left(u\right)$ in $E_{ℒ}.$ In view of embeddings (2.6), we may discuss three cases:

• •

  $\mathrm{\Psi }$ maps $E_{ℒ}$ into $E_{ℒ}^{\prime }$;

• •

  $\mathrm{\Psi }$ maps $E_{ℒ}$ into $L_{\rho }^2$; here by the composition ${ℒ}^{-1}\mathrm{\Psi }$ we mean ${ℒ}^{-1}j\mathrm{\Psi }$;

• •

  $\mathrm{\Psi }$ maps $L_{\rho }^2$ into $L_{\rho }^2$; here by ${ℒ}^{-1}\mathrm{\Psi }$ we mean ${ℒ}^{-1}j\mathrm{\Psi }{j}_0$.

Our first results are existence and uniqueness theorems, the first in terms of $\mathrm{\Psi }$ and the second in terms of $\mathrm{\Phi }$.

The next result gives a variational characterization of the solution guaranteed by the previous theorems.